CEJM 2(1) (2004) 1{18
On zeta-functions associated to certain cusp forms.I A. Laurin·cikas1a ¤, J. Steuding2y 1
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania 2 Mathematisches Seminar, Johann Wolfgang Goethe-UniversitÄat Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany
Received 7 October 2003; accepted 24 October 2003 Abstract: In the paper the asymptotics for Dirichlet polynomials associated to certain cusp forms are obtained. ® c Central European Science Journals. All rights reserved. Keywords: cusp form, normalized eigenform, Dirichlet polynomials MSC (2000): 11M41, 11N37 a
Partially supported by Lithuanian Foundation of Studies and Science
1
Introduction and statement of results
Let F (z) be a holomorphic cusp form of weight · for the full modular group SL(2; Z), i.e. the function F (z) is holomorphic in the half-plane Im z > 0, it satis¯es the functional equation
F
Ã
az + b cz + d
!
= (cz + d)· F (z)
for all
lim F (z) = 0:
Im z!1
y
Ba B @
c
and
¤
0
E-mail:
[email protected] E-mail:
[email protected]
1
bC
d
C 2 SL(2; Z); A
2
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
Moreover, we assume that F (z) is a normalized eigenform for all Hecke operators. In particular, F (z) has the Fourier series expansion F (z) =
1 X
c(m)e2¼imz ;
where
c(1) = 1:
m=1
For instance, ¢(z) :=
1 X
1 Y
¿ (m)e2¼imz = e2¼iz
m=1
(1 ¡
e2¼inz )24 ;
n=1
is a normalized eigenform of weight 12. Ramanujan [17] made the following three important conjectures about the so-called Ramanujan ¿ -function ¿ (n): (i) ¿ (n) is multiplicative, (ii) ¿ (n) satis¯es ¿ (pk+1 ) = ¿ (p)¿ (pk ) ¡ p11 ¿ (pk¡1 ) 11
for prime numbers p and integers k ¶ 2, and (iii) j¿ (p)j µ 2p 2 for any prime p. Mordell [16] proved the ¯rst two conjectures in 1917, and Deligne [4] gave his celebrated proof of the third one in 1971; see [19] for a nice survey on Ramanujan’s work and further developments. Let s = ¾ + it be a complex variable. The function ’(s; F ) =
1 X c(m)
m=1
ms
is called the zeta-function attached to the cusp form F (z). The series for ’(s; F ) converges absolutely for ¾ > ·+1 , and de¯nes there a holomorphic function. It is well known that 2 the function ’(s; F ) is analytically continuable to an entire function. Since F (z) is a normalized eigenform, ’(s; F ) has the following Euler product expansion ’(s; F ) =
Y p
Ã
®(p) ps
1¡
!¡1 Ã
1¡
¯(p) ps
!¡1
;
(1)
where here and in the sequel p denotes always a prime number, the product is taken over all primes p, and the numbers ®(p) and ¯(p) satisfy c(p) = ®(p) + ¯(p): P. Deligne obtained in [4] the estimates j®(p)j µ p
µ ¡1 2
;
j¯(p)j µ p
µ ¡1 2
(2)
(which proved the third of Ramanujan’s conjectures on ¿ (n)). Hence, the Euler product (1) converges for ¾ > ·+1 and, consequently, ’(s; F ) has no zeros in this half-plane. 2 The idea of applying probabilistic methods to investigations on the value-distribution of zeta-functions dates back to Harald Bohr. In [1] he noted that the value-distribution of the Riemann zeta{function ³(s) is governed by some probabilistic laws. The ¯rst results in this direction were obtained in [2], [3]. Later the probabilistic theory of zeta-functions was developed by A. Wintner, V. Borchsenius, A. Selberg, P.D.T.A. Elliott, A. Ghosh,
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
3
D. Joyner, D. Hejhal, E. Stankus, B. Bagchi, K. Matsumoto, R. Garunk·stis, W. Schwarz, · zevi·cien_e, the authors, and others. The majority of their results R. Ka·cinskait_e, R. Sle· can be found in [7], [9], [10], [14], [20]. The ¯rst probabilistic result on zeta-function attached to cusp forms was obtained in [8], where a limit theorem in the sense of the weak convergence of probability measures in the space of analytic functions was proved. An interesting and di±cult problem is the value-distribution of the function ’(s; F ) near and on the so-called critical line ¾ = ·2 . In these cases some power norming for ’(s; F ) is necessary, and the problem reduces to ¯nd the asymptotics for the moments of a certain normalization of ’(s; F ); see [9] for an analogous result for ³(s). In virtue of the Montgomery-Vaughan theorem [15] the latter problem is closely related to ¯nd the asymptotics of the corresponding Dirichlet polynomials. Therefore, the aim of this note is to obtain such asymptotics. Let w 6= 0 be an arbitrary complex number and ¾ > ·+1 . De¯ne a branch of the 2 w multi-valued function ’ (s; F ) by ’w (s; F ) = expfw log ’(s; F )g (
= exp ¡ w =
Y p
=
Y p
(
X p
Ã
Ã
log 1 ¡
Ã
Ã
exp ¡ w log 1 ¡ Ã
1¡
®(p) ps
!¡w Ã
1¡
®(p) ps ®(p) ps
!
!
¯(p) ps
Ã
¯(p) ps
+ log 1 ¡ Ã
+ log 1 ¡
!¡w
¯(p) ps
!!)
!!)
(3)
:
Here the multi-valued functions log(1 ¡ z) and (1 ¡ z)¡w in the region jzj < 1 are de¯ned by continuous variation along any path in this region, from the values log(1 ¡ z)jz=0 = 0 and (1 ¡ z)¡w jz=0 = 1, respectively. Let, as usual, ¡(z) denote Euler’s gamma-function. Then, for jzj < 1, 1 X ¡(w + k) k (1 ¡ z)¡w = z : k=0 ¡(w)k! De¯ne, for k = 1; 2; : : :,
dw (pk ) =
¡(w + k) w(w + 1) : : : (w + k ¡ = ¡(w)k! k!
Then we deduce from Eq. 3 that, for ¾ > ’w (s; F ) =
pks
l=0
:
·+1 , 2
1 1 YX dw (pk )®k (p) X dw (pl )¯ l (p) p k=0
1)
pls
=
1 YX gw (pk ) p k=0
pks
=
1 X gw (m)
m=1
ms
;
where gw (m) is the multiplicative function given by gw (pk ) =
k X l=0
for k = 1; 2; : : :.
dw (pl )® l (p)dw (pk¡l )¯ k¡l (p)
(4)
4
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
First we consider the mean value X
h(m);
h(m) = gw2 (m)m1¡· :
where
m·x
Such functions were studied by many authors; see, for example, [22],[12],[13]. However, in our applications the parameter w is not ¯xed, and therefore we need the uniformity in w. We suppose that w is a complex number satisfying jwj µ 12 and Re w 2 > 0. De¯ne m(h; x) =
Y
p·x
Ã
1+
1 X h(p® )
p®
®=1
!
and
Ã
!
log2 y R(y) = max jwj ; ; y 2
x where y µ x and log2 y ¶ 4 log x log2 log x. For brevity, let X = log and denote by log y °0 Euler’s constant. We write f (x) ½ g(x), resp. f (x) = O(g(x)), with a positive function g(x), if lim supx!1 jf(x)j is bounded. Further, f (x) = o(g(x)) denotes that g(x)
limx!1
jf (x)j g(x)
= 0.
Theorem 1. Let jwj µ
1 2
and Re w 2 > 0. Then uniformly in w, x and y, as x ! 1, Ã
2
Ã
jwj2 e¡°0 w x o(jwj2 ) h(m) = m(h; x)X 1+O ¡(w 2 ) log x y m·x X
Ã
!
Ã
!!
!
jwj2 xm(jhj; x) ¡1+o(jwj2 ) x 2 +O X +o m(jhj; x)X jwj R(y): log x log x
Theorem 1 can be easily applied to obtain an asymptotic formula for Dirichlet polynomials with coe±cients gw2 (m). Such polynomials appear in the investigations of the moments Z
0
T
j’(¾T + it; F )j2kT dt;
where ¾T ! ·2 + 0 and kT ! 0, as T ! 1. Let, similarly to the case of the Riemann zeta-function, see [9], ¾T =
· 1 + 2 log T
and
1
wT = u(2 log log T )¡ 2 ;
where u 6= 0. Theorem 2. Uniformly in u, 0 < juj µ u0 , as T ! 1, 2 X gw (m) T
m·T u2
m2¾T
u2
= e 2 (1 + o(1)):
Note that the numbers e 2 are the moments of the lognormal distribution law. But the lognormal distribution function cannot be de¯ned by its moments; see [11].
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
2
5
Auxiliary results
First, we recall Rankin’s result on the coe±cients c(m). Let cp = c(p)p Lemma 3. As x ! 1,
X
c2p =
p·x
1¡ µ 2
.
x (1 + o(1)): log x
This lemma is an immediate consequence of Theorem 2 from [18]. For the proof of Theorem 1 we will apply the method of di®erence { di®erential equations; see [21], Chapter III.5, and [12], [13]. We will consider the mean values of h(m) over positive integers free of small prime factors as well as over positive integers free of large positive prime factors. For this we need estimates for certain sums over primes. Lemma 4. As x ! 1,
X
h(p) =
p·x
and
X
x (w 2 + o(jwj2 )); log x
h(p) log p = x(w 2 + o(jwj2 )):
p·x
Proof. In view of Eq. 4 gw (p) = dw (p)(®(p) + ¯(p)) = dw (p)c(p) = wc(p): Therefore, h(p) = w 2 c2p , and the ¯rst assertion of the lemma follows from Lemma 3. Partial summation yields the second asymptotic equality of the lemma. Lemma 5. Let jwj µ 12 . Then, as x ! 1, X
p jh(p® )jjh(p¯ )j log p® ½ jwj4 x;
p¬ + ·x ®;¯¸1
and
X
p jh(p® )j log p® ½ jwj2 x:
p¬ ·x ®¸2
Proof. We write dw (m) =
Y
dw (p® );
p¬ jjm
where p® jjm means that p® jm but p®+1 does not divide m. Let ` be a positive integer, then it is known, see [6] and also [9], that dw` (m) =
X
m1 :::m` =m
dw (m1 ) : : : dw (m` ):
6
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
This, Eq. 2 and Eq. 4 show that, for k ¶ 1, jgw (pk )j µ p µp
(µ ¡ 1)k 2
(µ ¡ 1)k 2
k X l=0 k X l=0
Since jwj µ
1 2
jdw (pl )jjdw (pk¡l )j djwj (pl )djwj (pk¡l ) = p
(µ ¡ 1)k 2
d2jwj (pk ):
(5)
we have, for k = 1; 2; : : : :, d2jwj (pk ) =
2jwj(2jwj + 1) : : : (2jwj + k ¡ k!
1)
µ 2jwj;
and by Eq. 5 jh(pk )j µ d22jwj (pk ) µ 4jwj2 µ 1: Hence, the ¯rst sum of the lemma does not exceed log x
X
jh(p® )jjh(p¯ )j ½ jwj4 log x
p¬ + ·x ®;¯¸1
X
X
p p· x ®+¯· log x
½ Bjwj4 log2 x
1
log p
p 1 ½ jwj4 x: p log p p· x X
The second sum of the lemma is estimated similarly. Lemma 6. Let Re w 2 > 0. Suppose that ½w (u) is for u ¶ 0 a continuous solution of the di®erence-di®erential equation u½0w (u) = w 2 ½w (u ¡
1);
and that ½w (u) = 0 for u < 0. Then Ã
2
½0w (u)
2
!
e¡°0 w w2 ¡1 uRe w ¡2 = u + O ; j¡(w 2 )j ¡(w 2 )
and
2
½00w (u)
uRe w ¡2 ½ : j¡(w 2 )j
Proof. The estimates of the lemma without the factor j¡(w 2 )j¡1 were obtained in [13]. Let, as in [13], for Re s > 0, G(s) =
Z
0
1
e¡su ½w (u)du:
Then it follows from the di®erence-di®erential equation of the lemma that G0 (s) w2 + 1 ¡ =¡ G(s) s
e¡s ¡ 1 2 w : s
(6)
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
7
This gives (
Z
)
e¡v ¡ 1 G(s) = e s exp ¡ w dv v 0 ( Ã !) s2 ¡°0 w2 ¡w2 ¡1 2 ¡ ::: =e s exp ¡ w ¡ s + : 2 ¡°0 w2 ¡w2 ¡1
2
s
(7)
In [13] the asymptotic expansion 0
1
(r)
a w 2 ¡r @ (r) ½(r) a0 + 1 + : : :A ; w (u) = (1 + o(1))u u
for r = 0; 1; 2; : : :, was proved. We chose " > 0 such that Re w 2 ¡ 2
w ½w (u) = ½(0) + a1 uw w (u) = a0 u
2
¡1
+ O(uRe w
2
" > 0. Then we have
¡1¡"
):
Substituting this in Eq. 6, we obtain, for s ! +0, Ã
!
a0 ¡(w 2 + 1) a1 ¡(w 2 ) ¡(Re w 2 ¡ ") G(s) = + +O : 2 2 2 sw +1 sw sRe w ¡" This and Eq. 7 show that 2
e¡°0 w a0 = ¡(w 2 + 1)
2
and
e¡°0 w w 2 a1 = ; ¡(w 2 )
which implies the assertion of the lemma.
3
The case of large prime factors
In this section we consider the sum S1 (x; y) =
X
h(m1 );
m 1 ·x
where all prime divisors of m1 are greater than y, and where y µ x. Let ½w (u), for u ¶ 0, be a continuous solution of the equation of Lemma 6 with initial condition ½w (u) = 1 for 0 µ u µ 1. Moreover, we set "(x) = max(jwj¡2 "2 (x); (log x)¡1 ); where "2 (x) = max("1 (x); (log x)¡1 ) and
¯ ¯ X ¯1 "1 (x) = sup ¯¯ h(p) log p ¡ y¸x ¯ y p·y
¯ ¯ ¯ 2¯ w ¯: ¯
It follows from Lemma 4 that "1 (x) ! 0, as x ! 1. We also suppose that "(x) ! 0, as x ! 1.
8
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18 1 2
Lemma 7. Let jwj µ
and Re w 2 > 0. Then uniformly in x; y and w,
x½0w (X ) y ¡ w2 log y log y à ! x"(y) log y jwj2 ¡1 jwj2 ¡1 m(jhj; x) 2 "(y) +O jwj x X + X : log y m(jhj; y) y
S1 (x; y) = 1 +
Proof. Since all prime factors of m1 are larger than y, it is not di±cult to see that by Lemmas 4 and 5 X
Z(x; y) =
h(m1 ) log m1 =
X
X
h(m1 )
m1 ·x=y 2
X
=
h(m1 )
m1 ·x=p ¬ (m1 ;p)=1
h(p® ) log p®
p¬ ·x=m1 p>y
m1 ·x=y
¡
X
h(m1 )
X
h(p® ) log p®
p¬ ·x p>y
m1 ·x
=
X
Ã
0
h(p® )h(p¯ ) log p®
p¬ + ·x=m1 p>y; p6 j m1
h(m1 ) w
m1 ·x=y
X
2
µ
x ¡ m1
Ã
¶
y + O jwj 1
2
Ã
x "(y) + m1
s
x m1
!!!
(8)
x X jh(m1 )j A +O @jwj4 : y m1 ·x=y 2 m1
On the other hand,
Z(x; y) = S1 (x; y) log x ¡
Z
x
1
S1 (u; y)
du : u
(9)
Hence, taking x = y t , we ¯nd by Eq. 8 and Lemma 4 tS1 (y t ; y) ¡
Z
0
t
S1 (y u ; y)du ½ jwj2 "(y) ½ jwj2 "(y)
y t X jh(m1 )j log y m1 ·y t m1 y t m(jhj; y t) : log y m(jhj; y)
This implies 1 (S1 (y t ; y) ¡ t
1) ¡
1 t2
Z
t
1
(S1 (y u ; y) ¡
1)du ½ jwj2 "(y)
y t m(jhj; y t ) ; t2 log y m(jhj; y)
and integrating yields 1 t
Z
1
t
(S1 (y u ; y) ¡
1)du ½ jwj2 "(y)
y t m(jhj; y t ) : t2 log2 y m(jhj; y)
From this, Eq. 8 - Eq. 10, using the estimate t=
jwj2 "(y)y t log x w 2 "2 (y)y t log y t ½ = 2 ; log y w "2 (y)y t log y log y
(10)
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
9
we deduce w2 y t tS1 (y ; y) = log y
X
t
Ã
m1 ·y t¡ 1
!
w2 y "(y)y t m(jhj; y t) S1 (y t¡1 ; y) + O jwj2 : log y log y m(jhj; y)
h(m1 ) ¡ m1
Hence, by some properties of the Stieltjes integral, we obtain ¡t
t
ty S1 (y ; y) ¡
w
2
Z
t¡1
0
y ¡u S1 (y u ; y)du ½ jwj2
Now let
8 > > <0
±(t) = >
> :1
and set
y ¡t S1 (y t; y) = y ¡t +
½0w (t) ¡ log y
if
0 µ t < 1;
if
t ¶ 1;
w 2 ±(t)
"(y) m(jhj; y t ) : log y m(jhj; y)
(11)
y 1¡t "(y) + F (t; y) : log y log y
(12)
Substituting Eq. 12 in Eq. 11 yields t½0 (t) ¡ + w log y
Z
t¡1 y 1¡t "(y) ¡ w2 ty w ±(t) + tF (t; y) uy ¡u du log y log y 0 Z t¡1 0 Z t¡1 1¡u Z ½w (u) y w 2 "(y) t¡1 2 4 ¡ w du + w du ¡ F (u; y)du log y log y log y 0 0 0 "(y) m(jhj; y t ) ½ jwj2 ; log y m(jhj; y) ¡t
2
and, since t½0w (t) = w 2 ½w (t ¡ tF (t; y) ¡
w
2
1), hence we have, for t ¶ 1, Z
t¡1
0
F (u; y)du
+(ty¡t + w 2 (1 ¡ ½ jwj2
t)
y 1¡t w2 + ±(t ¡ log y log2 y
1)(1 ¡
y 2¡t ))
log y "(y)
m(jhj; y t ) : m(jhj; y)
(13)
Clearly, for t ¶ 1 and y ! 1, ¯ ¯ ¯ ¡t ¯ty + w 2 (1 ¡ ¯
y 1¡t w 4 ±(t ¡ 1)(1 ¡ t) + log y log2 t
Therefore Eq. 13 becomes, for t ¶ 1, tF (t; y) ¡
w
2
Z
t¡1
0
¯
Ã
y 2¡t ) ¯¯
!
jwj2 1 ¯µ +O : ¯ y log2 y
) log2 y F (u; y)du ½ jwj + : m(jhj; y) y 2 m(jhj; y
t
(14)
This implies ¡jwj2
t
jF (t; y)j ¡
2 ¡jwj2 ¡1
jwj t
Z
0
t
jF (u; y)jdu ½ jwj t
2
) t¡jwj ¡1 log2 y + ; m(jhj; y) y
2 ¡jwj2 ¡1 m(jhj; y
t
10
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
and after integration Z
t
1
2
v ¡jwj jF (v; y)jdv ¡ ¡jwj2
½ (1 ¡
t
jwj2
Z
1
Ã
t
(v ¡jwj
2 ¡1
!
Z
0
m(jhj; y t ) log2 y ) + : m(jhj; y) yjwj2
v
jF (u; y)jdu)dv
Since F (u; y) = 0 for 0 µ u < 1, we ¯nd Z
¡jwj2
t
t
jF (u; y)jdu ½ (1 ¡
0
t
¡jwj2
Ã
!
m(jhj; y t) log2 y ) + : m(jhj; y) yjwj2
Thus, for t ¶ 1, jwj
2
Z
t
0
2 jwj2
jF (u; y)jdu ½ jwj t
2 m(jhj; y t ) jwj2 log y +t : m(jhj; y) y
This and Eq. 14 give 2 ) jwj2 ¡1 log y +t : m(jhj; y) y
2 jwj2 ¡1 m(jhj; y
F (t; y) ½ jwj t
t
Substituting this in Eq. 12, we obtain with regard to t =
4
log x log y
= X the lemma.
The case of small prime factors
In this section we will consider the sums over positive integers m2 free of prime divisors greater than y. Let X h(m2 ) S2 (x; y) = : m 2 >x m 2 Lemma 8. Let jwj µ 12 . Then uniformly in x, y and w, S2 (x; y) ½ m(jhj; y) expf¡ X log X + O(jwj2 X )g: Proof. Let 0 < ± < 12 . Then, since h(pk ) ½ jwj2 , we ¯nd X jh(m2 )j µ m2 ¶±
S2 (x; y) ½
½ x¡±
X jh(m2 )j
1¡± x m2 m 2 ! Ã ! 1 Y X Y jh(p® )j jh(p)j ¡± ¡± ½x ½x 1+ 1 + 1¡± : ®(1¡±) p ®=1 p p·y p·y
m2
m2 >x
Ã
Moreover, we observe that by Lemma 4 ¡1
m (jhj; y)
Y
p·y
8 <X
½ exp : n
p·y
Ã
jh(p)j 1 + 1¡± p
jh(p)j
Ã
1 p1¡±
¡
!
9
8
19
0
! < X jh(p)j log p = 1 = A ½ exp O @± 1¡± : ; p ; p p·y o
n
o
½ exp O(jwj2 y ¡± + jwj2 y ± ) ½ exp O(jwj2 y ± ) :
(15)
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
This result and Eq. 15 with ± = Now let
log X log y
11
yield the lemma. X
S3 (x; y) =
h(m2 ):
m 2 ·x
Lemma 9. Let jwj µ 12 . Then uniformly in x, y and w, S3 (x; y) ½
jwj2 m(jhj; y) expf¡ X log X + O(jwj2 X )g: log x
Proof. Let, as in the proof of Lemma 8, 0 < ± < 12 . Then X
S3 (x; y) ½
jh(m2 )j
m2 ·x
µ
x m2
¶1¡±
½ x1¡±
X jh(m2 )j
m2 ·x
m1¡± 2
:
Hence, repeating the proof of Lemma 8, we obtain the estimate S3 (x; y) ½ xm(jhj; y) expf¡ X log X + O(jwj2 X)g:
(16)
To obtain the estimate of the lemma we study the sum S4 (x; y) =
X
jh(m2 )j log m2 :
m 2 ·x
Similarly to the proof of Lemma 7 we have
S4 (x; y) =
X
m2 ·x
0 B B
jh(m2 )j B B
@p¬
X
1
X
jh(p® )j log p® ¡
p¬ + ·x=m 2 p·y; p6 j m2
·x=m2 p·y
C C
jh(p® )jjh(p® )j log p® C C: A
Hence, using Lemma 5, we ¯nd Ã
X
µ
¶
x jh(m2 )j min S4 (x; y) ½ jwj ;y + m2 m2 ·x 2
½ jwj
2
X
jh(m2 )j y +
m2 ·x=y
½ jwj2 y
s
x m2
!
x m2
+ jwj2 x
! X
jh(m2 )j m2 x=y<m·x
jwj2 x X jh(m2 )j jh(m2 )j + p y m2 ·x=y m2 m2 ·x=y X
+jwj2 x
X
m2 ¸x=y
½ jwj2 y
Ã
s
jh(m2 )j m2
X jh(m2 )j jwj2 x jh(m2 )j + p m(jhj; y) + jwj2 x : y m2 m2 ·x=y m 2 ¸x=y X
This result along with Lemma 8 and Eq. 16 give S4 (x; y) ½ jwj2 xm(jhj; y) expf¡ X log X + O(jwj2 X)g: Hence, in summing by parts, we obtain the lemma.
12
5
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
Proof of Theorems 1 and 2
We begin with a relation between m(h; y) and m(h; x). Lemma 4 implies X h(p)
= w 2 log X + o(jwj2 log X ):
p
y
Therefore, from the de¯nition of m(h; x) it follows that Ã
1 Y X m(h; x) h(p® ) = 1+ ® m(h; y) ®=1 p y
!
8 Ã Ã !!9 < X jwj2 = h(p) = exp : +O ; p p2 y
8 9Ã < X h(p) = jwj2 = exp : 1 + O ; y y
= Xw
2
+o(jwj2 )
jwj2 y
1+O
(17)
:
Clearly, in view of Eq. 17 ¯
¯
Ã
Ã
jwj2 m(jhj; x) ¯¯ m(h; x) ¯¯ 2 2 ¶¯ ¯ = X Re w +o(jwj ) 1 + O m(jhj; y) ¯ m(h; y) ¯ y
!!
:
(18)
Now we are in position to give the proofs of the theorems. Proof (of Theorem 1). From the de¯nition of S1 (x; y) we have, for x=y < m µ x, S1
µ
¶
x ; y = 1: m
Therefore, X
h(m) =
X
h(m2 )
m2 ·x
m·x
h(m1 ) =
m1 ·x=m 2
X
=
X
µ
h(m2 ) S1
m2 ·x=y
µ
¶
x ;y ¡ m2
X
h(m2 )S1
m2 ·x
¶
1 +
X
µ
x ;y m2
¶
h(m2 ):
(19)
m2 ·x
By Lemma 7 X
µ
h(m2 ) S1
m2 ·x=y
x = log y
X
m2 ·x=y
µ
¶
x ;y ¡ m2
1
¶
µ x ¶ h(m2 ) 0 log m2 ¡ ½ m2 w log y
w2
y log y
X
h(m2 )
0 µ ¶ "(y) jwj2 ¡1 m(jhj; x) x"(y) log y jwj2 ¡1 +O @ jwj2 x X + X
log y
m(jhj; y)
Since ½0w (1) = w 2 , we have x log y
X
m2 ·x=y
µ x ¶ h(m2 ) 0 log m2 ½ m2 w log y
(20)
m2 ·x=y
y
X
m2 ·x=y
1
jh(m2 )j A : m2
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
w2 x = log y w2 x = log y =
X
h(m2 ) x + m2 log y
Z
X
h(m2 ) x + m2 log y
Z
m2 ·x=y
m2 ·x=y
xm(h; y) 0 ½w (X ) ¡ log y w2 + x log y
1
h(m2 ) ¡ m 2 ·x=y m2 w2 x log y
X
m2 ·x=y u X
1
w 2 xm(h; y) log y
X
xm(h; y) 0 = ½w (X ) ¡ log y
X
0
X
@m(h; y) ¡ 0
x log y
Z
X
h(m2 ) ¡ m2
m2 >x=y
h(m2 ) 0 d½w (u) m2
X
1
@
m2 >x=y u
1
h(m2 ) A 0 d½w (u) m2
m2 >x=y u
X
13
1
h(m2 ) A 00 ½w (u)du m2
x log y
Z
X
1
0 @
X
m2 >x=y u
1
h(m2 ) A 00 ½w (u)du: m2
Therefore, in view of Eq. 19 and Eq. 20, X
m·x
h(m) =
w2 x log y
xm(h; y) 0 ½w (X ) ¡ log y Z
x ¡ log y +
X
1
X
µ
X
m 2 >x=y u
h(m2 ) + O
m 2 ·x
Ã
X
m2 >x=y
h(m2 ) m2
¶
h(m2 ) 00 ½w (u)du ¡ m2
w2 y log y
X
h(m2 )
m2 ·x=y
jwj2 x"(y) jwj2 ¡1 X m(jhj; x) log y
(21)
!
x"(y) log y jwj2 ¡1 + X m(jhj; x) : y Since ¡¡1 (w 2 ) = w 2 (1 + O(jwj2 ));
(22)
Lemma 6, Eq. 17 and Eq. 18 yield Ã
2
xm(h; y) 0 e¡°0 w w 2 ¡1 xm(h; y) 2 xm(jhj; y) ½w (X ) = X + O jwj2 X Re w ¡2 2 log y ¡(w ) log y log y à à !! ¡°0 w 2 2 jwj e xm(h; x) o(jwj2 ) = X 1+O 2 ¡(w ) log x y à ! 2 jwj xm(jhj; x) ¡1+o(jwj2 ) +O X : log x
!
(23)
All other terms in Eq. 21 must be estimated. By Lemmas 8, 9 and Eq. 17, Eq. 18 w 2 x X h(m2 ) log y m2 >x=y m2
½
Ã
jwj2 xm(jhj; y) jwj2 log x=y log x=y log x=y ½ exp ¡ log +O log y log y log y log y 2 jwj xm(jhj; x) ½ expf¡ c1 Xg; log x
!¾
(24)
14
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
X
h(m2 ) ½
m2 ·x
jwj2 xm(jhj; x) expf¡ c2 X g; log x
(25)
and jwj2 xm(jhj; x) w2 y X h(m2 ) ½ expf¡ c3 Xg; log y m2 ·x=y log x
(26)
with some positive constants c1 , c2 and c3 . Through similar reasoning, we ¯nd x log y
Z
X
1 2
µ
X
m 2 >x=y u
¶
h(m2 ) 00 ½w (u)du m2 Z
jwj xm(jhj; y) X 2 expf¡ X + uguRe w ¡2 du log y 1 Ã ! Z X 2 jwj xm(jhj; y) Re w 2 ¡2 u Re w2 ¡3 ½ ¡ expf¡ X g + expf¡ Xg X e u du log y 1 jwj2 xm(jhj; x) ¡1 ½ X : log x ½
(27)
Moreover, jwj2 x"(y) jwj2 ¡1 jwj2 x"(y) jwj2 X m(jhj; x) = X m(jhj; x); log y log x and
x"(y) log y jwj2 ¡1 x"(y) log2 y jwj2 X m(jhj; x) = X m(jhj; x): y y log x
Substituting this and Eq. 23 - 27 in Eq. 21, we obtain the theorem. Proof (of Theorem 2). Denote by logk x the k-times iterated logarithm of x. Put x = T and y = yT = T
1 log 2 T
p
in Theorem 1, then it follows that à !! 2 µq ¶o(jwj2 ) à jwj2 e¡°0 w T h(m) = m(h; T ) log2 T 1+O ¡(w 2 ) log T yT m·T X
Ã
µ
¶
!
1¡o(jwj ) jwj2 T m(jhj; T ) q +O log2 T log T Ã ! µ ¶jwj2 T m(jhj; T ) q +o log2 T R(yT ) : log T
Moreover, since by Lemma 4 X h(p)
p·T
p
= (w 2 + o(jwj2 )) log2 T;
2
(28)
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
15
we ¯nd that 8 <X
Ã
Ã
jwj2 h(p) m(h; T ) = exp : log 1 + +O p p2 p·T = (log T )w
2 +o(jwj2 )
!!9 = ;
(1 + O(jwj2 ))
w2
= (log T ) (1 + o(jwj2 log2 T )eo(jwj
2
log 2 T )
);
and 2
2
m(jhj; T ) = (log T )jwj (1 + o(jwj2 log2 T )eo(jwj
log 2 T )
);
as T ! 1. This result along with Eq. 28 and the estimate ¡¡1 (w 2 ) ½ jwj2 show that 2
e¡°0 w T 2 2 2 h(m) = (log T )w ¡1 + o(jwj4 T (log2 T )Re w ¡1 log2 T eo(jwj log 2 T ) ) 2 ¡(w ) m·T X
Ã
2
+O jwj T (log T ) +o(T (log T )
jwj2 ¡1 o(jwj2 log 2 T )
e
jwj2 ¡1 o(jwj2 log 2 T )
e
µq
µq
log2 T
log2 T
¶jwj2
¶¡1 !
(29)
R(yT )):
Since jh(pk )j µ 4jwj2 µ 1 and h(m) is a multiplicative function, it follows that h(m) ½ jwj2 for m > 1. Therefore, 2 X gw (m)
m2¾T
m·T
X
=
m·log 22 T
X gw2 (m) gw2 (m) + m2¾T m2¾T log 2 T <m·T 2
X
2
= 1 + O(jwj log3 T ) +
log 22 T <m·T
gw2 (m) : m2¾T
(30)
Using Eq. 28 and summing by parts, we obtain X
log 22 T <m·T
=T
gw2 (m) = m2¾T
·¡2¾T ¡1
Ã
µ
X
log 22 T <m·T 2
2
+o(T (log T )w +(2¾T + 1 ¡
2 ¡1
·)
w 2 ¡1 o(jwj2 log 2 T )
e
2
eo(jwj
Z
log 2 T )
log 22 T
+o(e
o(jwj2 log 2 T )
log 2 T )
µq
log2 T
log2 T
¶jwj2
¶¡1 !
R(yT ))
q
)
Z
T log 22
T
(log u)jwj
( log4 T )¡1
log2 T
¶jwj2
Z
2 ¡1
u·¡2¾T ¡1 du
T
log 22 T
(log u)
R(ylog 22 T ))
Z
jwj2 ¡1
T
log 22 T
¶
(31)
e¡°0 w 2 (log u)w ¡1 u·¡2¾T ¡1 du 2 ¡(w )
+o(jwj log2 T e o(jwj2
µq
µq
2
T
o(jwj2 log 2 T )
4
+O jwj2 e
m2¾T +1¡·
e¡°0 w T 2 2 2 (log T )w ¡1 + o(jwj4 T (log T ))Re w ¡1 log2 T eo(jwj log2 T ) 2 ¡(w )
+O jwj T (log T )
Ã
h(m)
!
u·¡2¾T ¡1 du
(log u)jwj
2 ¡1
u·¡2¾T ¡1 du
16
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
It is easily seen that Z
T
log 22
T
(log u)w
= (2¾T ¡ = (2¾T ¡
·)
¡w2
·)
¡w2
µ
·)
¡w2
+O (2¾T ¡ = (2¾T ¡
u·¡2¾T ¡1 du Z
2
¡(w ) ¡
·)
2
Z
log 22
1
¡(w ) ¡ ¡Re w 2
e¡v v w ·)
(2¾T ¡
·)
((2¾T ¡
2 ¡1
¡w2
Z
(2¾T ¡·) log log 22 T
2
¡1
¡w2
2 ¡1
e¡v v w
2 ¡1
dv
dv
Z
(2¾T ¡·) log log 22 T
0
·) log T )
w ¡2 (log log22 T )
¡(w 2 ) ¡
dv
0
e¡v v w
(2¾T ¡·) log T
·)¡1 (log T )Re w
+(2¾T ¡
T
(2¾T ¡
2
·)
¡w2
(2¾T ¡·) log T
(2¾T ¡·) log
·)¡w
¡ (2¾T ¡ = (2¾T ¡
2 ¡1
Z
Re w 2 ¡1
w2
0
1
(v w
¡v
2 ¡1
e dv
+ O((2¾T ¡
2
+ O(v Re jwj ))dv
¶
·)(log log22 T )1+jwj
)
2
(32)
Obviously, Z
T
log 22
2 ¡1
T
(log u)jwj
u·¡2¾T ¡1 du ½ (2¾T ¡
2
·)¡jwj ¡(jwj2 ):
(33)
Now, taking into account the estimate Eq. 22 and 2
e¡°0 w = 1 + O(jwj2 ); we deduce the assertion of the theorem from Eq. 30-33 with w = p
6
u . 2 log 2 T
Concluding remarks
Grupp [5] studied Rankin’s mean-square formula (see Lemma 3) in arithmetic progressions. He proved that, for coprime integers a and q, X
pµ x p² a m o d
c2p log p =
x (1 + o(1)); ’(q)
q
where ’ is Euler’s totient. By partial summation, X
pµ x p² a m od
c2p =
x (1 + o(1)): ’(q) log x
q
Now let  be a Dirichlet character mod q. Then Grupp’s result implies X
p·x
c2p Â(p) =
X
pµ x p² a m od
a mod q
8 > > <
=> > :
X
Â(a)
x (1 log x
o
³
c2p = q
X x (1 + o(1)) Â(a) ’(q) log x a mod q
+ o(1))
if  = Â0 ;
´
otherwise;
x log x
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
17
where Â0 denotes the principal character mod q. Thus in our observations, with regard to Lemma 3, we can replace the Fourier coe±cients c(m) by twists c(m)Â(m) with real characters Â(m) and the assertions of Theorem 1 and 2 remain valid (since then Â2 = Â0 ). Note that such twisted zeta-functions associated to cusp forms are automorphic with respect to certain congruence subgroups.
References Ä [1] H. Bohr: "Uber Diophantische Approximationen und ihre Anwendung auf Dirichletsche Reihen, besonders auf die Riemannsche Zetafunktion",In: Proc. 5th Congress of Scand. Math., 1923, Helsingfors, pp. 131{154. Ä [2] H. Bohr and B. Jessen: "Uber die Wertverteilung der Riemannschen Zetafunktion, Erste Mitteilung", Acta Math., Vol. 54, (1930), pp. 1{35. Ä [3] H. Bohr and B. Jessen: "Uber die Wertverteilung der Riemannschen Zetafunktion, Zweite Mitteilung", Acta Math., Vol. 58, (1932), pp. 1{55. [4] P. Deligne: "La conjecture de Weil", Inst. Hautes Etudes Sci. Publ. Math., Vol. 43, (1974), pp. 273{307. [5] F. Grupp: "Eine Bemerkung zur Ramanujanschen ¿ -Funktion", Arch. Math., Vol. 43, (1984), pp. 358{363. [6] D.R. Heath-Brown: "Fractional moments of the Riemann zeta-function", J. London Math. Soc., Vol. 24(2), (1981), pp. 65{78. [7] D. Joyner: Distribution Theorems of L-functions, Longman Scienti¯c, Harlow, 1986. [8] A. Ka·ce_ nas and A. Laurin·cikas: "On Dirichlet series related to certain cusp forms", Lith. Math. J., Vol. 38, (1998), pp. 64{76. [9] A. Laurin·cikas: Limit Theorems for the Riemann Zeta-function, Kluwer, Dordrecht, 1996. [10] A. Laurin·cikas and R. Garunk·stis: The Lerch Zeta-Function, Kluwer, Dordrecht, 2002. [11] R. Leipnik: "The lognormal distribution and strong nonuniqueness of the moment problem", Teor. Veroyatn. Primenen., Vol. 26, (1981), pp. 863{865. [12] B.V. Levin and A.S. Fainleib: "On one method of summing of multiplicative functions", Izv. AN SSSR, ser. matem., Vol. 31, (1967), pp. 697{710. [13] B.V. Levin and A.S. Fainleib: "Application of certain integral equations to problems of the number theory", Uspechi matem. nauk, Vol. 22(3), (1967), pp. 119{197. [14] K. Matsumoto: "Probabilistic value-distribution theory of zeta-functions",S¹ ugaku, Vol. 53, (2001), pp. 279{296. [15] H.L. Montgomery and R.C. Vaughan: "Hilbert’s inequality", J. London Math. Soc., Vol. 8(2), (1974), pp. 73{82. [16] L. Mordell: "On Mr. Ramanujan’s empirical expansions of modular functions", Proc. Camb. Phil. Soc., Vol. 19, (1917), pp. 117{124. [17] S. Ramanujan: "On certain arithmetic al functions", it Trans. Camb. Phil. Soc., Vol. 22, (1916), pp. 159{184.
18
A. Laurinµcikas, J. Steuding / Central European Journal of Mathematics 2(1) (2004) 1{18
[18] R.A. Rankin: "An -result for the coe±cients of cusp forms", Math. Ann., Vol. 203, (1973), pp. 239{250. [19] R.A. Rankin: "Ramanujan’s tau-function and its generalizations", Ramanujan revisited (Urbana-Champaign, Ill. 1987), Academic Press, Boston, 1988, pp. 245{ 268. [20] A. Selberg: "Old and new conjectures and results about a class of Dirichlet series", Collected Papers, Vol. 2, Springer-Verlag, Berlin, 1991, pp. 47{63. [21] G. Tenenbaum: Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995. Ä [22] E. Wirsing: "Das asymptotische Verhalten von Summen Uber multiplikative Funktionen", Math. Ann., Vol. 143, (1961), pp. 75{102.
CEJM 2(1) (2004) 19{49
Classi¯cation of discrete derived categories Grzegorz Bobi¶ nski1¤ , Christof Gei¼2y , Andrzej Skowro¶ nski1z 1
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toru¶ n, Poland 2 Instituto de Matem¶aticas, UNAM, Ciudad Universitaria, 04510 Mexico D.F., Mexico
Received 8 July 2003; accepted 25 September 2003 Abstract: The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over nite dimensional algebras. c Central European Science Journals. All rights reserved. ® Keywords: derived category, Euler form, Auslander{Reiten quiver, gentle algebra MSC (2000): 18E30, 16G20, 16G60, 16G70
Introduction and main results
z
y
¤
Throughout the paper K denotes a ¯xed algebraically closed ¯eld. By an algebra we mean a connected ¯nite dimensional K-algebra (associative, with an identity) and by a module a ¯nite dimensional right module. For an algebra A, we denote by mod A the category of A-modules and by D b (mod A) the derived category of bounded complexes of A-modules. By an equivalence of two derived categories we mean an equivalence of triangulated categories [10]. Recall from [6, 12] that an A-module T is called a tilting (respectively, cotilting) module provided Ext2A (T ; ¡ ) = 0 (respectively, Ext2A (¡ ; T ) = 0), Ext1A (T ; T ) = 0 and the number of pairwise nonisomorphic indecomposable direct summands of T equals the rank of the Grothendieck group K0 (A) of A. Two algebras A and B are called tilting-cotilting equivalent if there exists a sequence of algebras A = A0 , A1 , . . . , Am , Am+1 = B and a sequence
[email protected] [email protected] [email protected]
20
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49 (i)
(i)
(i)
of modules TAi (0 µ i µ m) such that Ai+1 = End TAi and TAi is either a tilting or a cotilting Ai -module [3]. It is well-known that if two algebras A and B are tilting-cotilting equivalent then the derived categories D b (mod A) and D b (mod B) are equivalent [10]. Following [21] a derived category Db (mod A) is said to be discrete if for every vector n = (ni )i2Z of natural numbers there are only ¯nitely many isomorphism classes of indecomposable objects in D b (mod A) of homology dimension vector n. An important class of discrete derived categories is formed by the derived categories D b (mod K¢) of the path algebras K¢ of Dynkin quivers ¢ (of types Am , Dn , E6 , E7 , E8 ), called derived categories of Dynkin type. It is known that a derived category D b (mod A) is equivalent to D b (mod K¢), for some Dynkin quiver ¢, if and only if A is tilting-cotilting equivalent to K¢. In particular, for two Dynkin quivers ¢ and ¢0 , the derived categories D b (mod K¢) and D b (mod K¢0 ) are equivalent if and only if ¢ and ¢0 have the same underlying graph. Recently D. Vossieck proved in [21] that the derived category D b (mod A) of an algebra A is discrete but not of Dynkin type if and only if A is Morita equivalent to the bound quiver algebra of a gentle bound quiver (in the sense of [2]) having exactly one cycle with di®erent numbers of clockwise and counterclockwise oriented relations. However, the classi¯cation of such derived categories has been an open problem. Denote by the set of all triples (r; n; m) of integers such that n ¶ r ¶ 1 and m ¶ 0. For each (r; n; m) 2 consider the quiver Q(r; n; m) of the form
1
®
®n¡ r¡ 2
1 ¡! ¢¢¢ ¡ ¡ ¡ ¡ ! n ¡
r¡
1
®0
&®n ¡ r¡ 1
%
®¡ m
®¡ 2
®¡ 1
(¡ m) ¡ ¡ ! ¢ ¢ ¢ ¡ ¡ ! (¡ 1) ¡ ¡ ! 0
n¡
®n¡ 1 -
r
.®n ¡ r n¡
1 Á¡ ¡ ¡
®n¡ 2
¢ ¢ ¢ Á¡ ¡ ¡ ¡ ®n¡ r+
n¡
r+1
1
the ideal I(r; n; m) in the path algebra KQ(r; n; m) of Q(r; n; m) generated by the paths ®n¡1 ®0 , ®n¡2 ®n¡1 , . . . , ®n¡r ®n¡r+1, and put ¤(r; n; m) = KQ(r; n; m)=I (r; n; m). Our ¯rst main result is the following. Theorem A. Let A be a connected algebra and assume that D b (mod A) is not of Dynkin type. The following conditions are equivalent: (i) D b (mod A) is discrete. (ii) D b (mod A) ’ D b (mod ¤(r; n; m)), for some (r; n; m) 2 . (iii) A is tilting-cotilting equivalent to ¤(r; n; m), for some (r; n; m) 2 . Moreover, for (r; n; m); (r0 ; n0 ; m0 ) 2 , D b (mod ¤(r; n; m)) ’ D b (mod ¤(r 0 ; n0 ; m0 )) if and only if (r; n; m) = (r 0 ; n0 ; m0 ).
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
21
Let f = f(r; n; m) 2 ; n > rg. We note that (r; n; m) 2 f if and only if ¤(r; n; m) is of ¯nite global dimension. We prove in Section 2 that, for each (r; n; m) 2 f , the algebra ¤(r; n; m) is tilting-cotilting equivalent to the bound quiver algebra A(r; n; m) = K¢(r; n; m)=J(r; n; m), where the quiver ¢(r; n; m) is of the form (¡ 1)
¾
Á¡2
¢¢¢
¾m ¡ 1
Á¡ ¡ ¡
(¡ m + 1)
¾1
-¾m
.
0
(¡ m)
°n¡ 1 -
.°0 (n ¡
°n ¡ 2
1) Á¡ ¡ ¡
°n¡ r
¢ ¢ ¢ Á¡ ¡ ¡
n¡
°n¡ r ¡ 1
r Á¡ ¡ ¡ ¡
°1
¢¢¢ Á ¡ 1
and J (r; n; m) is the ideal in K¢(r; n; m) generated by the paths °n¡2 °n¡1 , °n¡3 °n¡2 , . . . , °n¡r¡1 °n¡r . The second aim of the paper is to describe the structure of discrete derived categories which are not of Dynkin type. For (r; n; m) 2 , we denote by ¡(D b (mod ¤(r; n; m))) the (Gabriel) quiver of the category of indecomposable objects in D b (mod ¤(r; n; m)), that is, the quiver whose vertices are the isomorphism classes of indecomposable objects in D b (mod ¤(r; n; m)) and arrows are given by the irreducible morphisms. We have the additional structure of a translation quiver in ¡(D b (mod ¤(r; n; m))) induced by Auslander{Reiten triangles [10, 11], hence ¡(D b (mod ¤(r; n; m))) is just the Auslander{ Reiten (translation) quiver of D b (mod ¤(r; n; m)). The quiver ¡(D b (mod ¤(r; n; m))) is stable if and only if (r; n; m) 2 f . The following theorem describes the structure of the quivers ¡(D b (mod ¤(r; n; m))). Theorem B. (i) For (r; n; m) 2 f , the quiver ¡(Db (mod ¤(r; n; m))) has exactly 3r components, namely 2r components X (0) , . . . , X (r¡1) , Y (0) , . . . , Y (r¡1) of type ZA1 , and (i) r components Z (0) , . . . , Z (r¡1) of type ZA1 we have ¿ m+r X = X[¡ r] 1 . For each X 2 X and for each Y 2 Y (i) we have ¿ n¡r Y = Y [r]. (ii) For (r; n; m) 2 n f , the quiver ¡(Db (mod ¤(r; n; m))) consists of precisely 2r components, namely r components X (0) , . . . , X (r¡1) of type ZA1 and r components L (0) , (i) . . . , L (r¡1) which are equioriented lines of type A1 we have ¿ m+r X = 1 . For each X 2 X X[¡ r], while the vertices of L (i) are projective-injective in ¡(D b (mod ¤(r; n; m))). Recall that n = r for (r; n; m) 2 n f . Theorem B implies in particular that ¤(r; n; m) and ¤(r0 ; n0 ; m0 ) are derived equivalent if and only if (r; n; m) = (r0 ; n0 ; m0 ). In contrast, the structure of the translation quiver ¡(D b (mod ¤(r; n; m))) reveals only the invariant r. For (r; n; m) 2 f , we have the Euler integral quadratic form ¤(r;n;m) and the (nonsymmetric) bilinear homological form h¡ ; ¡ i¤(r;n;m) de¯ned on K0 (D b (mod ¤(r; n; m))) ’ K0 (¤(r; n; m)) ’ Zn+m . We have the following.
22
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
Theorem C. (i) Let (r; n; m), (r 0 ; n0 ; m0 ) 2 f . The bilinear forms h¡ ; ¡ i¤(r;n;m) and h¡ ; ¡ i¤(r0 ;n0 ;m0 ) are Z-equivalent if and only if r ² r 0 (mod 2) and fm + r; n ¡ rg = fm0 + r0 ; n0 ¡ r 0 g. Moreover, if r is even then h¡ ; ¡ i¤(r;n;m) is Z-equivalent to the bilinear ~ m+r;n¡r . form of a hereditary algebra of Euclidean type A (ii) Let (r; n; m) 2 f . If r is odd then the Euler form ¤(r;n;m) is positive de¯nite of Dynkin type Dn+m . If r is even then ¤(r;n;m) is positive semi-de¯nite of Dynkin type An+m¡1 and corank 1.
1
Preliminaries
1.1. Let R be a locally bounded category over K [7]. We denote by mod R the category of all ¯nite dimensional contravariant functors from R to the category of K-vector spaces. If R is bounded (the number of objects in R is ¯nite), then mod R is equivalent to the L category mod A of ¯nite dimensional right modules over the algebra A = R formed by the quadratic matrices a = (ayx )x;y2R such that ayx 2 R(x; y). Conversely, to each basic L algebra A we can attach the bounded category R with A ’ R whose objects are formed by a complete set E of orthogonal primitive idempotents e of A, R(e; f ) = f Ae and the composition is induced by the multiplication in A. We shall identify a bounded category L R with its associated basic algebra R. Recall also that every locally bounded category R is the bound quiver category K Q=I, where Q = QR is the (locally ¯nite) quiver of R and I is an admissible ideal in the path category KQ of Q. In particular, every ¯nite dimensional K-algebra ¤ is Morita equivalent to a bound quiver algebra KQ¤ =I . For a locally bounded category R = KQ=I and a vertex i of Q, we shall denote by ei the corresponding primitive idempotent of R, by SR (i) the corresponding simple R-module, and by PR (i) (respectively, IR (i)) the projective cover (respectively, injective envelope) of SR (i) in mod R. Following [19] a locally bounded category R is said to be special biserial if R ’ KQ=I, where the bound quiver (Q; I) satis¯es the following conditions: (1) The number of arrows in Q with a prescribed source or target is at most 2. (2) For any arrow ® of Q there are at most one arrow ¯ and at most one arrow ° such that ®¯ and °® are not in I. 1.2. For a locally bounded category R we shall denote by ¡(mod R) the Auslander{ Reiten quiver of mod R and by ¿R and ¿R¡ the Auslander{Reiten translations D Tr and Tr D, respectively. We shall identify the vertices of ¡(mod R) with the corresponding indecomposable R-modules. By a component of ¡(mod R) we mean a connected component of ¡(mod R). 1.3. For an algebra ¤ we denote by D b (mod ¤) the bounded derived category of the abelian category of ¯nite dimensional ¤-modules. It has the structure of a triangulated category in the sense of Verdier [20]. The corresponding translation functor D b (mod ¤) ! D b (mod ¤) assigns to each complex X in D b (mod ¤) its shift X[1]. Accordingly, the distinguished triangles in D b (mod ¤) are of the form X ! Y ! Z ! X [1]. We shall often
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23
identify a module from mod ¤ with the corresponding complex in D b (mod ¤) concentrated in degree zero. The homology dimension vector of a complex X from D b (mod ¤) is the vector h-dim X = (dimK H i (X ))i2Z , where H i (X ) is the i-th homology space of X . Following [21] the derived category Db (mod ¤) is said to be discrete provided for every vector n = (ni )i2Z of natural numbers there are only ¯nitely many isomorphism classes of indecomposable complexes in D b (mod ¤) of homology dimension vector n. Recall also that by a result due to J. Rickard [16] two derived categories D b (mod A) and D b (mod B) are equivalent (as triangulated categories) if and only if A = EndDb (mod B) (T ) for a tilting complex T in D b (mod B), that is, a perfect (consisting of ¯nite dimensional projective modules) complex T with HomDb (mod B) (T ; T [i]) = 0 for all i 6= 0 such that the additive category add T of T generates D b (mod B) as a triangulated category. 1.4. The repetitive category [13] of a bounded category (algebra) ¤ is the sel¯njective ^ whose objects are formed by the pairs (n; x) = xn , x 2 ¤, locally bounded category ¤ ^ n ; yn ) = fng £ ¤(x; y), ¤(x ^ n+1 ; yn ) = fng £ D¤(y; x), and ¤(x ^ p ; yq ) = 0 if n 2 Z, and ¤(x ^ p 6= q; q + 1, where DV denotes the dual space HomK (V; K). The repetitive category ¤ was introduced as a Galois covering of the trivial extension T (¤) = ¤ n D¤ of ¤ by its ^ of ¯nite dimensional right ¤-modules ^ injective cogenerator D¤. Then the category mod ¤ can be regarded as the category of ¯nite dimensional Z-graded modules over T (¤). We ^ as a family M = (Mn )n2Z of modules from mod ¤ such view every module M in mod ¤ ^ that M (xn ) = M n (x) for each x 2 ¤ and n 2 Z. The stable module category mod ¤ is a triangulated category where the suspension functor ¡ serves as the translation ^ ! mod ¤, ^ and hence the distinguished triangles in mod ¤ ^ are of the form functor mod ¤ X ! Y ! Z ! ¡ X. We will usually denote ¡ X by X [1]. The Auslander{Reiten ^ is of the form ¿ = º 2 , where º is the Nakayama translation induced translation in mod ¤ ^ (see [10] for details). We have by the canonical shift xn 7! xn+1, x 2 R, n 2 Z, in ¤ ^ which sends a ¤-module X into a ¤-module ^ the canonical inclusion mod ¤ ! mod ¤ M = (Mn ) concentrated at degree 0 (that is, M0 = X and Mn = 0, n 6= 0). An essential role in our investigations will be played by the Happel functor ^ F : D b (mod ¤) ! mod ¤ which is full, faithful, exact, and sends a complex X = (X i )i2Z concentrated in degree ^ 0 to the ¤-module Y = (Yi )i2Z concentrated in degree 0 with Y0 = X 0 , see [10, 14] for details. Moreover, F is an equivalence of triangulated categories if and only if gl: dim ¤ < 1 [10, 11]. In general, by the image of F we will mean the triangulated subcategory of ^ generated by objects of the from F (X), X 2 D b (mod ¤). Note that if Y 2 mod ¤ ^ mod ¤ is nonzero and Y belongs to the image of F then Y [n] 6’ Y for n 6= 0. 1.5. Recall that two ¯nite dimensional algebras A and B are called tilting-cotilting equivalent if there is a sequence of algebras A = A0 , A1 , . . . , Am , Am+1 = B and a (i) (i) (i) sequence of modules TAi , (0 µ i µ m) such that Ai+1 = End TAi and TAi is either a
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tilting or a cotilting Ai -module. Observe that two Morita equivalent algebras are tiltingcotilting equivalent, because every projective generator is a tilting module. Further, every algebra A is tilting-cotilting equivalent to its opposite algebra Aop because the injective cogenerator DA of mod A is a cotilting A-module and Aop = EndA DA. We need in our considerations APR-tilting modules and APR-cotilting modules introduced in [4]. Namely, for an algebra A = KQ=I and a simple projective noninjective A-module SA (i), L the module T i = ¿A¡ SA (i)©( j2Q0 nfig PA (j)) is a tilting A-module, called the APR-tilting module associated to SA (i). Dually, for each simple injective nonprojective A-module L SA (i) the module i T = ¿A SA (i) © ( j2Q0 nfig IA (j)) is a cotilting A-module called the APR-cotilting module associated to SA (i). Finally, recall that if A and B are tiltingcotilting equivalent algebras then the derived categories D b (mod A) and D b (mod B) are equivalent but in general the converse is not true. 1.6. The one-point extension (respectively, coextension) of an algebra A by an A-module M will be denoted by A[M ] (respectively, by [M ]A). Let A = KQ=I and i be a sink of Q. Following [13] the re°ection Si+ A of A is de¯ned to be the quotient of the one-point extension A[IA (i)] by the two-sided ideal generated by the idempotent ei . Then the sink i of Q is replaced in the quiver of Si+ A by a source i0 . Dually, for a source j of Q, the re°ection Sj¡ A of A at j is the quotient of the one-point coextension [PA (j)]A by the two-sided ideal generated by the idempotent ej . Moreover, the source j of Q is replaced in the quiver of Sj¡ A by a sink j 0 . It has been proved in [22] that Si+ A (respectively, Sj¡ A) is tilting-cotilting equivalent to A. 1.7. Assume ¤ = KQ=I is a bound quiver algebra of ¯nite global dimension. Then the Cartan matrix C¤ = (dim K HomA (P¤ (i); P¤ (j)))i;j2Q0 is invertible over Z, and we have a nonsymmetric bilinear form h¡ ; ¡ i¤ : K0 (¤) £ K0 (¤) ! Z given by hx; yi = xC¤¡ t yt for x; y 2 K0 (¤) = ZQ0 . It has been proved by C. M. Ringel [17] that for modules X and Y from mod ¤ we have hdim X; dim Y i¤ =
X
(¡ 1)i dimK Exti¤ (X; Y );
i¸0
where dim Z denotes the dimension vector of a module Z in mod ¤. The associated integral quadratic form ¤ : K0 (¤) ! Z, given by ¤ (x) = hx; xi¤ , for x 2 K0 (¤), is called the Euler form of ¤. Using the isomorphism K0 (¤) ’ K0 (D b (mod ¤)) induced by the natural inclusion K0 (¤) » K0 (D b (mod ¤)) we can consider ¤ as the form de¯ned on K0 (D b (mod ¤)). It is known that if an algebra A is tilting-cotilting equivalent to ¤ (respectively, D b (mod A) ’ D b (mod ¤)) then the Euler forms ÂA and ¤ are Z-equivalent. Moreover, there exists a Z-invertible map ¾ : K0 (A) ! K0 (¤) such that h¾x; ¾yi¤ =
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
25
hx; yi¤ . Finally, we note that if ¤ is positive semi-de¯nite then rad ¤ = fx 2 K0 (¤) j Â(x) = 0g is a subgroup of K0 (¤) such that K0 (¤)= rad ¤ is torsionfree and the form induced on K0 (¤)= rad ¤ by ¤ is Z-equivalent to the Euler form ÂH , where H is the path algebra K¢ of a Dynkin quiver ¢ uniquely determined by ¤ , called the Dynkin type of ¤ . The rank of rad ¤ is called the corank of ¤ . The Z-equivalence class of ¤ is uniquely determined by its corank and Dynkin type (see [5]).
2
Gentle one-cycle algebras
The purpose of this section is to prove the equivalence of the conditions (i), (ii) and (iii) in Theorem A. Following [2] a bound quiver algebra KQ=I is said to be gentle if the bound quiver (Q; I) satis¯es the following conditions: 1) Q is connected and the number of arrows in Q with a prescribed source or sink is at most two, 2) I is generated by a set of paths in Q of length two, 3) For any arrow ® 2 Q1 there are at most one ¯ 2 Q1 and one ° 2 Q1 such that ®¯ and °® do not belong to I, 4) For any arrow ® 2 Q1 there are at most one » 2 Q1 and ´ 2 Q1 such that ®» and ´® belong to I. Examples of gentle algebras are the algebras tilting-cotilting equivalent to the hereditary ~ n , classi¯ed respectively in [1] and [2]. algebras of type An and A By a gentle one-cycle algebra we mean a gentle algebra A = KQ=I whose quiver contains exactly one cycle, or equivalently jQ0 j = jQ1 j. Observe that the bound quiver (Q; I) of a gentle one-cycle algebra A = KQ=I consists of a single cycle together with some branches, each of which is the bound quiver of an algebra tilting-cotilting equivalent to a hereditary algebra of type At , that is, a full connected ¯nite bound subquiver of the in¯nite tree
’
Ã
’
Ã
’
Ã
’
Ã
à ’
Ã
’ Ã
’
Ã
’
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G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
bound by all possible relations ’Ã = 0 = Ã’; also, each branch is joined to the cycle at a single point, which we shall call the root of the branch. It has been proved by J. Nehring [15] that the trivial extension ¤ n D¤ of a non-simply connected algebra ¤ is of polynomial growth if and only if ¤ is Morita equivalent to a gentle one-cycle algebra. Finally, we say that a gentle one-cycle algebra A = KQ=I satis¯es the clock condition provided in the unique cycle of (Q; I) the number of clockwise oriented relations equals the number of counterclockwise oriented relations. The following two theorems give characterizations of gentle one-cycle algebras in terms of the derived categories. Theorem 2.1 ([2]). For an algebra ¤ the following conditions are equivalent: ~ n. (i) D b (mod ¤) ’ D b (mod K¢) for a quiver ¢ of Euclidean type A ~ n. (ii) ¤ is tilting-cotilting equivalent to a hereditary algebra of type A (iii) ¤ is Morita equivalent to a gentle one-cycle algebra satisfying the clock condition. Theorem 2.2 ([21]). The derived category D b (mod ¤) of an algebra ¤ is discrete but not of Dynkin type if and only if ¤ is Morita equivalent to a gentle one-cycle algebra not satisfying the clock condition. Observe that the algebras ¤(r; n; m), (r; n; m) 2 , de¯ned in the introduction are gentle one-cycle algebras not satisfying the clock conditions. Recall also that two tiltingcotilting equivalent algebras have equivalent derived categories, and two Morita equivalent algebras are trivially tilting-cotilting equivalent. Hence, in order to show the equivalence of the conditions (i), (ii) and (ii) in Theorem A, it remains to prove the following fact. Proposition 2.3. Let A be a gentle one-cycle algebra which does not satisfy the clock condition. Then there is a triple (r; n; m) 2 such that A is tilting-cotilting equivalent to ¤(r; n; m). Proof 2.4. Let A = K Q=I, where the bound quiver (Q; I) contains exactly one cycle and satis¯es the conditions (1){(4) of gentle algebra. A path of length two in Q belonging to I is called a zero-relation. We shall prove that there exists a sequence of algebras A = A0 , A1 , . . . , As , As+1 = ¤(r; n; m), for some (r; n; m) 2 , such that the algebras Ai and Ai+1 , 0 µ i µ s, are tilting-cotilting equivalent. This will be done in several steps. (a) In the ¯rst step we prove that A is tilting-cotilting equivalent to a gentle onecycle algebra A1 = KQ(1) =I (1) such that all external branches of the unique cycle are not bound, and consequently are linear quivers without zero-relations. Assume that one of the external branches of (Q; I) is bound be a zero-relation. By passing, if necessary, to the opposite algebra, we may assume that (Q; I) is of the following form Q0A ®
®l¡ 1
1 ¡ ¢¢¢ Á ¡ al¡1 Á¡ ¡ a1 Á¡ a2 Á
®
¡ l al+1 Q00A al Á
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
27
where ®l ®l¡1 2 I, ®l¡1 ®l¡2 62 I, . . . , ® 2 ®1 62 I, one of Q0A and Q00A is a branch, while
L
the other contains the cycle. We de¯ne a module TA = b2Q0 T (b), where T (ai ) = P (al )=P (ai ), for i 2 f1; : : : ; l ¡ 1g, and T (b) = P (b) for b 2 Q0 n fa1 ; : : : ; al¡1 g. Then TA is a tilting A-module and B = End TA = KQB =J, where the bound quiver (QB ; J ) has the form ¯
¯1
¡ l a1 Á ¡ ¢¢¢ Á ¡ al¡1 Á¡ al+1 Q00B Q0B al Á
Q0B = Q0A is bound by the same relations as Q0A , while Q00B = Q00A is bound by the same relations as Q00A . Moreover, the linear quiver al Á al¡1 Á ¢ ¢ ¢ Á a1 Á al+1 is not bound, and º¯1 2 J , for some º 2 Q00B = Q00A , if and only if º®l 2 I. We refer for details to the proof of [3, Lemma 2.4]. Observe that we have replaced the branch of (Q; I) containing the sink a1 by a branch having the same number of vertices, but exactly one zero-relation less. Thus by an obvious induction on the number of zero-relations occurring in the branches of (Q; I) we reduce A = KQ=I to a gentle one cycle algebra A1 = K Q(1) =I (1) whose branches are not bound by zero-relations. (b) The second step in our procedure consists in replacing the algebra A1 = K Q(1) =I (1) by a gentle one-cycle algebra A2 = KQ(2) =I (2) , tilting-cotilting equivalent to A1 , and whose all branches are equioriented linear quivers without zero-relations. This is done by a suitable iterated application of APR-tilting (respectively, APR-cotilting) modules at the simple projective (respectively, simple injective) modules corresponding to sinks (respectively, sources) of the linear branches (Q(1) ; I (1) ). (c) In the third step we replace the algebra A2 = KQ(2) =I (2) by a gentle one-cycle algebra A3 = KQ(3) =I (3) which is tilting-cotilting equivalent to A2 , all zero-relations are on the unique cycle of (Q(3) ; I (3) ), and the branches of (Q(3) ; I (3) ) are equioriented linear quivers. We have some cases to consider. Assume ¯rst that (Q(2) ; I (2) ) admits a bound subquiver of the form c Á ®
a1 ¡! a2 ¡! ¢ ¢ ¢ ¡! at ¡! b ¯&
d where b, c, d lie on the cycle, ®¯ 2 I (2) , and a1 is a source of Q(2) . Suppose the cycle of (Q(2) ; I (2) ) contains a bound subquiver °
¾
b ! c ! ¢ ¢ ¢ ! u ¡! v ¡! w
28
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49 °
with °¾ 2 I (2) , and the quiver b ! c ! ¢ ¢ ¢ ! u ¡! v is not bound. Then the iterated re°ection Sa¡t ¢ ¢ ¢ Sa¡2 Sa¡1 A2 is a gentle one-cycle algebra given by the bound quiver obtained from (Q(2) ; I (2) ) by replacing the branch a1 ! a2 ! ¢ ¢ ¢ ! at by a subpath of an equioriented branch v ! ¢ ¢ ¢ ! a1 ! a2 ! ¢ ¢ ¢ ! at rooted to the cycle in the middle ° ¾ point of the path u ¡! v ¡! w belonging to I (2) . Moreover, Sa¡t ¢ ¢ ¢ Sa¡2 Sa¡1 A2 is tiltingcotilting equivalent to A2 (see 1.6). Assume now that the cycle of (Q(2) ; I (2) ) contains a subquiver of the form b ! c ! c1 ! ¢ ¢ ¢ ! cq Á cq+1 which is not bound, and possibly is of the reduced form b = c¡1 Á c0 = c. Then Sa¡t ¢ ¢ ¢ Sa¡2 Sa¡1 A2 is a gentle one-cycle algebra given by the bound quiver obtained from (Q(2) ; I (2) ) by replacing the branch a1 ! a2 ! ¢ ¢ ¢ ! at by a subpath of an equioriented line ° ¾ ¡ cq Á ¡ cq+1 at Á ¢ ¢ ¢ Á a2 Á a1 Á ¢ ¢ ¢ Á u Á bound by ¾° = 0, with cq and cq+1 lying on the cycle, and the remaining ones not on the cycle. Further, assume that (Q(2) ; I (2) ) contains a bound quiver of the form (§; R) d Á ´
¡ b2 Á ¡ ¢¢¢ Á ¡ br Á ¡ a b1 Á »-
c with »´ 2 I (2) , and a, c, d lying on the cycle. Then the Auslander{Reiten quiver ¡(mod A2 ) admits a full translation subquiver ¿ ¡ P (b1 )
P (b1 ) &
%
¿ ¡r+1 P (b1 )
¢¢¢
&
% ..
P (b2 ) &
..
. &
...
.
%
%
&
% ... %
¿ ¡ P (br )
P (br ) &
%
P (a)
%
¿ ¡r+1 P (b2 )
¿ ¡ P (br¡1 ) &
&
¿ ¡r P (b1 )
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
29
L
where ¿ ¡r P (b1 ) is the direct summand of the radical of P (c). Let TA2 = x2(QA )0 T (x), 2 where T (bi ) = ¿ ¡r+i¡1 P (bi ) for i 2 f1; : : : ; rg, and T (x) = P (x) for x 2 (QA2 )0 n fb1 ; : : : ; br g. Then TA2 is a tilting A2 -module and End TA2 is given by the bound quiver obtained from (Q(2) ; I (2) ) by replacing the bound quiver (§; R) by the following linear quiver ¡ br Á ¡ ¢¢¢ Á ¡ b2 Á ¡ b1 Á ¡ c d¡ ¡ aÁ without relations. Observe that End TA2 is a gentle one-cycle algebra, and is clearly tilting-cotilting equivalent to A2 . Finally assume that (Q(2) ; I (2) ) contains a bound subquiver (¢; J ) of the form e #® °
¾
a1 ¡! a2 ¡! ¢ ¢ ¢ ¡! at ¡! d ¡! br ¡! ¢ ¢ ¢ ¡! b2 ! b1 #¯ c with r; t ¶ 1, c, d, e lying on the cycle, and ®¯, °¾ 2 I (2) . Then ¡(mod A2 ) admits a full translation subquiver ¿ ¡ P (b1 )
P (b1 ) &
%
¿ ¡r+1 P (b1 )
¢¢¢
&
% ..
P (b2 ) &
..
. &
..
.
&
%
%
&
%
¿ ¡r+1 P (b2 ) % ..
¿ ¡ P (br¡1 )
.
&
¿ ¡r P (b1 )
.
%
¿ ¡ P (br )
P (br ) &
%
P (d) % P (c)
& ¿ ¡ P (c)
where ¿ ¡r P (b1 ) = rad P (at ), . . . , P (ai ) = rad P (ai¡1 ), 2 µ i µ t ¡ 1, and P (d)=P (c) = L 0 0 ¡r+i¡1 rad P (e). Let TA0 2 = P (bi ) for i 2 f1; : : : ; rg, x2(QA 2 )0 T (x), where T (bi ) = ¿ 0 0 and T (x) = P (x) for x 2 (QA2 )0 n fb1 ; : : : ; br g. Then TA2 is a tilting A2 -module and End TA0 2 is given by the bound quiver obtained from (Q(2) ; I (2) ) by replacing the bound
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G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
quiver (¢; J ) by the bound quiver of the form e #® a1 ¡! a2 ¡! ¢ ¢ ¢ ¡! at ¡! b1 ¡! b2 ¡! ¢ ¢ ¢ ! br ¡! d #¯ c bound only by ®¯ = 0. Therefore, applying the above procedure to all branches of (Q(2) ; I (2) ) which are not rooted to the cycle in the middle point of a zero-relation (lying entirely on the cycle), we obtain the required gentle one-cycle algebra A3 = KQ(3) =I (3) , tilting-cotilting equivalent to A2 , and whose all zero-relations lie on the cycle. (d) The fourth step in our procedure consists in replacing A3 by a gentle one-cycle algebra A4 = KQ(4) =I (4) such that all (equioriented) branches of (Q(4) ; I (4) ) are oriented toward the cycle, that is, have a source not lying on the cycle. Assume (Q(3) ; I (3) ) contains a bound subquiver of the form e ®#
d ¡! br ¡! ¢ ¢ ¢ ! b2 ¡! b1 ¯#
c
L
0 0 Taking as above the tilting A3 -module TA0 3 = x2(QA 3 )0 T (x), where we put T (bi ) = ¿ ¡r+i¡1 P (bi ) for i 2 f1; : : : ; rg, and T 0 (x) = P (x) for x 2 (QA3 )0 n fb1 ; : : : ; br g, we obtain a gentle one-cycle algebra End TA0 3 given by the bound quiver obtained from (Q(3) ; I (3) ) by replacing the above bound subquiver by the following one
e ®#
b1 ¡! b2 ¡! ¢ ¢ ¢ ! br ¡! d ¯#
c and bound only by ®¯ = 0, and which is tilting-cotilting equivalent to A3 . Applying the iterated re°ections (as above) to all branches of (Q(3) ; I (3) ) which are not oriented toward the cycle, we obtain the required gentle one-cycle algebra A4 = KQ(4) =I (4) .
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
31
(e) The ¯fth step in our procedure consists of removing in (Q(4) ; I (4) ) all consecutive zero-relations oriented in opposite directions on the cycle, together with (possible) branches rooted in the midpoints of those relations. Assume (Q(4) ; I (4) ) admits a full subquiver of the form c #® ¯
a1 ¡ ¡
¢¢¢ ¡ ¡
¡ at Á ¡ ¢¢¢ Á ¡ al at¡1 Á
b1 ¡ ¡
¢¢¢ ¡ ¡
¡ bs Á ¡ ¢¢¢ Á ¡ bk bs¡1 Á
. a0 = b0 ¾
"° d bound only by ®¯ = 0 and °¾ = 0, the vertices c, at, . . . , a1 , a0 = b0 , b1 , . . . , bs , d lie on the cycle, and possibly l = t or k = s. Let H be the path algebra of the full linear subquiver of the above quiver formed by all vertices except c and d. Then H is a hereditary algebra of Dynkin type Al+k¡1 and the Auslander{Reiten quiver ¡(mod H) contains a complete section § containing the simple modules S(at ) and S(bs ), belonging to the opposite border orbits in ¡(mod H ). Let TA0 4 be the direct sum of modules lying on §, considered as A4 -modules. Consider the A4 -module TA4 = TA0 4 ©
M
P (x):
(4) x2Q0 n(QH )0
Then TA4 is a tilting A4 -module and End TA4 is a gentle one-cycle algebra given by the bound quiver obtained from (Q(4) ; I (4) ) by replacing the above bound subquiver by a quiver of the form c ! u1 ! u2 ! ¢ ¢ ¢ ! ui ! w Á vj Á ¢ ¢ ¢ Á v2 Á v1 Á d with i + j = l + k, and not bound. Therefore, we incorporated the linear quivers al ! ¢ ¢ ¢ ! at+1 and bk ! ¢ ¢ ¢ ! bs+1 inside the cycle and erased simultaneously the two zero-relations with midpoints at and bs (thus a clockwise and a counterclockwise zerorelations on the cycle). Applying systematically the above procedure we erase completely all the consecutive zero-relations of opposite directions on the cycle. Thus we obtain a gentle one-cycle algebra A5 = KQ(5) =I (5) , where all zero-relations in (Q(5) ; I (5) ) are either clockwise oriented or counterclockwise oriented zero-relations on the cycle, all branches of (Q(5) ; I (5) ) are lines oriented toward to the cycle and rooted in the midpoints of zerorelations, and A5 is tilting-cotilting equivalent to A4 .
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(f) Our next objective is to replace A5 by a gentle one-cycle algebra A6 = KQ(6) =I (6) , tilting-cotilting equivalent to A5 , and such that all zero-relations in (Q(6) ; I (6) ) are clockwise oriented zero-relations on the cycle. Suppose all zero-relations (Q(5) ; I (5) ) are counterclockwise oriented zero-relations on the cycle. Observe that the opposite algebra Aop 5 op is tilting-cotilting equivalent to A5 (see 1.5). Moreover, A5 is a gentle one-cycle algebra where all zero-relations are clockwise oriented zero-relations on the cycle but all (equioriented) branches are oriented outside the cycle. Applying now the procedure described in (d), we obtain the required gentle one-cycle algebra A6 = KQ(6) =I (6) , obtained from Aop 5 by reversing orientations of all arrows in the branches. (g) We now replace A6 by a gentle one-cycle algebra A7 = K Q(7) =I (7) , tilting-cotilting equivalent to A6 , such that all zero-relations in (Q(7) ; I (7) ) are consecutive clockwise oriented zero-relations on the cycle, and all branches are oriented toward the cycle. Assume that the cycle of (Q(6) ; I (6) ) admits a full bound subquiver § of the form ®
¯
a ¡! b ¡! c = u0 ¡
u1 ¡
¢¢¢ ¡
ul¡1 ¡
°
¾
ul = d ¡! e ¡! f
with l ¶ 0, and bound only by ®¯ = 0 = °¾. We have two cases to consider. Suppose ¯rst that the above walk contains a subquiver of the form ui¡1 ! ui Á ui+1 , for some i 2 f0; : : : ; l ¡ 1g (where u¡1 = b). Consider the path algebra H of the quiver given by the vertices c = u0 , u1 , . . . , ul¡1 , ul = d. Then ¡(mod H) admits a complete section of the form P (c) & ..
. & V %
..
.
% P (d) Denote by TA0 6 the direct sum of modules, considered as A6 -modules, lying on this section, (6) by P the direct sum of all projective A6 -modules P (x), for x 2 Q0 n (QH )0 , and put TA6 = TA0 6 © P . Then B = End TA6 is a gentle one-cycle algebra K¢=J , where (¢; J ) is obtained from (Q(6) ; I (6) ) by replacing § by a quiver §0 of the form ®
¯
°
¾
a ¡! b ¡! c Á v1 Á ¢ ¢ ¢ Á vt ! vt+1 ! ¢ ¢ ¢ ! vl¡1 ! vl = d ¡! e ¡! f and bound only by ®¯ = 0 = °¾. Let C = Sv¡l ¢ ¢ ¢ Sv¡t+ 1 Sv¡1 ¢ ¢ ¢ Sv¡t B be the iterated re°ection. Then C is a gentle one-cycle algebra K¢0 =J 0 , where (¢0 ; J 0 ) is obtained from
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
33
(¢; J ) by replacing the above quiver §0 by a quiver §00 of the form a
vl ®&
% b ¯&
..
.
..
.
% c
vt+1 ´
»&
%
vt %½
!&
e
vt¡1 ¾&
& f
& v1 bound by ®¯ = 0, ¯» = 0, »´ = 0, ½! = 0, and º½ = 0 for the arrow º in Q(6) (if exists) with sink e and di®erent from °. Observe that the vertices a, b, c, vt, e and f lie on the cycle of (¢0 ; J 0 ), while the quivers vt¡1 ! ¢ ¢ ¢ ! v1 and vt+1 ! ¢ ¢ ¢ ! vl are branches. Applying now the procedure from (c) we may replace the algebra C by a gentle one-cycle algebra D = K¢00 =J 00 , where (¢00 ; J 00 ) is obtained from (¢0 ; J 0 ) by replacing the above quiver §00 by the quiver §000 ®
¯
°
’
¾
¡ e ¡! f a ¡! b ¡! c ¡! vl ! ¢ ¢ ¢ ! vt+1 ! vt Á vt¡1 Á ¢ ¢ ¢ Á v1 Á Ã
bound by ®¯ = 0 = ¯°. Moreover, if we have in (¢0 ; J 0 ) a path wp ! ¢ ¢ ¢ ! w2 ! w1 ¡ ! e then also Ã’ 2 J 00 . Finally, applying again the procedure from (c) we may replace D by a gentle one-cycle algebra E = K¢000 =J 000 , tilting-cotilting equivalent to D (hence also to A6 ) given by a bound quiver obtained from the bound quiver (¢00 ; J 00 ) by insertion the path wp ! ¢ ¢ ¢ ! w1 ! e into the cycle. Observe that in our process we replaced the zero-relation °¾ = 0 in (Q(7) ; I (7) ) by a zero-relation ¯° = 0 which is consecutive to ®¯ = 0, all zero-relations in (¢000 ; J 000 ) are clockwise oriented zero-relations on the cycle, and all branches are lines oriented toward the cycle and rooted to the cycle in the midpoints of zero-relations. Assume now that § is the equioriented quiver ®
¯
°
¾
a ¡! b ¡! c = u0 ! u1 ! ¢ ¢ ¢ ! ul¡1 ! ul = d ¡! e ¡! f;
34
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
with l ¶ 0 and bound only by ®¯ = 0 = °¾. Observe that we may have branches in (Q(6) ; I (6) ) rooted to the cycle in the vertices b and e. Denote by § the subquiver of (Q(6) ; I (6) ) consisting of § and the branch xk ! xk¡1 ! ¢ ¢ ¢ ! x1 ! x0 = b rooted to the cycle in the vertex b, where possible k = 0 (§ = §) if such a branch do not exist. Then the Auslander{Reiten quiver ¡(mod A6 ) admits a full translation subquiver of the form P (xk ) % .
&
..
..
.
%
&
P (x1 ) %
&
..
% ..
..
.
%
%
& ..
P (d) &
..
. &
..
.
&
%
S(d)
&
.
% .. &
%
S(ul¡1 )
..
. &
.
.
.
%
S(u1 )
% ..
&
&
%
Nk
%
% ..
&
° &
%
S(c)
&
.
°
° %
I(c)
%
° %
&
.
° &
&
& ¢¢¢
%
&
..
..
&
..
&
&
.
.
&
..
%
%
..
I(u1 )
%
%
°
° &
&
°
%
.
%
°
°
P (ul¡1 )
&
&
°
%
%
% ..
%
° &
I(ul¡1 )
%
° &
&
..
°
.
&
.
. &
..
. &
&
P (u1 )
.
&
P (c) %
% ..
P (b) %
I(d)
.
% N1
&
%
S(b) & P (a)
Let M be the direct sum of the indecomposable A6 -modules I(ul¡1 ), . . . , I(u1 ), I(c), Nk , . . . , N 1 , S(b) (respectively, I(ul¡1 ), . . . , I(u1 ), I(c), S(b), if § = §). Further, denote by (6) P the direct sum of the indecomposable projective A6 -modules P (z), for all z 2 Q0 n fd; ul¡1 ; : : : ; u1 ; c; b; x1 ; : : : ; xk¡1 g, and put T = M © P . Observe that T is a direct sum (6) of jQ0 j pairwise nonisomorphic indecomposable A6 -modules. Moreover, it follows from our choice of M that we have Ext1A6 (T ; T ) = Ext1A6 (M; T ) = D HomA6 (T ; ¿A6 M ) = 0,
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
35
and HomA6 (D(A6 ); ¿A6 T ) = HomA6 (D(A6 ); ¿A6 M ) = 0, and so pdA6 T µ 1. Thus T is a tilting A6 -module. A simple checking shows that F = EndA6 (T ) is a gentle one-cycle algebra K¢=J , where (¢; J ) is obtained from (Q(6) ; I (6) ) by replacing the subquiver § by the subquiver ¯
®
°
¾
a ¡! b Á x1 Á ¢ ¢ ¢ Á xk ! u0 ! ¢ ¢ ¢ ! ul¡1 ¡! d ¡! e ¡! f if § 6= §, and by the subquiver ¯
®
°
¾
a ¡! b ! u0 ! ¢ ¢ ¢ ! ul¡1 ¡! d ¡! e ¡! f is § = §, and bound only by zero-relations ¯° = 0 = °¾ (in both cases). Observe that in this process we replaced the zero-relation ®¯ = 0 by the zero-relation ¯° = 0 which is consecutive to °¾ = 0, and inserted the branch xk ! ¢ ¢ ¢ ! x1 ! x0 = b into the cycle, if such a subquiver of (Q(6) ; I (6) ) exists. Iterating the above two types of procedures, we obtain a gentle one-cycle algebra A7 = KQ(7) =I (7) , tilting-cotilting equivalent to A6 , and such that all zero-relations of (Q(7) ; I (7) ) are consecutive clockwise oriented zero-relations on the cycle, and all branches of (Q(7) ; I (7) ) are lines oriented toward the cycle and rooted to the cycle in midpoints of zero-relations. (h) Assume now that the cycle of (Q(7) ; I (7) ) is not an oriented cycle with I (7) generated by all paths of length 2 on it. We shall prove that then A7 is tilting-cotilting equivalent to an algebra A8 = A(r; n; m) = K¢(r; n; m)=J (r; n; m), where ¢(r; n; m) is the quiver (¡ 1)
¾
Á¡2
¢¢¢
¾m¡1
Á¡ ¡ ¡
(¡ m + 1)
¾1
-¾m
.
0
(¡ m)
°n¡ 1 -
.°0 (n ¡
°n ¡ 2
1) Á¡ ¡ ¡
°n¡ r
¢ ¢ ¢ Á¡ ¡ ¡
n¡
°n¡ r ¡ 1
r Á¡ ¡ ¡ ¡
°1
¢¢¢ Á ¡ 1
for some n > r ¶ 1 and m ¶ 0, equivalently (r; n; m) 2 f , and J(r; n; m) is generated by the paths °n¡r¡1 °n¡r , . . . , °n¡2 °n¡1 . It follows from our assumption that the cycle of (Q(7) ; I (7) ) admits a subquiver ¯r
¯r¡ 1
¯2
¯1
¯0
®
ar+1 ¡ ! ar ¡ ¡ ! ar¡1 ! ¢ ¢ ¢ ! a3 ¡ ! a2 ¡ ! a1 ¡ ! a0 ¡ ¡ b with r ¶ 1 and such that ¯r ¯r¡1 , . . . , ¯2 ¯1 , ¯1 ¯0 2 I (7) are all zero-relations in (Q(7) ; I (7) ). Moreover, beside the cycle, we may have in the quiver (Q(7) ; I (7) ) lines oriented toward the cycle and rooted to the cycle in the vertices ar , . . . , a2 , a1 . We ¯rst show that A7 is tilting-cotilting equivalent to a gentle one-cycle ¤ = K ¢=J where (¢; J ) has the same bound cycle as (Q(7) ; I (7) ) but additionally at most one external line, and such a line is oriented toward the cycle and rooted in the vertex a1 . Thus we shall insert all lines rooted
36
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
in the vertices ar , . . . , a2 into a line rooted in a1 . Suppose t is the maximal element from f1; : : : ; rg such that there is a nontrivial line rooted in the vertex ar , and assume t ¶ 2. Let wp ! ¢ ¢ ¢ ! w2 ! w1 be the branch rooted to the cycle in at , that is, there exists an arrow w1 ! at di®erent from ¯t . Taking the iterated re°ection Sw¡1 Sw¡2 ¢ ¢ ¢ Sw¡p A7 we obtain a gentle one-cycle algebra given by the bound quiver obtained from (Q(7) ; I (7) ) by replacing »
the line wp ! ¢ ¢ ¢ ! w2 ! w1 ! at by the line at¡1 ¡! wp ! ¢ ¢ ¢ ! w2 ! w1 , and ´ moreover we create a zero-relation ´» = 0 if there exists in (Q(7) ; I (7) ) an arrow c ¡! at¡1 di®erent from ¯t¡1 . Applying now the corresponding procedures from (c) and (d) we may replace the algebra Sw¡1 Sw¡2 ¢ ¢ ¢ Sw¡p A7 by a gentle one-cycle algebra having the same bound cycle as (Q(7) ; I (7) ) but the lines rooted only in the vertices at¡1 , . . . , a1 . Hence, by an obvious induction we obtain the required gentle one-cycle algebra ¤ = K¢=J. Suppose ° (¢; J ) admits a subquiver xk ! xm¡1 ! ¢ ¢ ¢ ! x1 ¡! x0 = a1 with ° 6= ¯1 . Applying now the constructions from (g), we may replace ¤ by a gentle one-cycle algebra ¤0 = K¢0 =J 0 , tilting-cotilting equivalent to ¤ (and hence to A7 ), such that (¢0 ; J 0 ) consists of a gentle cycle bound by r consecutive clockwise oriented zero-relations and having m consecutive counterclockwise oriented arrows. Applying now APR-tiling and APRcotilting modules at the simple projective and simple injective ¤0 -modules respectively, we obtain an algebra A8 isomorphic to an algebra A(r; n; m) = K¢(r; n; m)=J (r; n; m), for some (r; n; m) 2 f , which is tilting-cotilting equivalent to A7 . We ¯nally note that A8 = A(r; n; m) = End TA9 , where A9 = ¤(r; n; m) is the algebra KQ(r; n; m)=I(r; n; m) described in the introduction and TA9 is the tilting A9 -module constructed in the second part of (g). In particular, A8 is tilting-cotilting equivalent to A9 = ¤(r; n; m). (i) Finally, assume that the cycle of (Q(7) ; I (7) ) has cyclic orientation and I (7) is generated by all paths of lengths 2 on the cycle. Applying arguments as above (changing of equioriented lines), we conclude that A7 is tilting-cotilting equivalent the gentle onecycle algebra A8 = KQ(8) =I (8) , where (Q(8) ; I (8) ) has the same bound cycle as (Q(7) ; I (7) ) but at most one external line, and this line is not bound and oriented toward the cycle. Observe that A8 is isomorphic to an algebra ¤(r; n; m) = KQ(r; n; m)=I(r; n; m). Therefore, we have proved that A is tilting-cotilting equivalent to ¤(r; n; m), for some (r; n; m) 2 . This ¯nishes the proof of the proposition.
3
Structure of ¡(D b (mod ¤(r; n; m)))
Fix (r; n; m) 2 and let ¤ = ¤(r; n; m). We also denote by Q the quiver Q(r; n; m). Our aim in this section is to describe the quiver ¡(D b (mod ¤)). In particular, we are interested in the action of the suspension functor on ¡(D b (mod ¤)). ^ which is full and Recall that we have the Happel functor F : D b (mod ¤) ! mod ¤ faithful. Moreover, F is an equivalence of triangulated categories if the global dimension of ^ is special biserial (see [2]) and the Auslander{ ¤ is ¯nite, that is, if r < n. We know that ¤ ^ consists of 2r components X (0) , . . . , X (r¡1) , Y (0) , . . . , Y (r¡1) of Reiten quiver of mod ¤ type ZA1 and r components Z (0) , . . . , Z (r¡1) of type ZA1 1 (see [9, Propostion (3.1)]). However, in order to determine which parts of them belong to the image of F we need a
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
37
more precise knowledge about their structure. This information will be also useful in the next section. ^ Let Q ^ be the quiver whose vertices are (i; k), First we give a precise description of ¤. ^ an arrow i 2 Z, k = ¡ m; : : : ; n ¡ 1. For each i 2 Z and k = ¡ m; : : : ; n ¡ 1, we have in Q ®i;k : (i; k) ! (i; k + 1). Next, for i 2 Z and k = n ¡ r + 1; : : : ; n ¡ 1, we have an arrow ^ Finally, we have an arrow ®¤ ®¤i;k : (i; k + 1) ! (i + 1; k) in Q. r + 1) ! i;n¡r : (i; n ¡ ^ for any i 2 Z. In all above formulas (i; n) denotes the vertex (i; 0). It (i + 1; ¡ m) in Q ^ is the quiver of ¤. ^ is known that Q Let !i;¡m be the path ®i;¡m ¢ ¢ ¢ ®i;n¡r and we put !i;k = ®i;k ¢ ¢ ¢ ®i;n¡r ®¤i;n¡r ®i+1;¡m ^ generated by all the ¢ ¢ ¢ ®i+1;k¡2 , k = ¡ m + 1; : : : ; n ¡ r + 1. Let I^ be the ideal in K Q relations of the forms ®i;k ®i;k+1 ; k = n ¡
r; : : : ; n ¡
1;
¤ ®¤i;k ®i+1;k¡1 ; k = n ¡ r + 1; : : : ; n ¡ 1; ¤ ¤ ®i;n¡r ®i+1;n¡1 if m = 0; ®i;¡1 ®¤i+1;n¡1 if m > 0; ®i;n¡r+1 ®¤i;n¡r+1 ¡ ® ¤i;n¡r !i+1;¡m if r > 1; ¤ ¤ ¡ ®i;k¡1 ®i;k ®i;k ®i+1;k¡1 ; k = n ¡ r + 2; : : : ; n ¤ ®i;n¡1 ®i+1;n¡1 ¡ ®i;0 !i;1 if r > 1; ®¤i;n¡1 !i+1;¡m ¡ ®i;0 !i;1 if r = 1;
®i;k !i;k+1 ®i+1;k ; k = ¡ m; : : : ; ¡ 1; 1; : : : ; n ¡
¡
1;
r:
^ ’ K Q= ^ I^ (see for example [18]). We may also identify ¤ with the full subcategory Then ¤ ^ formed by (0; k), k = ¡ m; : : : ; n ¡ 1. of ¤ ^ we denote by M! the corresponding string ¤-module. ^ For each string ! in ¤ If ! = e(i;k) is the trivial path at the vertex (i; k) then we write Mi;k instead of Me(i;k ) . (k)
Fix k 2 f0; : : : ; r ¡ 1g. We denote the vertices of X (k) by Xi;j , i µ j, i; j 2 Z, in (k) (k) (k) (k) (k) (k) such a way that ¿ Xi;j = Xi¡1;j¡1 and we have arrows Xi;j ! Xi;j+1 and Xi;j ! Xi+1;j (k) (provided i + 1 µ j). Similarly, we denote the vertices of Yi;j , i ¶ j, i; j 2 Z, in such a (k)
(k)
(k)
(k)
(k)
(k)
way that ¿ Yi;j = Yi¡1;j¡1 and we have arrows Yi;j ! Yi+1;j and Yi;j ! Yi;j+1 (provided (k) i ¶ j + 1). Finally, we denote the vertices of Z (k) by Zi;j , i; j 2 Z, in such a way (k) (k) (k) (k) (k) (k) that ¿ Zi;j = Zi¡1;j¡1 and we have arrows Zi;j ! Zi+1;j and Zi;j ! Zi;j+1 . Using the general Auslander{Reiten theory for special biserial algebras the above numbering can be arranged in such a way we have the following chains of morphism coming from the natural ordering of strings (k)
(k)
(k)
(k)
(k)
Xi;i ¡ ! Xi;i+1 ¡ ! ¢ ¢ ¢ ¡ ! Zi;i¡1 ¡ ! Zi;i ¡ ! Zi;i+1 ¡ ! (k)
(k)
¢ ¢ ¢ ¡ ! Xi;i¡1 [1] ¡ ! Xi¡1;i¡1 [1]; (k)
(k)
(k)
(k)
(k)
Yi;i ¡ ! Yi+1;i ¡ ! ¢ ¢ ¢ ¡ ! Zi¡1;i ¡ ! Zi;i ¡ ! Zi+1;i ¡ ! (k) (k) ¢ ¢ ¢ ¡ ! Yi¡1;i [1] ¡ ! Yi¡1;i¡1 [1]:
38
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
Moreover, we have distinguished triangles (k)
(k)
(k)
(k)
Xi;i+d ¡ ! Zi;j ¡ ! Zi+d+1;j ¡ ! Xi;i+d [1];
(1)
(k) Yi+d;i
(2)
¡ !
(k) Zi;j
¡ !
(k) Zi;j+d+1
¡ !
(k) Xi+d;i [1]; (k)
(k+1)
(k)
which will play an important role. We may also assume that Xi;j [1] = Xi;j , Yi;j [1] = (k+1) (k) (k+1) Yi;j and Zi;j [1] = Zi;j for k = 0; : : : ; r ¡ 2. (We will see in Lemmas 3.1 and 3.2 that (r¡1) (0) (r¡1) (0) with this convention we have Xi;j [1] = Xi+r+m;j+r+m and Yi;j [1] = Yi+r¡n;j+r¡n .) The above numbering is uniquely determined by the above conditions if we assume that (0)
Z0;0 = S¤ (0); and thus (0) X0;0 [1]
=
8 > > <M!¡ 1;0
if m = 0 and r = 1
M
if m = 0 and r > 1
®¡ 1;n¡ r+ 1 > > :S (¡ 1) if m > 0; ¤ 8 > > M®¤0;1 if r = 1 = n; > > > <S (n ¡ 1) if r = 1 and n > 1; ¤ (0) Y0;0 [1] = > M!0;n¡ 1 if r = 2; > > > > :M®¤ if r > 2:
(3)
(4)
0;n¡ 2
(k) (k) It is known (see [9]) that the modules Xi;i and Yi;i are of the form Mi;k , k = ¡ m; : : : ; ¡ 1; 1; : : : ; n ¡ r, M®i;k , M ®¤i;k , k = n ¡ r + 1; : : : ; n ¡ 1, and M!i;k , k = ¡ m; : : : ; n ¡ r + 1. ^ we need to Thus in order to describe the action of the suspension functor on ¡(mod ¤)
calculate the action of ¿ = ¿¤^ and the suspension functor on the above modules. ^ and our convention M [¡ 1] = M we can easily Using the above description of ¤ calculate the following: M i;k [¡ 1] = M!i;k + 1 ; k = ¡ m; : : : ; ¡ 1; m ¶ 1; M i;k [¡ 1] = M!i;k + 1 ; k = 1; : : : ; n ¡ M®i;n¡ r+ 1 [¡ 1] = M!i+
1;¡ m
; r ¶ 2;
M®i;k [¡ 1] = M®i+
1;k¡ 1
; k =n¡
M®¤i;k [¡ 1] = M®¤i;k + 1 ; k = n ¡
r; n ¶ r + 1;
r + 2; : : : ; n ¡ r + 1; : : : ; n ¡
1; r ¶ 3; 2; r ¶ 3;
M®¤i;n¡ 1 [¡ 1] = M!i;1 ; r ¶ 2; M!i;k [¡ 1] = Mi+1;k ; k = ¡ m; : : : ; ¡ 1; m ¶ 1; M!i;0 [¡ 1] =
(
M®i+
1;n¡ 1
r¶2
M!i+
1;¡ m
r=1
;
M!i;k [¡ 1] = Mi+1;k ; k = 1; : : : ; n ¡ M!i;n¡ r+ 1 [¡ 1] =
(
M!i;1
r=1
M®¤i;n¡ r +
r¶2
1
:
r; n ¶ r + 1;
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
39
Since ¿ = º 2 , ºMi;k = Mi¡1;k , ºM®i;k = M®i¡ 1;k , ºM®¤i;k = M®¤i¡ 1;k and ºM !i;k = M!i¡ 1;k , we can calculate the rules for ¿ which are a little bit more tricky and we will not present them in all details. Note that we have M0;k = S¤ (k), k = ¡ m; : : : ; ¡ 1; 1; : : : ; n ¡ r, M!0;¡ m = P¤ (¡ m) and M®0;k = P¤ (k), k = n ¡ r + 1; : : : ; n ¡ 1. As the result we get ¿ S¤ (k)[j] = S¤ (k + 1)[j]; k = ¡ m; : : : ; ¡ 2; m ¶ 2; ¿ S¤ (1)[j] =
(
P¤ (n ¡
1)[j] r ¶ 2
P¤ (m)[j]
r=1
; m ¶ 2;
¿ S¤ (k)[j] = S¤ (k + 1)[j]; k = 1; : : : ; n ¡ ¿ S¤ (n ¡
1; n ¶ r + 2;
r)[j] = S¤ (1)[j + r]; n ¶ r + 1;
¿ P¤ (¡ m)[j] =
8 > >
1)[j ¡
m = 0; r = 1
1]
¤ > > :S (¡ m)[j ¡ 1] ¤
¿ P¤ (k)[j] = P¤ (k ¡ ¿ P¤ (n ¡
r¡
1)[j ¡
r + 1)[j] = P¤ (¡ m)[j ¡
m = 0; r ¶ 2 ; m ¶ 0; m¶1
1]; k = n ¡
r + 2; : : : ; n ¡
1; r ¶ 3;
1]; r ¶ 2:
Each module of one of the forms Mi;k , with k = ¡ m; : : : ; ¡ 1, M®i;k , with k = n ¡ r + 1; : : : ; n ¡ 1, M!i;k , with k = ¡ m; : : : ; ¡ 1, is the shift of one of the modules S¤ (k), k = ¡ m; : : : ; 0, P¤ (¡ m), P¤ (k), k = n ¡ r + 1; : : : ; n ¡ 1. It follows from the formulas M!i;¡ m [¡ 2k + 1] = Mi+k;¡m+k¡1 ; k = 1; : : : ; m; M!i;¡ m [¡ 2k] = M!i+
k;¡ m + k
; k = 1; : : : ; m;
M !i;¡ m [¡ 2m ¡
k] = M®i+
m+
M!i;¡ m [¡ 2m ¡
r] = M!i+
m + 1;¡ m
1;n ¡ k
; k = 1; : : : ; r ¡
1;
:
Taking into account the above calculations and the assumption (3) we get the following statement about the components X (k) . Lemma 3.1. We have the following formulas (k)
Xq(r+m)+m;q(r+m)+m = P¤ (¡ m)[qr + k]; (k)
Xq(r+m)+p;q(r+m)+p = S¤ (¡ 1 ¡ (k)
Xq(r+m)¡p;q(r+m)¡p = P¤ (n ¡
p)[qr + k ¡ p)[qr + k ¡
1]; p = 0; : : : ; m ¡ p]; p = 1; : : : ; r ¡
(k)
1; m > 0;
1;
(k)
k = 0; : : : ; r ¡ 1, q 2 Z. In particular, Xi;j [r] = ¿ ¡m¡r Xi;j for any k = 0; : : : ; r ¡ i; j 2 Z, i µ j.
1,
Similarly as above, one can show that for r < n each module of one of the forms M i;k , k = 1; : : : ; n ¡ r, M ®¤i;k , k = n ¡ r + 1; : : : ; n ¡ 1, M!i;k , k = 1; : : : ; n ¡ r + 1, is the shift of one of the modules S¤ (k), k = 1; : : : ; n ¡ r. On the other hand, if r = n we have M®¤i;k [r] = M®¤i;k ; k = n ¡ r + 1; : : : ; n ¡
1; r ¶ 2;
40
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
M!i;1 [r] = M!i;1 ; and ¿ M ®¤i;k = M®¤i¡ 1;k+ 2 ; k = n ¡
r + 1; : : : ; n ¡
3; r ¶ 4;
¿ M®¤i;n ¡ 2 = M!i¡ 1;1 ; r ¶ 3; ¿ M®¤i;n ¡ 1 = M®¤i¡ 1;n¡ r + 1 ; r ¶ 2; ¤ ¿ M!i;1 =
(
M!i¡ 1;1
r = 1; 2
M®¤i¡ 1;n¡ r +
r¶3
2
:
Hence, we get the following information about the components Y (k) using the assumption (4). Lemma 3.2. If r < n then we have the following formulas (k)
Yq(n¡r)+p;q(n¡r)+p = S¤ (n ¡
r¡
p)[r ¡
2¡
qr + k]; p = 0; : : : ; n ¡
r¡
1;
(k)
If r = n then the modules Yi;i , k = 0; : : : ; r ¡ 1, i 2 Z, coincide with the modules M®¤i;k , (k)
(k)
k = n ¡ r + 1; : : : ; n ¡ 1, M!i;1 , i 2 Z. In both cases we get Yi;j [r] = ¿ n¡r Yi;j for any k = 0; : : : ; r ¡ 1, i; j 2 Z, i ¶ j. Part (i) of Theorem B follows immediately from the above lemmas, since the Happel functor F is an equivalence if r < n. For part (ii) note ¯rst that the components X (0) , (k) . . . , X (r¡1) are contained in the image of F . It follows, because each module Xi;i is the (k) (k) 6 j, are iterated extension of some Xl;l shift of a ¤-module and we the Xi;j , i = . (0) (r¡1) On the other hand, we have Y [r] ’ Y for Y 2 Y _¢ ¢ ¢_Y , hence the components (0) (r¡1) Y , ..., Y are not contained in the image of F . Using triangles (k) (k) (k) (k) Xi;¡1 ¡ ! Zi;0 ¡ ! Z0;0 = S¤ (0)[k] ¡ ! Xi;0 [1]; i < 0; (k)
(k)
(k)
(k)
X0;i ¡ ! Z0;0 = S¤ (0)[k] ¡ ! Zi+1;0 ¡ ! X0;i [1]; i ¶ 0; (k)
we get that the modules Zi;0 , k = 0; : : : ; r ¡ using triangles (k)
(k)
(k)
1, i 2 Z, belong to the image of F . Finally, (k)
Y¡1;j ! Zi;j ! Zi;0 ! Y¡1;j [1]; j > 0; j < 0; i 2 Z; (k) (k) (k) (k) Yj¡1;0 ! Zi;0 ! Zi;j ! Yj¡1;0 [1]; j > 0; i 2 Z; (k)
we obtain that the modules Zi;j , k = 0; : : : ; r ¡ image of F .
4
1, i; j 2 Z, j 6= 0, do not belong to the
Properties of the Euler form
Fix (r; n; m) 2 f and put ¤ = ¤(r; n; m). We will also use notation introduced in the previous section. Our aim in this section is to describe the properties of the Euler
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
41
form  = ¤ and dimension vectors of indecomposable objects in D b (mod ¤). We put h¡ ; ¡ i = h¡ ; ¡ i¤ . One can easily calculate that hx; yi¤ =
n¡1 X
xi yi ¡
i=¡m
n¡1 X
xi yi+1 +
i=¡m
r+1 X £
(¡ 1)
k
n+1¡k X
¤
xi yi+k ;
i=n¡r
k=2
where yn = y0 and yn+1 = y1 . Consequently n¡1 X
¤ (x) = hx; xi¤ =
n¡1 X
x2i ¡
i=¡m
xi xi+1 +
i=¡m
r+1 X £
(¡ 1)k
n+1¡k X i=n¡r
k=2
¤
xi xi+k ;
where xn = x0 and xn+1 = x1 . We introduce the following notation: (0) si = ¡ dim Xi;i ; i = 0; : : : ; m + r ¡
ti = ¡
(0) dim Yi;i ;
i = 0; : : : ; n ¡
r¡
1; 1;
h1 = s0 + ¢ ¢ ¢ + sm+r¡1 ; h2 = t0 + ¢ ¢ ¢ + tn¡r¡1 : (0)
(0)
Since the objects Xi;i and Yi;i have been described in Lemmas 3.1 and 3.2 we can give more direct formulas for si and ti . In particular, we have h2 = h1 if r is even. We will write just h for this common value in this case. If r is odd then h2 = ¡ h1 ¡ 2e0 , where ei = dim S¤ (i), i = ¡ m; : : : ; n ¡ 1. Moreover, we get the following basis in K0 (¤) d1 = e 0 ; di = si¡2 ; i = 2; : : : ; m + r; di = ti¡m¡r¡1 ; i = m + r + 1; : : : ; m + n ¡
1;
dm+n = h1 : In order to describe the dimension vectors of indecomposable objects in D b (mod ¤) we introduce the following construction. The shift functor T : D b (mod ¤) ! D b (mod ¤) acts on ¡(D b (mod ¤)) in a natural way. Let § = §¤ be the quiver obtained from ¡(D b (mod ¤)) by dividing by T 2 . Since dim X = dim X[2] with each vertex x of § we can associate the dimension vector of the corresponding object of D b (mod ¤), which we will call the dimension vector of x. (k) (k+1) (k) (k+1) Assume ¯rst r is even. Recall, we assumed that Xi;j [1] = Xi;j , Yi;j [1] = Yi;j (k) (k+1) and Zi;j [1] = Zi;j , k = 0; : : : ; r ¡ 2. Finally, from Lemmas 3.1 and 3.2 we get (r¡1)
Xi;j
(0)
(r¡1)
[1] = Xi+r+m;j+r+m and Yi;j
(0)
[1] = Yi¡r+n;i¡r+n . As the consequence, using tri-
(r¡1)
(0)
angles (1) and (2) we obtain that Zi;j [1] = Zi+r+m;j¡r+n . Hence, we get that in this case § is the disjoint union of four stable tubes, two of them of rank m + r and two ~ n¡r;m+r . The dimension vectors of them of rank n ¡ r, and two components of type ZA of vertices lying on the mouth of tubes of rank m + r are by de¯nition s0 , . . . , sm+r¡1 and ¡ s0 , . . . , ¡ sm+r¡1 , respectively, while the dimension vectors of vertices lying on the
42
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
mouth of tubes of rank n ¡ r are t0 , . . . , tn¡r¡1 and ¡ t0 , . . . , ¡ tn¡r¡1 . Finally, using the ~ n¡r;m+r sections triangles (1) and (2) for i = 0 = j we get in the components of type ZA of the forms e0 ! e0 + s0 ! ¢ ¢ ¢ ! e0 + s0 + ¢ ¢ ¢ + sm+r¡2 &
& e0 + t0 ! ¢ ¢ ¢ ! e0 + t0 + ¢ ¢ ¢ + tn¡r¡2 ! e0 + h
and ¡ e0 ! ¡ e0 ¡
s0 ! ¢ ¢ ¢ ! ¡ e 0 ¡
s0 ¡
¢¢¢ ¡
sm+r¡2
&
;
& ¡ e0 ¡
t0 ! ¢ ¢ ¢ ! e 0 ¡
t0 ¡
¢¢¢ ¡
tn¡r¡2 ! ¡ e0 ¡
h
respectively, where we replaced the vertices by their dimension vectors. We get the following description of dimension vectors of indecomposable objects in b D (mod ¤) in this case. Lemma 4.1. If r is even and X is an indecomposable object in the derived category D b (mod ¤) then dim X is of one of the forms ph; p 2 Z; ph + ph +
k+l¡1 X i=k k+l¡1 X
si ; 0 µ k µ m + r ¡
1; 0 < l µ m + r ¡
ti ; 0 µ k µ n ¡
1; 0 < l µ n ¡
r¡
r¡
1; p 2 Z; 1; p 2 Z;
i=k
§ (e0 + ph +
k¡1 X i=0
si +
l¡1 X
ti ); 0 µ k µ m + r ¡
1; 0 µ l µ n ¡
r¡
1; p 2 Z;
i=0
where sm+r+i = si and tn¡r+i = ti . On the other hand, if x is one of the above dimension vectors then: (a) there exist up to shift n + m isomorphism classes of indecomposable objects X in D b (mod ¤) such that dim X = x if x = ph, p 2 Z, (b) there exists a uniquely determined up to shift indecomposable object X in D b (mod ¤) such that dim X = x, otherwise. Proof 4.2. It follows from the well-known properties of stable tubes and quivers of the ~ p;q . form ZA Suppose now r is odd. Similarly as above we get now that the quiver § consists of two tubes of ranks 2(m + r) and 2(n ¡ r), respectively, and one component of type ~ 2(n¡r);2(m+r) . The vertices lying on the mouth of the tube of rank 2(m + r) have ZA dimension vectors s0 , . . . , sm+r¡1 , ¡ s0 , . . . , ¡ sm+r¡1 , the vertices lying on the mouth of
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
43
the tube of rank 2(n ¡ r) have dimension vectors t0 , . . . , tn¡r¡1 , ¡ t0 , . . . , tn¡r¡1 , and in the component of type ZA2(n+m) we have a section ! e0 + s0 ! ¢ ¢ ¢ ! e0 + s0 + ¢ ¢ ¢ + sm+r¡2 !
e0
e0 + h1
#
#
e 0 + t0
e0 + s1 + ¢ ¢ ¢ + sm+r¡1
# .. .
# .. .
#
#
e0 + t0 + ¢ ¢ ¢ + tn¡r¡2
e0 + sm+r¡1
#
#
e0 + h2 ! e0 + t1 + ¢ ¢ ¢ + tn¡r¡1 ! ¢ ¢ ¢ ! e0 + tn¡r¡1 !
;
e0
where again we replaced vertices by their dimension vectors. By the same arguments as above we get the following. Lemma 4.3. If r is odd and X is an indecomposable object in D b (mod ¤) then dim X is of one of the forms § §
k+l¡1 X i=k k+l¡1 X
si ; 0 µ k µ m + r ¡
1; 0 < l µ m + r ¡
1;
ti ; 0 µ k µ n ¡
1; 0 < l µ (n ¡
1;
i=k k¡1 X
e0 +
si +
i=0
l¡1 X
r¡
ti ; 0 µ k µ 2(m + r) ¡
r) ¡
1; 0 µ l µ 2(n ¡
r) ¡
1;
i=0
and 0, where sm+r+i = ¡ si , s2(m+r)+i = si , i = 0; : : : ; m + r ¡ 1, tn¡r+i = ¡ si , s2(n¡r)+i = si , i = 0; : : : ; n ¡ r ¡ 1. On the other hand, if x is one of the above dimension vectors then there exists up to shift in¯nitely many indecomposable objects in D b (mod ¤) with dimension vector x. Let ¡ = ¡m+r;n¡r be the path algebra of the quiver 2
Á¢¢¢ Á
m+r
.
n + m:
1 -
. m + r + 1Á¢¢¢ Án + m ¡
1
44
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
We have the following. Lemma 4.4. Assume r is even. Let ¾ : K0 (¤) ! K0 (¡) be the map given by ¾(di ) = dim S¡ (i); i = 1; : : : ; m + n ¡ ¾(dm+n ) =
m+n X
1;
dim S¡ (j):
j=1
Then ¾ is the isomorphism of K0 (¤) and K0 (¡) such that h¾x; ¾yi¡ = hx; yi for x; y 2 K0 (¤). Proof 4.5. It is easily to check by direct calculations that the vectors ¾di , i = 1; : : : ; m+ n, form a basis of K0 (¡) and h¾di ; ¾dj i¡ = hdi ; dj i¤ , i; j = 1; : : : ; m + n. The following description of  is the immediate consequence of the above lemma. Corollary 4.6. If r is even then  is Z-equivalent to the form of the Euclidean diagram ~ n+m¡1 . In particular,  is positive semide¯nite with corank 1 and of Dynkin of type A type An+m¡1 . We also get the following description of dimension vectors of indecomposable objects in D b (mod ¤) in terms of the Euler form. Proposition 4.7. Let r be even. If x is the dimension vector of an indecomposable object in D b (mod ¤) then ¤ (x) 2 f0; 1g. On the other hand, given x 2 K0 (¤) we have: (a) if Â(x) = 0 then there exist up to shift n + m isomorphism classes of indecomposable objects X in D b (mod ¤) such that dim X = x, (b) if Â(x) = 1 then there exists a uniquely determined up to shift indecomposable object X in D b (mod ¤) such that dim X = x. Proof 4.8. The proposition follows from the description of dimension vectors of indecomposable objects in D b (mod ¤) presented in Lemma 4.1, the formula for the isomorphism ¾ : K0 (¤) ! K0 (¡) given in Lemma 4.4, and well-know description of 0-roots and 1-roots of the form ¡ . Now we turn our attention to the case r odd. Proposition 4.9. If r is odd then  is Z-equivalent to the form of the Dynkin diagram of type Dn+m , hence is positive de¯nite. Proof 4.10. Since r is odd we can rewrite  in the form ¡1 n¡r¡1 n¡1 X X X ¤ 1£ 2 2 2 Â(x) = x¡m + (xi ¡ xi+1 ) + (xi ¡ xi+1 ) + x2i +(xn¡r ¡ xn¡r+1+¢ ¢ ¢+x1 )2 : 2 i=¡m i=1 i=n¡r+1
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
45
Hence Â(x) ¶ 0 and Â(x) = 0 if and only if the following equations are satis¯ed
xi ¡ xi ¡
x¡m = 0;
xi+1 = 0; i = ¡ m; : : : ; ¡ 1; xi+1 = 0; i = 1; : : : ; n ¡ xi = 0; i = n ¡
xn¡r ¡
r¡
1;
r + 1; : : : ; n ¡
1;
xn¡r+1 + ¢ ¢ ¢ + x1 = 0:
As a consequence we get x¡m = ¢ ¢ ¢ = x0 = 0 and there exists a 2 Z such that xi = a, i = 1; : : : ; n ¡ r. Finally, taking into account the last equation, we get 2a = 0, and so a = 0. Hence  is positive de¯nite. Using the same arguments as above we can show that, for each a 2 Zn+m with Pn+m n+m of the system i=1 ai is even, there exists a unique solution x 2 Z x¡m = a1 ;
xi ¡
xi+1 = ai+m+2 ; i = ¡ m; : : : ; ¡ 1;
xi ¡
xi+1 = ai+m+1 ; i = 1; : : : ; n ¡ xi = ai+m ; i = n ¡
xn¡r ¡
r¡
r + 1; : : : ; n ¡
(5)
1; 1;
xn¡r+1 + ¢ ¢ ¢ + x1 = an+m :
In particular,  has exactly 2(n + m ¡ 1)(n + m) roots. Indeed, Â(x) = 1 if and only if x is a solution of the system (5), where jak j = jal j = 1 for some k < l and ai = 0, i 6= k; l. As the consequence we get that  is of type Dn+m since the Dynkin type of a positive de¯nite form is uniquely determined by the number of roots. The connection of the Euler form  with dimension vectors of indecomposable objects in D b (mod ¤) is described by the following proposition. Proposition 4.11. Let r be odd. If x is a dimension vector of an indecomposable object in D b (mod ¤), then Â(x) 2 f0; 1; 2g. Moreover, for each 1-root x of Â, there exists an indecomposable object X in D b (mod ¤) such that dim X = x. Proof 4.12. Since we have a description of the dimension vectors of indecomposable objects in D b (mod ¤) given in Lemma 4.3, by direct calculations we obtain Â(0) = 0; Â(§ Â(§
k+l¡1 X
si ) = 1; 0 µ k µ m + r ¡
i=k k+m+r¡1 X i=k
1; 0 < l < m + r ¡
si ) = 2; 0 µ k µ m + r ¡
1;
1;
46
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
Â(§ Â(§
k+l¡1 X
ti ) = 1; 0 µ k µ n ¡
i=k k+n¡r¡1 X
Â(e0 +
i=k k¡1 X i=0
r¡
si ) = 2; 0 µ k µ n ¡
si +
l¡1 X
1; 0 < l < n ¡ r¡
r¡
1;
1;
ti ) = 1; 0 µ k µ 2(m + r) ¡
1; 0 µ l µ 2(n ¡
r) ¡
1;
i=0
where sm+r+i = ¡ si , s2(m+r)+i = si , i = 0; : : : ; m + r ¡ 1, tn¡r+i = ¡ si , s2(n¡r)+i = si , i = 0; : : : ; n ¡ r ¡ 1, and hence the ¯rst part follows. The second part also follows, since we have exactly 2(m +r)(m +r ¡ 1) di®erent dimension vectors of indecomposable objects in D b (mod ¤) which are 1-roots. The statement of the above proposition is not true for 2-roots of Â, that is in general not each 2-root is a dimension vector of an indecomposable object in D b (mod ¤). Indeed, ¡ ¢ we have 24 m+n + 2(m + n) 2-roots of Â, while there are only 2(m + n) dimension vectors 4 of indecomposable objects in D b (mod ¤) which are 2-roots (these numbers are equal if and only if m + n < 4). We ¯nish our consideration by pointing out how much information can be derived from the knowledge of the Auslander{Reiten quiver and the bilinear Ringel from. Let © = ©¤ be the Coxeter transformation of ¤. Moreover, for nonzero integers a and b, denote by gcd(a; b) the greatest common divisor of a and b, and by lcm(a; b) the least common multiplicity of a and b. Lemma 4.13. Let r be odd. Then there are m + n ¡ 2 + 2 gcd(m + r; n ¡ r) ©-orbits of 1-roots of Â. There are m + r ¡ 1 ©-orbits with 2(m + r) elements, n ¡ r ¡ 1 ©-orbits with 2(n ¡ r) elements and 2 gcd(m + r; n ¡ r) ©-orbits with 2 lcm(m + r; n ¡ r) elements. Proof 4.14. Using the formula ©(dim X ) = dim ¿Db (mod ¤)X , which holds for any object X 2 D b (mod ¤), and the knowledge of the Auslander{Reiten quiver D b (mod ¤) we easily get the following ©s0 = ¡ sm+r¡1 ; ©si = si¡1 ; i = 1; : : : ; m + r ¡
1;
©t0 = ¡ tn¡r¡1 ; ©ti = ti¡1 ; i = 1; : : : ; n ¡
r¡
1:
It follows immediately from the above formulas that, for each l = 1; : : : ; m + r ¡ 1, P l 6= m + r, the vectors k+l¡1 si , k = 0; : : : ; 2(m + r) ¡ 1, form a ©-orbit, and, for each i=k Pk+l¡1 l = 1; : : : ; n ¡ r ¡ 1, the vectors ti , k = 0; : : : ; 2(n ¡ r) ¡ 1, form a ©-orbit. i=k Recall that we use the convention sm+r+i = ¡ si , s2(m+r)+i = si , i = 0; : : : ; m + r ¡ 1, tn¡r+i = ¡ si , s2(n¡r)+i = si , i = 0; : : : ; n ¡ r ¡ 1. P In order to analyze the action of © on the dimension vectors of the form e0 + k¡1 i=0 si +
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
Pl¡1
i=0 ti
47
note that 2(m+r)¡2
X
©e0 = e0 + sm+r¡1 + tn¡r¡1 = e0 +
2(n¡r)¡2
si +
i=0
since ©¡1 e0 = e0 + s0 + t0 . Consequently, we get ©(e0 + ©(e0 + ©(e0 +
k¡1 X i=0
si +
k¡1 X i=0 l¡1 X i=0 l¡1 X i=0
si ) = e0 + ti ) = e0 + ti ) = e0 +
k¡2 X
si +
i=0 m+r¡2 X i=0 k¡2 X i=0
ti ;
i=0
n¡r¡1 X
ti ; k ¶ 1;
i=0 l¡2 X
si +
si +
X
ti ; l ¶ 1;
i=0 l¡2 X
ti ; l; k ¶ 1:
i=0
Note that the above dimension vectors are in a natural correspondence with the elements of the set R = R2(m+r);2(n¡r) = f0; : : : ; 2(m + r) ¡ 1g £ f0; : : : ; 2(n ¡ r) ¡ 1g. According to the above formulas the action of © induces the action on R given by the formula (i; j) 7! (i ¡ 1; j ¡ 1), where the result on the ¯rst coordinate is taken modulo m + r and the result on the second coordinate is taken module n ¡ r. It is an easy combinatorics to notice that this action has exactly gcd(2(m + r); 2(n ¡ r)) = 2 gcd(m + r; n ¡ r) orbits, each of them with 2 4(m+r)(n¡r) = 2 lcm(m + r; n ¡ r) elements. gcd(m+r;n¡r) Note that it follows from the above lemma that in general the bilinear form h¡ ; ¡ i is not Z-equivalent to the form h¡ ; ¡ iD , where D is a hereditary algebra of type Dn+m . Indeed, there are m + n orbits of the action of ©D on 1-roots of ÂD and each orbit has exactly 2(m + n ¡ 1) elements. Proposition 4.15. If (r0 ; n0 ; m0 ) 2 f then the bilinear forms h¡ ; ¡ i and h¡ ; ¡ i¤(r0 ;n0 ;m0 ) are Z-equivalent if and only if r ² r 0 (mod 2) and either m+r = m0 +r 0 and n¡ r = n0 ¡ r 0 or m + r = n0 ¡ r0 and n ¡ r = m0 + r 0 . Proof 4.16. It follows from Corollary 4.6 and Proposition 4.9 that the bilinear forms h¡ ; ¡ i and h¡ ; ¡ i¤(r0 ;n0 ;m0 ) can be Z-equivalent only if r ² r 0 (mod 2). If r and r0 are even then the claim follows from Lemma 4.4, since the bilinear forms of the algebras ¡p;q and ¡p0 ;q 0 are Z-equivalent if and only if either p = p0 and q = q 0 or p = q 0 and q = p0 . Assume now that both r and r 0 are odd. If neither one of the conditions formulated in the proposition is satis¯ed then using the previous lemma we get that the actions of the corresponding Coxeter transformations on 1-roots di®er, hence the forms h¡ ; ¡ i and h¡ ; ¡ i¤(r0 ;n0 ;m0 ) cannot be Z-equivalent. Finally, assume that either m + r = m0 + r 0 and n ¡ r = n0 ¡ r0 or m + r = n0 ¡ r0 and n ¡ r = m0 + r 0 . Then n0 + m0 = n + m. If m + r = m0 + r0 and n ¡ r = n0 ¡ r0 then the map G : K0 (¤) ! K0 (¤(r0 ; n0 ; m0 )) given by G(di ) = d0i is an isomorphism of abelian groups such that hGx; Gyi¤(r0 ;n0 ;m0 ) = hx; yi, where d01 , . . . , d0n+m is the basis of K0 (¤(r0 ; n0 ; m0 ))
48
G. Bobi´nski et al. / Central European Journal of Mathematics 2(1) (2004) 19{49
de¯ned in the analogous way as the basis d1 , . . . , dn+m of K0 (¤). Similarly, if m + r = n0 ¡ r0 and n ¡ r = m0 + r 0 then we de¯ne the map H : K0 (¤) ! K0 (¤(r 0 ; n0 ; m0 )) by the formulas H(d1 ) = d01 ; H(di ) = d0m0 +r0 +i¡1 ; i = 2; : : : ; m + r; H(di ) = d0i¡m¡r+1 ; i = m + r + 1; : : : ; m + n ¡ H(dn+m ) = ¡ d0n+m ¡
1;
2d01 :
A direct checking shows that H is the required isomorphism. An important information which follows from the above proposition is the following. Given (r0 ; n0 ; m0 ) 2 f such that the Auslander{Reiten quivers of D b (mod ¤) and D b (¤(r; n; m)) are isomorphic as the translation quivers, and the bilinear forms h¡ ; ¡ i and h¡ ; ¡ i¤(r0 ;n0 ;m0 ) are Z-equivalent, then either (r0 ; n0 ; m0 ) = (r; n; m) or (r0 ; n0 ; m0 ) = (r; m + 2r; n ¡ 2r). Obviously the second possibility may appear only if n ¶ 2r.
Acknowledgments This work has been done during the visit of the second named author at the Nicholaus Copernicus University in Toru¶n. The authors gratefully acknowledge support from Polish Scienti¯c Grant KBN No 2PO3A 012 14 and Foundation for Polish Science. The second named author acknowledges also support from habilitation grant of DFG (Germany).
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CEJM 2(1) (2004) 50{56
The generalized Boardman homomorphisms Dominique Arlettaz¤ Institut de math¶ ematiques, Universit¶e de Lausanne, CH{1015 Lausanne, Switzerland
Received 4 November 2003; accepted 25 November 2003 Abstract: This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism bn : º n (X) ! H n (X; Z), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations º n (X) ! E n (X), F n (X) ! (E ^ F )n (X), F n (X) ! H n (X; º 0 F ) and F n (X) ¡ ! H n+t (X; º t F ) for other cohomology theories E ¤ (¡ ) and F ¤ (¡ ). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F . ® c Central European Science Journals. All rights reserved. Keywords: Boardman homomorphism, cohomotopy groups, generalized cohomology theories MSC (2000): Primary: 55 N 20 , 55 Q 55; Secondary: 55 Q 45 , 55 S 45
1
Introduction and statement of the results
The classical Boardman homomorphism for a spectrum X is a homomorphism bn : ¼ n (X) ¡ ! H n (X; Z) between the cohomotopy groups and the ordinary integral cohomology groups of X de¯ned for all integers n as follows: (§ n i)¤
¤
bn : ¼ n (X ) = [X; §n S] ¡ ¡ ¡ ¡ ¡ ! [X; §n H(Z)] = H n (X; Z) is induced by a generator i : S ! H(Z) of ¼0 H(Z) ¹= Z, where S denotes as usual the sphere spectrum and H (G) the Eilenberg-MacLane spectrum associated with the abelian group G.
[email protected]
D. Arlettaz / Central European Journal of Mathematics 2(1) (2004) 50{56
51
This short paper provides a couple of results establishing very general approximations of the size of the kernel and of the cokernel of the Boardman homomorphism and of its generalizations. Proofs are presented in the next section. Similar results have been obtained in [4] for the stable Hurewicz homomorphism. When X is an (m ¡ 1)-connected d-dimensional spectrum, bn is trivial if n < m or n > d, an isomorphism if n = d and an epimorphism if n = d ¡ 1. For m µ n µ d ¡ 1, one can easily show, using the Atiyah-Hirzebruch spectral sequence H s (X ; ¼¡t S) ) ¼ s+t (X) that the kernel and the cokernel of bn are of ¯nite exponent: the ¯rst goal of this paper is even to provide universal upper bounds for their exponent (see Theorem 1.2). De¯nition 1.1. For j µ 0 let ½j = 1, and for j ¶ 1 let ½j be the exponent of the j-th homotopy group ¼j S of the sphere spectrum S. Then de¯ne ½¹i =
i Y
½j for i ¶ 1.
j=1
Notice that a prime number p divides ½¹i if and only if p µ
i+3 . 2
The main theorem of the paper is the following: Theorem 1.2. If X is an (m¡ 1)-connected spectrum of ¯nite dimension d, the Boardman homomorphism bn : ¼ n (X) ! H n (X ; Z) has the property that (a) ½¹d¡n ker bn = 0 for m µ n µ d ¡ 1 and (b) ½¹d¡n¡1 coker bn = 0 for m µ n µ d ¡ 2. The second purpose of this note is to generalize this theorem in several ways. First, notice that it is possible to de¯ne a Boardman homomorphism by replacing the ordinary cohomology by other cohomology theories. De¯nition 1.3. Let E be a connective (i.e., (¡ 1)-connected) ring spectrum with identity i : S ! E, and E ¤ (¡ ) the corresponding cohomology theory. For all integers n and for any spectrum X, the E-Boardman homomorphism b bn : ¼ n (X) ! E n (X ) is the homomorphism n
bbn : ¼ n (X) = [X; §n S] ¡ ¡ (§¡ ¡ i)¡ ¤ ! [X; §n E] = E n (X) induced by i. Observe that if X is of ¯nite dimension d, then b bn is trivial for n > d.
Remark. If E = H(Z), the H(Z)-Boardman homomorphism is the classical Boardman homomorphism bn : ¼ n (X) ! H n (X ; Z). If E and X are ¯nite spectra and if X is ¯nite dimensional, we are able to produce universal bounds for the exponent of the kernel of b bn .
Theorem 1.4. Let E be a ¯nite connective ring spectrum such that ¼ 0 (E) is in¯nite cyclic, E ¤ (¡ ) the corresponding cohomology theory, and assume that X is a ¯nite (m¡ 1)-
52
D. Arlettaz / Central European Journal of Mathematics 2(1) (2004) 50{56
connected d-dimensional spectrum. Then the E-Boardman homomorphism b bn : ¼ n (X ) ! E n (X ) satis¯es (½m¡n+1 ½m¡n+2 ¢ ¢ ¢ ½d¡n ) ker b bn = 0
for any integer n µ d. In particular, b bn is injective if n = d and ½¹d¡n ker b bn = 0 if m µ n µ d ¡ 1. One can also extend De¯nition 1.3 and Theorem 1.4 as follows (see [6], p. 291). De¯nition 1.5. Let E be a connective ring spectrum with identity i : S ! E, F any spectrum, and F ¤ (¡ ), respectively (E ^ F )¤ (¡ ), the cohomology theory associated with F , respectively with E ^F . For all integers n and for any spectrum X , the (F and E ^F )Boardman homomorphism b bn : F n (X) ! (E ^ F )n (X) is the homomorphism )¤ bbn : F n (X) = [X; §n (S ^ F )] ¡ §¡ ¡(i^¡ id ¡ ! [X; §n (E ^ F )] = (E ^ F )n (X ) n
induced by i ^ id, where id denotes the identity of F .
De¯nition 1.6. For a connective spectrum F , let ½j (F ) = 1 if j µ 0 and, if j ¶ 1, let us call ½j (F ) the order of the Postnikov invariant k j+1 (F ) of F in the group H j+1 (F [j ¡ 1]; ¼j F ), where F [j ¡ 1] is written for the (j ¡ 1)-st Postnikov section of F (recall that ½j (F ) is ¯nite by Theorem 1.5 of [2]). Then de¯ne ½¹i (F ) =
i Y
½j (F ) for i ¶ 1.
j=1
Theorem 1.7. Let E be a ¯nite connective ring spectrum such that ¼ 0 (E) is in¯nite cyclic, F a connective spectrum, and F ¤ (¡ ), respectively (E ^ F )¤ (¡ ), the cohomology theory associated with F , respectively with (E ^ F ). Furthermore, assume that X is a ¯nite (m ¡ 1)-connected d-dimensional spectrum. Then the (F and E ^ F )-Boardman homomorphism b bn : F n (X) ! (E ^ F )n (X) ful¯lls (½m¡n+1(F )½m¡n+2 (F ) ¢ ¢ ¢ ½d¡n (F )) ker b bn = 0
for any integer n µ d. In particular, b bn is injective if n = d and ½¹d¡n (F ) ker b bn = 0 if m µ n µ d ¡ 1. Remark. In the case where F is the sphere spectrum S, this gives the statement of Theorem 1.4, because ½j (S) = ½j according to Theorem 1.3 of [4]. There is another way to generalize the de¯nition of the Boardman homomorphism and the assertion of Theorem 1.2. De¯nition 1.8. Let F be a connective spectrum and F ¤ (¡ ) the corresponding cohomology theory. For all integers n and for any spectrum X , the generalized F -Boardman homomorphism e bn : F n (X) ! H n (X ; ¼0 F ) is the homomorphism
D. Arlettaz / Central European Journal of Mathematics 2(1) (2004) 50{56
53
n
®0 )¤ ebn : F n (X) = [X; §n F ] ¡ (§ ¡ ¡ ¡ ¡ ! [X; §n H (¼0 F )] = H n (X; ¼0 F )
induced by the 0-th Postnikov section ®0 : F ! H(¼0 F ) of F (i.e., such that (®0 )¤ : ¼0 F ! ¼0 H(¼0 F ) is an isomorphism). Notice that e bn is trivial if n < m or n > d.
Remark. If F = S, the generalized S-Boardman homomorphism is exactly the classical Boardman homomorphism bn : ¼ n (X) ! H n (X; Z). In this case, we get universal bounds for the exponent of the cokernel of the homomorphism e bn , assuming that X is ¯nite dimensional.
Theorem 1.9. Let F be a connective spectrum, F ¤ (¡ ) its associated cohomology theory, and consider an (m ¡ 1)-connected spectrum Xof ¯nite dimension d. Then the generalized F -Boardman homomorphism e bn : F n (X) ! H n (X; ¼0 F ) satis¯es (a) e bn is an isomorphism if n = d and an epimorphism if n = d ¡ 1, (b) ½¹d¡n¡1 (F ) coker e bn = 0 for m µ n µ d ¡ 2.
It is actually possible to extend De¯nition 1.8 by constructing a family of generalized F -Boardman homomorphisms as follows. De¯nition 1.10. Let F be a connective spectrum, F ¤ (¡ ) the corresponding cohomology theory, and X an (m ¡ 1)-connected d-dimensional spectrum. Then, for any integer n µ d and any integer t ¶ 0 such that m ¡ n µ t µ d ¡ n, there exists a generalized F -Boardman homomorphism ebn;t : F n (X) ¡ ! H n+t (X; ¼tF ) ;
which is de¯ned as follows. The cohomological version of Lemma 4.2 of [2] implies that F n (X) ¹= F (m ¡ n; d ¡ n]n (X), where F (m ¡ n; d ¡ n] is the spectrum with the property that ¼i F (m ¡ n; d ¡ n] = 0 if i < m ¡ n or i > d ¡ n and ¼i F ¹= ¼i F (m ¡ n; d ¡ n] if m ¡ n µ i µ d ¡ n (see Section 4 of [2], Section 2.2 of [5], or Section 4 of [7]). Then, for any integer t ¶ 0 with m ¡ n µ t µ d ¡ n, the homomorphism ¸d¡n;t : F (m ¡ n; d ¡ n] ! §t H(¼t F ) introduced in Theorem 1.5 of [3] induces the homomorphism n
¸d¡ n ;t )¤ ebn;t : F n (X) ¹= [X; §n F (m ¡ n; d ¡ n]] (§ ¡ ¡ ¡ ¡ ¡ ¡ ! [X; §n+t H(¼t F )] = H n+t (X ; ¼t F ) :
Observe that e bn;0 = e bn .
The exponent of the cokernel of these homomorphisms is also universally bounded.
Theorem 1.11. Let F be a connective spectrum, F ¤ (¡ ) its associated cohomology theory, and X an (m ¡ 1)-connected spectrum of ¯nite dimension d. Then the generalized F -Boardman homomorphisms e bn;t : F n (X) ! H n+t (X; ¼t F ) satisfy (½t(F )½t+1 (F ) ¢ ¢ ¢ ½d¡n (F )) coker e bn;t = 0
for any n µ d and for any t ¶ 0 with m ¡
nµtµd¡
n.
54
2
D. Arlettaz / Central European Journal of Mathematics 2(1) (2004) 50{56
Proofs
Since Theorem 1.2 is based on the arguments involved in the proofs of Theorems 1.4 and 1.9, we shall prove it at the end of this section. Proof of Theorem 1.4. Since E and X are ¯nite, we may conclude that E n (X) = [X; §n E] ¹= [X; §n (S ^ E)] ¹= [E ¤ ^ X; §n S] = ¼ n (E ¤ ^ X) ; where E ¤ is the dual of E (see [1], p. 195), and that the E-Boardman homomorphism b bn is in fact bbn : ¼ n (X) ¹= ¼ n (S ^ X) ¡ ! ¼ n (E ¤ ^ X ) ¹= E n (X ) ;
induced by i¤ ^ id, where i¤ : E ¤ ! S is the dual of i : S ! E and id the identity X ! X . Observe that, according to the cohomological version of Lemma 4.1 of [2], ¼ n (X) ¹= S[d ¡ n]n (X ) since X is of ¯nite dimension d. The spectrum E ¤ ^ X is also ¯nite dimensional: let us call d0 its dimension (notice that d0 ¶ d) and deduce similarly that ¼ n (E ¤ ^ X) ¹= S[d0 ¡ n]n (E ¤ ^ X). Then consider the commutative diagram S[d ¡ d¡n M
n]n (X)
? ? n y ª1
H
t=m¡n
S[d ¡
n+t
(X; º t S)
? ? n y ª1
n]n (X)
¹=
¹=
S[d0 ¡ 0 ¡n dM
? ? n y ª2
H
t=m¡n
¹=
n]n (X)
S[d0 ¡
n+t
(X; º t S)
? ? n y ª2
bbn
¡ ¡ ¡ !
´
¡ ¡ ¡ !
S[d0 ¡ 0 ¡n dM
n]n (E ¤ ^ X) ? ? n y ª2
H n+t (E ¤ ^ X; º t S)
t=m¡n
n]n (X)
in which ´ is also induced by i¤ ^ id : E ¤ ^ X ! S ^ X , and where the vertical arrows are the homomorphisms introduced in Section 6 of [3] with the property that the composition ªn1 ©n1 is multiplication by (½m¡n+1 ½m¡n+2 ¢ ¢ ¢ ½d¡n ). This follows from the fact that the order ½j (S) of the k-invariant k j+1 (S) of the sphere spectrum S is equal to the exponent ½j of ¼j (S) (see Theorem 1.3 of [4]). Since ¼0 E ¤ ¹= ¼ 0 (E) ¹= Z, it turns out that Hj (X ; Z) ¹= Hj (X; Z) « ¼0 E ¤ is a direct summand of Hj (E ¤ ^ X ; Z) for any j and thus that the homomorphism ´ is split injective. Consequently, the commutativity of the diagram shows that the kernel of b bn is contained in the kernel of the composition ªn1 ©n1 : this produces the assertion of Theorem 1.4. Proof of Theorem 1.7. As above, the ¯niteness of E and X implies that (E ^ F )n (X ) = [X; §n (E ^ F )] ¹= [E ¤ ^ X; §n F ] = F n (E ¤ ^ X) and that
bbn : F n (X) ¹= F n (S ^ X ) ¡ ! F n (E ¤ ^ X) ¹= (E ^ F )n (X)
is induced by i¤ ^ id : E ¤ ^ X ! S ^ X. Then, let us use again the argument developed in the proof of Theorem 1.4 with the spectrum F instead of S and the integers ½j (F ) instead of ½j .
D. Arlettaz / Central European Journal of Mathematics 2(1) (2004) 50{56
55
The proof of Theorem 1.9 is based on the next lemma, the proof of which is analogous to that of Lemma 3.3 of [4]. Lemma. In the Atiyah-Hirzebruch spectral sequence E 2s;t ¹= H s (X ; ¼¡tF ) ) F s+t (X) converging towards the F -cohomology of any (m ¡ 1)-connected d-dimensional spectrum n;0 ! E1 X, the edge homomorphism F n (X) ! ,! E2n;0 ¹= H n (X; ¼0 F ) is exactly the generalized F -Boardman homomorphism e bn for all integers n such that m µ n µ d.
Proof of Theorem 1.9. Assertion (a) is trivial. Now, take n such that m µ n µ d ¡ 2. Since F is connective and X is d-dimensional, the E2 -term of the Atiyah-Hirzebruch n;0 spectral sequence satis¯es E2s;t ¹= H s (X ; ¼¡t F ) = 0 if s > d, and consequently E1 = n;0 n;0 n e Ed¡n+1. By the previous lemma, the image of b is Ed¡n+1 , which is the subgroup of n;0 ! Ern+r;1¡r for E2n;0 ¹= H n (X; ¼0 F ) consisting of the kernel of the di®erentials dn;0 r : Er n;0 n;0 2 µ r µ d ¡ n, and coker e bn ¹= E2 =Ed¡n+1. However, it follows from Proposition 2.6 of n;0 [2] that ½r¡1 (F ) dr = 0. This implies that the product ½¹d¡n¡1 (F ) kills coker e bn . Proof of Theorem 1.2. According to Theorem 1.3 of [4], ½j (S) = ½j and Theorem 1.9 provides the second assertion of Theorem 1.2 For X ¯nite, the ¯rst assertion of Theorem 1.2 is exactly the statement of Theorem 1.4 for E = H(Z). In the general case, the above lemma for the Atiyah-Hirzebruch spectral sequence E 2s;t ¹= H s (X ; ¼¡t S) ) ¼ s+t (X) implies that the kernel of the classical Boardman homomorphism bn : ¼ n (X ) ! H n (X; Z) is killed by the product of the exponents of the groups E2n¡t;t for n ¡ d µ t µ ¡ 1, hence by the product ½¹d¡n . Proof of Theorem 1.11. By de¯nition of e bn;t and by Theorem 6.2 of [3], there is a homomorphism £n;t : H n+t (X; ¼t F ) ! F n (X) such that the composition £n ;t
H n+t (X; ¼t F ) ¡ ¡ ¡ ¡ ¡ ! F n (X)
ebn;t
¡ ¡ ¡ ¡ ¡ ! H n+t (X; ¼t F )
is multiplication by the product (½t(F )½t+1 (F ) ¢ ¢ ¢ ½d¡n (F )) on H n+t(X ; ¼t F ). This provides the assertion of Theorem 1.11.
References [1] J.F. Adams: Stable homotopy and generalised homology, The University of Chicago Press, Chicago, 1974. [2] D. Arlettaz: "The order of the di®erentials in the Atiyah-Hirzebruch spectral sequence", K-Theory, Vol. 6, (1992), pp. 347{361. [3] D. Arlettaz: "Exponents for extraordinary homology groups", Comment. Math. Helv., Vol. 68, (1993), pp. 653{672. [4] D. Arlettaz: "The exponent of the homotopy groups of Moore spectra and the stable Hurewicz homomorphism", Canad. J. Math., Vol. 48, (1996), pp. 483{495. [5] C.R.F. Maunder: "The spectral sequence of an extraordinary cohomology theory", Math. Proc. Cambridge Philos. Soc., Vol. 59, (1963), pp. 567-574
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D. Arlettaz / Central European Journal of Mathematics 2(1) (2004) 50{56
[6] R.M. Switzer: Algebraic topology - homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, 1975. [7] J.W. Vick: "Poincar¶e duality and Postnikov factors", Rocky Mountain J. Math., Vol. 3, (1973), pp. 483-499
CEJM 2(1) (2004) 57{66
Oscillation Results for Second Order Nonlinear Di® erential Equations Jozef D¾ urina1¤, Dá¹ a Lacková2y 1
2
Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University, B. Nì mcovej 32, 042 00 Ko¹ ice, Slovakia Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University, B. Nì mcovej 32, 042 00 Ko¹ ice, Slovakia
Abstract: In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral di¬erential equations of the form ³ ³ £ ¡ ¢£ ¡ ¢¤0 ´0 ¤´ r(t)Á x(t) x(t) + p(t)x ½ (t) + q(t)f x ¼ (t) = 0: Easily veri able criteria are obtained that are also new for di¬erential equations without neutral term i.e. for p(t) ² 0. ® c Central European Science Journals. All rights reserved. Keywords: Neutral equation, oscillatory solution MSC (2000): 34C10
1
Introduction
In this paper we deal with the oscillatory behavior of the solutions of the following neutral di®erential equation
³
¡
¢£
¡
y
¤
r(t)Ã x(t) x(t)+p(t)x ¿ (t)
[email protected] [email protected]
¢¤0 ´0
³ £
+q(t)f x ¾(t)
¤´
= 0:
(E)
58
J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
Throughout this paper we suppose that the following conditions (H1){(H6) hold. (H1) r(t), q(t) 2 C([t0 ; 1)) are positive; (H2) p(t) 2 C([t0 ; 1)), 0 µ p(t) µ p < 1; (H3) ¿ (t) 2 C([t0 ; 1)), ¿ (t) µ t, lim ¿ (t) = 1; t!1
(H4) ¾(t) 2 C 1 ([t0 ; 1)), ¾(t) µ t, lim ¾(t) = 1, ¾ 0 (t) ¶ 0; t!1
(H5) Ã(u) 2 C((¡ 1; 1)), 0 < m µ Ã(u) µ M ; (H6) f (u) 2 C((¡ 1; 1)) is nondecreasing, f 2 C 1 (R ¡ f0g) and uf (u) > 0 for u 6= 0. The problem of oscillatory behavior of the neutral di®erential equations is of both theoretical and practical interest. Recently, many results regarding particular cases of (E) have been published (see enclosed references). The authors generalized and extended some known oscillatory results. Notably, Wang & Yu in [14] considered general di®erential equations and obtained some oscillation criteria for the second order neutral di®erential equations. The aim of this paper is to establish some new oscillatory results for (E), which are new even for the corresponding ordinary delay di®erential equations (i.e. p(t) ² 0) and improve those results presented in [1]{[14]. By a solution of (E) we mean a function x 2 C 1 [Tu ; 1), Tu ¶ t0 , which has the ¡ ¢£ ¡ ¢¤0 property r(t)à x(t) x(t) + p(t)x ¿ (t) 2 C 1 [Tu ; 1) and satis¯es (E) on [Tu ; 1). We consider only those solutions u(t) of (E) which satisfy supfju(t)j : t ¶ T g > 0 for all T ¶ Tu . We assume that (E) possesses such a solution. As usual, a solution of (E) is said to be oscillatory if it has arbitrarily large zeros on [t0 ; 1) and (E) is said to be oscillatory if every solution of this equation is oscillatory. For the sake of convenience, we assume that all functional inequalities, used in this paper, hold eventually, that is, they are satis¯ed for all su±ciently large t.
2
Oscillation
The following theorems provide su±cient conditions for oscillation of all solutions of (E) with respect to properties of the function f (u). Theorem 2.1. Assume that f 0 (u) is nondecreasing in (¡ 1; ¡ t¤ ) and non-increasing in (t¤ ; 1), t¤ ¶ 0. Let R(t) =
Z
t
t0
Z
1
µ
Z
£
1
³
1 ds ! 1 r(s)
£
¤´
q(s)f § N R ¾(s)
¤
R ¾(t) q(t) ¡
4(1 ¡
ds = 1
as
t!1; for all
N >0;
M ¾ 0 (t) p)R[¾(t)]r[¾(t)]f 0 (§ KR[¾(t)])
for some K > 0. Then (E) is oscillatory.
(1)
¶
dt = 1 ;
(2) (3)
J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
59
Proof. Let K > 0 be such that (3) holds. Assuming the converse, we admit that (E) has an eventually positive solution x(t). The case, when x(t) < 0 can be treated by the same arguments. Set ¡ ¢ z(t) = x(t) + p(t)x ¿ (t) : (4) Then z(t) > 0, z(t) ¶ x(t) and moreover, (E) can be rewritten as
³
¡
¢
0
r(t)Ã x(t) z (t)
³
¡
¤
£
¢
Consequently, r(t)Ã x(t) z 0 (t)
´0
´0
³ £
+ q(t)f x ¾(t)
¤´
=0:
(5)
< 0 and taking into account (H1) and (H5), we obtain
¡
¢
that either z 0 (t) > 0 or z 0 (t) < 0. It is easy to see that the condition r(t)Ã x(t) z 0 (t) < 0 together with (1) and (H5) lead to z(t) ! ¡ 1 as t ! 1. This contradiction a±rms that z 0 (t) > 0 and moreover
£
¤
x ¾(t) = z ¾(t) ¡
where p¤ = 1 ¡
£
¶ z ¾(t)
¤³
¡
¢ ³ £
p ¾(t) x ¿ ¾(t)
1¡
£
p ¾(t)
p. From (H6) we have
¤´
³
¡
£
¤
¶ f p z ¾(t)
¢
´0
³
£
¢
¶ p¤ z ¾(t) ;
¤´
r(t)Ã x(t) z 0 (t)
De¯ne
¡
³
£
¤
£
¤´
¤ r(t)Ã(x(t))z 0(t) f (p¤ z[¾(t)])
¡
¢
¤ ³ £
p ¾(t) z ¿ ¾(t)
¤´
(6)
¤´
+ q(t)f p¤ z ¾(t)
w(t) = R ¾(t)
£
¶ z ¾(t) ¡
³ £
f x ¾(t) and then (5) implies
¤´
µ0:
(7)
:
£
¤ £
¤
Then w(t) > 0. Using the fact that r(t)Ã x(t) z 0 (t) µ M r ¾(t) z 0 ¾(t) , one gets in view of (E)
£ ¤ ¾ 0 (t) r(t)Ã(x(t))z 0 (t) ¡ R ¾(t) q(t) r[¾(t)] f (p¤ z[¾(t)]) £ ¤ r(t)Ã(x(t))z 0 (t) 0 ³ ¤ £ ¤´ ¤ 0£ ¤ ¡ R ¾(t) f p z ¾(t) p z ¾(t) ¾ 0 (t) ¤ 2 f (p z[¾(t)]) 0 £ ¤ ¾ (t) p¤ ¾ 0 (t)f 0 (p¤ z[¾(t)]) 2 µ w(t) ¡ w (t) ¡ R ¾(t) q(t) : R[¾(t)]r[¾(t)] M R[¾(t)]r[¾(t)]
w 0 (t) µ
It is easy to verify that
w 0 (t) µ ¡
£ ¤ M ¾ 0 (t) ¡ R ¾(t) q(t) 4p¤ R[¾(t)]r[¾(t)]f 0 (p¤ z[¾(t)]) · ¸2 p¤ ¾ 0 (t)f 0 (p¤ z[¾(t)]) M w(t) ¡ : M R[¾(t)]r[¾(t)] 2p¤ f 0 (p¤ z[¾(t)])
60
J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
Therefore w 0 (t) µ
£
M ¾ 0 (t) ¡ 4p¤ R[¾(t)]r[¾(t)]f 0 (p¤ z[¾(t)])
¡
¤
R ¾(t) q(t) :
¢
(8)
¡
¢
We shall prove that r(t)Ã x(t) z 0 (t) ! 0 as t ! 1. To show it, we let r(t)Ã x(t) z 0 (t) ! ¡ ¢ 2L as t ! 1, 0 < L < 1. Then, since r(t)Ã x(t) z 0 (t) is decreasing, we see that ¡ ¢ r(t)Ã x(t) z 0 (t) ¶ 2L. Integrating this inequality from t1 to ¾(t), we obtain
¤ 2L ³ £ z ¾(t) ¶ z(t1 ) + R ¾(t) ¡ M £
¤
´
¤ L £ R ¾(t) : M
£
¤´
R(t1 ) ¶
Integrating (7) from t1 to 1 and using the last estimate, we get
¡
¢
0
r(t1 )Ã x(t1 ) z (t1 ) ¶ ¶
Z
1
Zt11
³
q(s)f p¤ z ¾(s) q(s)f
t1
µ
¡
ds
¶
¤ p¤ L £ R ¾(s) ds : M ¢
This contradicts (2) and we conclude that r(t)Ã x(t) z 0 (t) ! 0 as t ! 1. Then for every ¸ > 0, there exists a t2 > t1 such that for all t ¶ t2
¡
¢
¸ : 2
r(t)Ã x(t) z 0 (t) µ In other words
¸ : 2m Dividing both sides by r(t) and then integrating from t1 to ¾(t), we get for ¸ = r(t)z 0 (t) µ
£
¤
z ¾(t) µ z(t1 ) +
¸ ³ £ R ¾(t) ¡ 2m
R(t1 )
¤´
µ
¤ K £ R ¾(t) : p¤
mK p¤
(9)
We claim that lim z(t) = 1. To prove it, assume that lim z(t) = 2c, 0 < c < 1. Thus
£
¤
t!1
t!1
z ¾(t) ¶ 2c. Then integrating (7) from t to 1 we have
¡
¢
0
r(t)Ã x(t) z (t) ¶
Z
1
t
Integrating once more from t1 to 1, we get 2c ¶ z(t1 ) +
Z
1
Z t1
³
£
q(s)f p¤ z ¾(s)
¤´
ds :
Z
1 ³ £ ¤´ 1 q(s)f p¤ z ¾(s) ds du r(u)Ã(x(u)) u Z 1 1 q(s)ds du r(u) u
f (p¤ c) 1 ¶ M Zt 1 f (p¤ c) 1 ¡ = R(s) ¡ M t1
¢
R(t1 ) q(s)ds :
R1 £
¤
On the other hand, since (3) implies R ¾(s) q(s)ds = 1, the previous inequality leads to a contradiction. Hence we conclude that lim z(t) = 1. Therefore, in view of (9)
£
¤
t!1
£
¤
t¤ < p¤ z ¾(t) µ KR ¾(t) :
J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
61
Combining the last inequality with (8), we get
£
M ¾ 0 (t) ¡ 4p¤ R[¾(t)]r[¾(t)]f 0 (KR[¾(t)])
w 0 (t) µ
Integrating from t1 to t, one can see that
Z tµ
w(t) µ w(t1 ) ¡
£
¤
M ¾ 0 (s) 4p¤ R[¾(s)]r[¾(s)]f 0 (KR[¾(s)])
R ¾(s) q(s) ¡
t1
¤
R ¾(t) q(t) :
¶
ds :
Letting t ! 1, we get w(t) ! ¡ 1. This contradiction completes the proof of Theorem 2.1. Remark. Theorem 2.1 generalizes Theorem 4.4.4 in [4] and Theorem 1 in [14]. Remark. Note that for f (u) ² u we can replace assumption (H5) by
f5) Ã(u) 2 C(¡ 1; 1), 0 < Ã(u) µ M (H
and Theorem 2.1 is still true.
For a particular case of (E), namely for the di®erential equation
¢h
¡
£
(r(t)Ã x(t) x(t)+p(t)x ¿ (t)
¤i0
¯ £ ¯
¤¯¯¯¡1 £
)0 +q(t)¯x ¾(t) ¯
Theorem 2.1 provides the following corollaries:
¤
x ¾(t) = 0 ;
(E¯ )
Corollary 2.2. Let 0 < ¯ < 1. Assume that (1) holds. If for some L > 0
Z
µ
1
£
R ¾(t) q(t) ¡
then (E¯ ) is oscillatory. Proof. Setting K =
³
¤
4(1¡p)¯L M
´ 1¡1
L¾ 0 (t) R ¯ [¾(t)]r[¾(t)]
¶
dt = 1 ;
we see that (10) implies (3). On the other hand, it
follows from (10) that for some L > 0 lim
t!1
µZ
£
t
¤
L
R ¾(s) q(s)ds ¡
t1
1¡
¯
R
1¡¯
Hence, for all N > 0 and t large enough
Z
t
¤
R ¾(s) q(s)ds ¡
t1
This means that
£
Rt
t1
(10)
R[¾(s)]q(s)ds R 1¡¯ [¾(t)]
L 1¡
>
¯
¾(t)
£
¶ ¤
=1:
¤
R1¡¯ ¾(t) > N :
L 1¡
£
¯
+
N R 1¡¯ [¾(t)]
:
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J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
Taking limits on both sides and using L’Hospital’s rule we obtain R 1+¯ [¾(t)]q(t)r[¾(t)] L > : 0 t!1 ¾ (t) 2 lim
Therefore for all large t
R 1+¯ [¾(t)]r[¾(t)]q(t) L > : ¾ 0 (t) 2
Obviously,
£ ¤´ L d³ R ¾(t) q(t) > ln R ¾(t) : 2 dt Integrating from t1 to 1 we get ¯
£
Z
¤
1
£
¤
R ¯ ¾(t) q(t)dt = 1
t1
and thus, (2) is satis¯ed for (E¯ ). The assertion of this corollary follows from Theorem 2.1. Corollary 2.3. Let 0 < ¯ < 1. Assume that (1) holds. Let lim inf t!1
R1+¯ [¾(t)]r[¾(t)]q(t) >0: ¾ 0 (t)
(11)
Then (E¯ ) is oscillatory. Proof. The reader will easily check that (11) implies (10). Remark. Theorem 2.1, Corollaries 2.2 and 2.3 are new also for the corresponding delay di®erential equation ³ £ ¡ ¢0 ¤´ r(t)x0 (t) + q(t)f x ¾(t) = 0 and generalize Chanturia and Kiguradze’s results [3].
Corollary 2.4. Let ¯ = 1. Assume that (1) holds. If
Z
1
µ
£
¤
R ¾(t) q(t) ¡
then (E¯ ) is oscillatory.
M ¾ 0 (t) p)R[¾(t)]r[¾(t)]
4(1 ¡
¶
dt = 1 ;
(12)
Proof. The reader will have no di±culty to show that (12) implies (2) and (3). Example 2.5. We consider
µ
1 1 + x2 (t)
µ
x(t) +
³1 2
¡
¡t
e
´ £
x t¡
where a > 0, 0 < ¸ < 1. We have Ã(u) = that condition (12) reduces to
j sin tj
1 1+u2
a>
¶¶ ¤ 0 0
+
µ 1, p(t) =
1 2¸
a x[¸t] = 0 ; t2 1 2
¡
t¶1
(13)
e¡t µ 12 . It is easy to see (14)
J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
63
and by Corollary 2.4, Eq.(13) is oscillatory if (14) holds. Corollary 2.6. Let ¯ = 1. Assume that (1) holds. Let lim inf t!1
R 2 [¾(t)]r[¾(t)]q(t) M > : ¾ 0 (t) 4(1 ¡ p)
Then (E¯ ) is oscillatory. The proof is omitted. Now we turn our attention to (E) with di®erent properties of the function f (u). Theorem 2.7. Let (1) hold. Assume that f 0 (u) is non-increasing in (¡ 1; ¡ t¤ ) and nondecreasing in (t¤ ; 1), t¤ ¶ 0. If for some K > t¤
Z
1
µ
¡
¢
R ¾(t) q(t) ¡
then (E) is oscillatory.
4(1 ¡
M ¾ 0 (t) p)R[¾(t)]r[¾(t)]f 0 (§ K)
¶
dt = 1 ;
(15)
Proof. On the contrary, we assume that x(t) is a positive solution of (E). Arguing as in R1 £ ¤ the proof of Theorem 2.1, we are led to (8). Since R ¾(t) q(t)dt = 1, then similarly as £ ¤ in the proof of Theorem 2.1, it can be shown that lim z(t) = 1. Then p¤ z ¾(t) > K for
¡
£
any K > t¤ and at the same time f 0 p¤ z ¾(t) with (8), we get
¡
0
¢
w (t) µ ¡ R ¾(t) q(t) +
4(1 ¡
¤¢
t!1
¶ f 0 (K). Combining the last inequality
M ¾ 0 (t) : p)R[¾(t)]r[¾(t)]f 0 (K)
The rest of the proof is similar to the proof of Theorem 2.1 and hence, it is omitted. Now we apply our previous result to Eq. (E¯ ). Corollary 2.8. Let (1) hold and ¯ > 1. If for some L > 0
Z then (E¯ ) is oscillatory.
1
µ
¡
¢
R ¾(t) q(t) ¡
L¾ 0 (t) R[¾(t)]r[¾(t)]
¶
dt = 1 ;
Proof. Direct calculation shows that (15) reduces to (16) for (E ¯ ). Corollary 2.9. Let (1) hold and ¯ > 1. If lim inf t!1
R 2 [¾(t)]r[¾(t)]q(t) >0; ¾ 0 (t)
(16)
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J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
then (E¯ ) is oscillatory. The proof is trivial and so it is left to the reader. To relax the monotonicity conditions imposed to f (u) and f 0 (u), let us consider the following neutral di®erential equation
µ
¡
r(t)Ã x(t)
¢³
£
x(t) + p(t)x ¿ (t)
¶ ¤´0 0
³ £
¤´
+ q(t)h x ¾(t)
=0;
¹ (E)
subject to conditions (H1) { (H5) and (H7) h(u) 2 C(¡ 1; 1), uh(u) > 0 for u 6= 0 . It is easy to see from the proofs of presented theorems and corollaries that those ¹ employing the following additional condition results can be reformulated for (E) h(u) ¶ Af (u) ;
A>0
to the assumptions of those theorems and corollaries. Remark. The method presented in this paper can be applied to the more general neutral di®erential equation of the form
Ã
¡
r(t)Ã x(t)
¢³
x(t) +
n X
£
pi (t)x ¿i (t)
i=1
¤´0
!0
+
m X
³ £
qj (t)fj x ¾j (t)
i=1
¤´
=0:
(E ¤ )
For the sake of simpler exposition, we stated and proved our result for (E). It is obvious how our results can be extended to (E¤ ) under suitable assumptions on the coe±cients and arguments involved. We ¯nish our paper with the following result, in which we relax the monotonicity condition imposed on f 0 (u). Theorem 2.10. Let (1) hold. If
Z
1
q(s)ds = 1 ;
(17)
then (E) is oscillatory. Proof. We assume that x(t) is an eventually positive solution of (E). Setting z(t) as in (4) and proceeding as in the proof of Theorem 2.1, we conclude that (7) holds. Set w(t) =
r(t)Ã(x(t))z 0 (t) : f (p¤ z[¾(t)])
Then w(t) > 0 and moreover w 0 (t) µ ¡ q(t) ¡
r(t)Ã(x(t))z 0 (t)f 0 (p¤ z[¾(t)])p¤ z 0 [¾(t)]¾ 0 (t) µ ¡ q(t) : f 2 (p¤ z[¾(t)])
J. D urina, D. Lackov / Central European Journal of Mathematics 2(1) (2004) 57{66
65
An integration of the last inequality yields w(t) µ w(t1 ) ¡
Z
t
q(s)ds :
t1
Then w(t) ! ¡ 1 as t ! 1 and this contradiction ¯nishes the proof. Example 2.11. We consider
³
r(t)e
¡jx(t)j
¡
x(t) + px(t ¡
¿)
¢0´0
³
2
£
+ q(t) ln 1 + x ¾(t)
¤´
£
¤
sgn x ¾(t) = 0 ;
which is a particular case to (E). By Theorem 2.10, this equation is oscillatory provided that (1) and (17) are satis¯ed.
Acknowledgements Research supported by VEGA-grant 1/0426/03.
References [1] D.D. Bainov and D.P. Mishev: Oscillation Theory for Neutral Di®erential Equations with Delay, Adam Hilger, Bristol, Philadelphia, New York, 1991. [2] M.Budincevic: Oscillation of second order neutral nonlinear di®erential equations , Novi Sad J. Math., Vol. 27, (1997), pp. 49{56. [3] T.A. Chanturija and I.T. Kiguradze: Asymptotic properties of solutions of nonautonomous ordinary di®erential equations, Nauka, Moscow, 1991. (Russian) [4] L.H. Erbe, Q. Kong, B.G. Zhang: Oscillation Theory for Functional Di®erential Equations, Adam Hilger, New York, Basel, Hong Kong, 1991. [5] L.H. Erbe and Q. Kong: Oscillation results for second order neutral di®erential equations , Funk. Ekvacioj, Vol. 35, (1992), pp. 545{557. [6] Q. Chuanxi and G. Ladas: Oscillations of higher order neutral di®erential equations with variable coe±cients , Math. Nachr., Vol. 150, (1991), pp. 15{24. [7] S.R. Grace and B.S. Lalli: Oscillation and asymptotic behavior of certain second order neutral di®erential equations , Radovi Mat., Vol. 5, (1989), pp. 121{126. [8] M.K. Grammatikopoulos, G. Ladas and A. Meimaridou: Oscillation and asymptotic behavior of higher order neutral equations with variable coe±cients , Chin. Ann. of Math, Vol. 9B, (1988), pp. 322{338. [9] K. Gopalsamy, B.S. Lalli and B.G. Zhang: Oscillation of odd order neutral di®erential equations , Czech. Math. J., Vol. 42, (1992), pp. 313{323. [10] I. Gy ri and G. Ladas: Theory of Delay Di®erential Equations with Applications, Clarendon Press, Oxford, (1991). [11] J.Hale: Theory of functional di®erential equations, Springer-Verlag, New York, 1977. [12] N. Parhi and P.K. Mohanty: Oscillation of neutral di®erential equations of higher order Bull. Inst. Math. Sinica, Vol. 24, (1996), pp. 139{150.
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[13] M. Ru i kov and E. p nikov : Comparison theorems for di®erential equations of neutral type , Fasc. Math, Vol. 128, (1998), pp. 141{148. [14] P. Wang, Y. Yu: Oscillation of second order order neutral equations with deviating argument Math. J. Toyama Univ., Vol. 21, (1998), pp. 55{66.
CEJM 2(1) (2004) 67{75
The representation dimension of domestic weakly symmetric algebras RafaÃl Bocian1¤, Thorsten Holm2y , Andrzej Skowro¶ nski1z 1
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toru¶ n, Poland 2 Otto-von-Guericke-UniversitÄat, Institut fÄur Algebra und Geometrie, Postfach 4120, 39016 Magdeburg, Germany
Received 16 October 2003; accepted 21 October 2003 Abstract: Auslander’s representation dimension measures how far a nite dimensional algebra is away from being of nite representation type. In [1], M. Auslander proved that a nite dimensional algebra A is of nite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are nite dimensional algebras of an arbitrarily large nite representation dimension. One of the exciting open problems is to show that all nite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed elds having simply connected Galois coverings. c Central European Science Journals. All rights reserved. ® Keywords: representation dimension, weakly symmetric algebra, domestic representation type, sel¯njective algebra of Euclidean type, derived equivalence MSC (2000): Primary 16D50, 16E10, 16G60; Secondary 16G10, 18E30
1
Introduction
z
y
¤
Throughout the paper we assume K to be an algebraically closed ¯eld and we consider ¯nite dimensional associative K-algebras. A fundamental distinction in the representation theory of algebras is given by the representation type. An algebra A is called of
[email protected] [email protected] [email protected]
68
R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75
¯nite representation type if there are only ¯nitely many indecomposable A-modules (up to isomorphism). The representation type can be further divided into tame and wild representation types. An algebra A is said to have tame representation type if for any ¯xed dimension d there are ¯nitely many A-K [X]-bimodules M 1 ; : : : ; M nd (free of ¯nite rank as right K[X ]-modules) such that, up to isomorphism, all but ¯nitely many indecomposable A-modules of dimension d are of the form Mi «K[X] K[X]=(X ¡ ¸) for some 1 µ i µ nd and ¸ 2 K. Roughly speaking, in the tame case all indecomposable modules of a ¯xed dimension occur in ¯nitely many one-parameter families, with ¯nitely many exceptions. Any algebra not of tame representation type is called wild. For a tame algebra A denote by ¹A (d) the smallest number of bimodules satisfying the above condition for the dimension d. In particular, A is of ¯nite representation type if and only if ¹A (d) = 0 for all d (by the validity of the second Brauer-Thrall conjecture). There is a well-known and important hierarchy of algebras of tame representation type: ° A has domestic representation type if there exists m ¶ 1 such that ¹A (d) µ m for all d; ° A is of polynomial growth if there exists a number m ¶ 1 such that ¹A (d) µ dm for all d; ° A is tame if ¹A (d) < 1 for all d. The representation type of an algebra is invariant under Morita equivalence, or more importantly, also preserved by stable equivalence [13], [14]. The present paper is concerned with sel¯njective algebras of domestic representation type. More precisely, we study a homological invariant of weakly symmetric algebras of b By de¯nition, Euclidean type. For an algebra B, we denote its repetitive algebra by B. b b is the a sel¯njective algebra of Euclidean type is an algebra of the form B=G where B repetitive algebra of a tilted algebra B of Euclidean type ¢ 2 f e m ; e n ; e 6 ; e 7 ; e 8 g, and b It has been proved in G is an admissible in¯nite cyclic group of automorphisms of B. [18] that the class of all sel¯njective algebras of Euclidean type coincides with the class of all representation-in¯nite domestic sel¯njective algebras which admit simply connected Galois coverings. Recently, the structure of all weakly symmetric algebras of Euclidean type has been described in [3], [4], [15]. Moreover, a derived equivalence and stable equivalence classi¯cation of all weakly symmetric algebras of Euclidean type was given in [2]. Recall that a sel¯njective algebra is called weakly symmetric if the socle and the top of any indecomposable projective module are isomorphic. The class of weakly symmetric algebras contains the class of symmetric algebras for which there exists a nondegenerate symmetric associative bilinear form. Further, the structure of all sel¯njective algebras which are socle equivalent to sel¯njective algebras of Euclidean type has also been described recently in [5]. Recall that two sel¯njective algebras A and ¤ are called socle equivalent if the factor algebras A= soc A and ¤= soc ¤ by the socles are isomorphic. In this paper we consider the representation dimension, a homological invariant de¯ned by M. Auslander around 1970 as a possible way of measuring the distance of an algebra to having ¯nite representation type. This is achieved by studying the global dimensions of endomorphism rings of modules (which are not too small). More precisely,
R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75 69
for any algebra A the representation dimension is de¯ned as repdim (A) := inffgldim(End A (N )) j N generator-cogeneratorg : Here, a ¯nitely generated module N is called a generator-cogenerator for A if all projective indecomposable and all injective indecomposable A-modules occur as direct summands of N . M. Auslander proved in [1] the fundamental result that repdim (A) µ 2 if and only if A is of ¯nite representation type. For many years, the representation dimension remained a mysterious invariant, and only recently have new advances been made in this area. For instance, O. Iyama showed that the representation dimension is always ¯nite [12]. By de¯nition, the representation dimension is invariant under Morita equivalence. Recently, C. Xi could prove that the representation dimension is even invariant under stable equivalence of Morita type [20]. In particular, for sel¯njective algebras the representation dimension is invariant under derived equivalences [16]. Note however, that for arbitrary algebras the representation dimension is not a derived invariant. Algebras of small representation dimension are of particular interest due to a result of K. Igusa and G. Todorov [11]: if repdim (A) µ 3 then the ¯nitistic dimension of A is ¯nite, that is, the notorious open ¯nitistic dimension conjecture holds for A. For background on the ¯nitistic dimension conjecture we refer to [21]. Despite these exciting new developments, the precise value of the representation dimension can still only be determined for very few classes of algebras. To the best of the authors’ knowledge, the following short list comprises all classes of algebras known up to now to have representation dimension at most 3. ° algebras of ¯nite representation type [1]; ° stably hereditary algebras [1], [20]; ° algebras A with rad(A)m+1 = 0 and A= rad(A)m of ¯nite representation type, in particular algebras with radical square zero [1]; ° Schur algebras of tame representation type [9]; ° local algebras of quaternion type [10]; ° special biserial algebras [6]. One of the fundamental open problems about representation dimensions is a good answer to the following question: which algebras have representation dimension at most 3? Very recently, R. Rouquier announced a proof that an exterior algebra of an n-dimensional vector space has representation dimension n + 1, thus providing the ¯rst examples of algebras with representation dimension at least 4. According to Auslander’s philosophy the representation dimension should measure the distance to ¯nite representation type. So one would expect that algebras of tame representation type should have small representation dimension. The present paper contributes to the long-term project of determining the representation dimension for all algebras of tame representation type. Our main result gives an answer for weakly symmetric algebras of domestic representation type.
70
R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75
Theorem. Let A be a sel¯njective algebra which is socle equivalent to a weakly symmetric algebra of Euclidean type. Then repdim (A) = 3. According to a conjecture of the third named author, every basic connected representation-in¯nite domestic sel¯njective algebra is socle equivalent to a sel¯njective algebra of Euclidean type. Hence, our main result probably asserts that the representation dimension of any domestic weakly symmetric algebra is at most 3.
2
Weakly symmetric algebras of Euclidean type
In this section we brie°y review the results of [2] on the derived equivalence classi¯cation of weakly symmetric algebras of Euclidean type. This will be an important reduction step in the proof of our main result, since for sel¯njective algebras the representation dimension is invariant under derived equivalence [20]. In order to state the derived equivalence classi¯cation we shall need the following families of basic algebras, de¯ned by quivers with relations.
A (¸)
®
¯
¸ 2 Kn f0g
® 2 = 0, ¯ 2 = 0, ®¯ = ¸¯®;
¯4
¯3
®2
¯2
A (p; q)
¯1
16p6q p+q >3
¯q ¯q¡3
¯q¡2
¯q¡1
®3
®4
®1 ®p ®p¡1
®p¡2
®p¡3
®1 ®2 : : : ®p ¯1 ¯2 : : : ¯q = ¯1 ¯2 : : : ¯q ®1 ®2 : : : ®p; ®p®1 = 0; ¯q ¯1 = 0; ®i ®i+1 : : : ®p ¯1 : : : ¯q ®1 : : : ®i¡1 ®i = 0; 2 6 i 6 p; ¯j ¯j+1 : : : ¯q ®1 : : : ®p ¯1 : : : ¯j¡1 ¯j = 0; 2 6 j 6 q;
R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75 71
¯4
¯3
¯2
¤ (n)
¯1
n>2
¯n
® ¯n¡3
¯n¡2
¯n¡1
®2 = (¯1 ¯2 : : : ¯n )2 ; ®¯1 = 0; ¯n ® = 0; ¯j ¯j+1 : : : ¯n ¯1 : : : ¯n ¯1 : : : ¯j¡1 ¯j = 0; 2 6 j 6 n;
¯4
¯3
¯2
®1
¡ (n)
¯1
n>1
¯n ¯n¡3
¯n¡2
®2 °2
°1
¯n¡1 2
®1 ®2 = (¯1 ¯2 : : : ¯n ) = °1 °2 ; ®2 ¯1 = 0; °2 ¯1 = 0; ¯n ®1 = 0; ¯n °1 = 0; ®2 °1 = 0; °2 ®1 = 0; ¯j ¯j+1 : : : ¯n ¯1 : : : ¯n ¯1 : : : ¯j¡1 ¯j = 0; 2 6 j 6 n: Then the following main result of [2] gives a complete derived equivalence classi¯cation of weakly symmetric algebras of Euclidean type. Theorem 2.1 ([2]). (1) For an algebra A the following statements are equivalent: (i) A is weakly symmetric of Euclidean type and its Cartan matrix CA is singular. (ii) A is derived equivalent to the trivial extension T (C) of a canonical algebra C of Euclidean type. (2) For a sel¯njective algebra A of Euclidean type the following statements are equivalent: (i) A is weakly symmetric and the Cartan matrix CA is nonsingular. (ii) A is derived equivalent to exactly one of the algebras A(¸), A(p; q), ¤(n) or ¡(n).
72
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R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75
Proof of the Theorem
In this section we are going to prove that all sel¯njective algebras socle equivalent to a weakly symmetric algebra of Euclidean type have representation dimension 3. Several of these algebras are special biserial. The representation dimension of special biserial algebras was determined in [6]. We shall also need the more general main result of [6] which we therefore state separately for further reference. First the following preliminary result is often useful when computing representation dimensions. Proposition 3.1 ([6], 1.2). Let A be a basic K-algebra, and let P be an indecomposable projective-injective A-module. Consider the factor algebra B := A= soc(P ) modulo the socle of P . If repdim (B) µ 3, then also repdim (A) µ 3. An algebra monomorphism f : A ! B is called a radical embedding if it maps the Jacobson radical of A onto the Jacobson radical of B. The main result of [6] is the following. Theorem 3.2 ([6], 1.1). Let A be an algebra such that there exists a radical embedding f : A ! B where B is of ¯nite representation type. Then repdim (A) µ 3. For special biserial algebras, explicit constructions of radical embeddings into uniserial (hence representation-¯nite) algebras are given in [6] (Sections 3 and 4). Corollary 3.3 ([6], 1.3). All special biserial algebras have representation dimension at most 3. Let A be a sel¯njective algebra which is socle equivalent to a weakly symmetric algebra ¤ of Euclidean type. Assume ¯rst that A is not isomorphic to a weakly symmetric algebra of Euclidean type. Then it follows from the main theorem of [5] that K is of characteristic 2, A is isomorphic to a one-parametric biserial symmetric algebra 0 (T ), for a Brauer graph T with exactly one loop, which is moreover socle equivalent to a one-parametric symmetric special biserial algebra ¤0 (T ) of Euclidean type. In particular, A= soc A ¹= ¤0 (T ) = soc ¤0 (T ) is special biserial. Hence, applying Proposition 3.1, Corollary 3.3 and [1], we conclude that repdim (A) = 3. Therefore, we may assume that A is a weakly symmetric algebra of Euclidean type. The structure theory of weakly symmetric algebras of Euclidean type depends on whether the Cartan matrix is singular or nonsingular (Theorem 2.1). Singular Cartan matrix case. Let A be a weakly symmetric algebra of Euclidean type with a singular Cartan matrix. By Theorem 2.1, A is derived equivalent to the trivial extension T (C) of a canonical algebra C of Euclidean type. We shall prove that repdim (T (C)) = 3. This will imply that A has representation dimension 3, because the representation dimension is a derived invariant for sel¯njective algebras [20]. If C is of Euclidean type e m then T (C) is a representation-in¯nite special biserial algebra
R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75 73
(see [3]), n and our claim o follows from Corollary 3.3. Assume that C is of Euclidean type, ¢ 2 e n ; e 6 ; e 7 ; e 8 . Then invoking [8] (see also [7], [17]) we conclude that C is a
tilted algebra of the form C = End H (T ), where H is a hereditary algebra of type ¢ with rad (H)2 = 0 and T is a preprojective tilting H-module. In particular, C is derived equivalent to H [7]. Then, according to J. Rickard’s result [16], T (C) is derived equivalent to T (H ). But T (H) is a symmetric algebra with (rad T (H ))3 = 0, and so T (H) = soc T (H) is a radical square zero algebra. Hence, applying Proposition 3.1 and [1], we conclude that repdim (T (H)) = 3. Nonsingular Cartan matrix case. Let A be a weakly symmetric algebra of Euclidean type with a nonsingular Cartan matrix. By Theorem 2.1, A is derived equivalent to one of the algebras A(¸), A(p; q), ¤(n) or ¡(n) de¯ned above. According to C. Xi’s theorem [20], the representation dimension is a derived invariant for sel¯njective algebras. Hence it su±ces to prove our theorem for each of the above normal forms A(¸), A(p; q), ¤(n) and ¡(n). The algebras A(¸), A(p; q) and ¤(n) are special biserial, so we can deduce from Corollary 3.3 that they have representation dimension 3. Note that the algebras ¡(n) are not special biserial. However, in order to show that repdim (¡(n)) = 3 we shall use Proposition 3.1 and Theorem 3.2, that is, we shall construct a radical embedding of some factor algebra of ¡(n) into an algebra of ¯nite representation type. The vertex of the quiver of ¡(n) with three incoming and outgoing arrows will be denoted by 1, with corresponding projective indecomposable ¡(n)-module P1 . We factor ~ out the socle of P1 and get the algebra ¡(n) := ¡(n)= soc P1 . This factor algebra is de¯ned by the same quiver, but with additional relations ®1 ®2 = 0, °1 °2 = 0 and (¯1 : : : ¯n )2 = 0. ~ By Proposition 3.1 it su±ces to show that ¡(n) has representation dimension 3. In order ~ to construct a radical embedding of ¡(n) into an algebra of ¯nite representation type we use the technique of splitting datum as introduced in [6]. We refer to [6] (Section 3) for ~ notations and details. At the vertex 1 of the quiver of ¡(n) we can distribute the incoming and outgoing edges as follows: E1 [ E2 := f¯n g [ f®2 ; °2 g and S1 [ S2 := f¯1 g [ f®1 ; °1 g. The conditions for a splitting datum are satis¯ed: all products in Ei Sj are zero for i 6= j ~ and all relations of ¡(n) are monomial. (Note that for the latter second condition we ~ actually needed to pass to the factor algebra ¡(n); the above construction is not a splitting datum for ¡(n).) Any splitting datum gives rise to a radical embedding into an algebra whose quiver is obtained by splitting the vertex under consideration. In our case, we get a radical ~ ~ sp given by the disjoint union of a cyclic quiver embedding of ¡(n) into the algebra ¡(n) with edges ¯1 ; : : : ; ¯n and relations (¯1 : : : ¯n )2 = 0, ¯j ¯j+1 : : : ¯n ¯1 : : : ¯n ¯1 : : : ¯j¡1 ¯j = 0 for 2 µ j µ n, and the quiver ¬
2
®
1
¬
1
®
2
~ sp with relations ®1 ®2 = 0, °1 °2 = 0, ®2 °1 = 0, °2 ®1 = 0. Note that this new algebra ¡(n) is special biserial and of ¯nite representation type [19].
74
R. Bocian, T. Holm, A. Skowro´nski / Central European Journal of Mathematics 2(1) (2004) 67{75
~ Then it follows from Theorem 3.2 that ¡(n) has representation dimension 3, as desired. This completes the proof of the Theorem.
Acknowledgments The authors acknowledge support from the Polish Scienti¯c Grant KBN No. 5 P03A 008 21.
References [1] M. Auslander: Representation dimension of artin algebras, Queen Mary College, Mathematics Notes, University of London, 1971, 1{179. Also in: Selected works of Maurice Auslander (eds. I. Reiten, S. Smal¿ and Â. Solberg) Part I, Amer. Math. Soc., 1999, pp. 505{574. [2] R. Bocian, T. Holm and A. Skowro¶nski: "Derived equivalence classi¯cation of weakly symmetric algebras of Euclidean type", Preprint, (2003). [3] R. Bocian and A. Skowro¶ nski: "Symmetric special biserial algebras of Euclidean type", Colloq. Math., Vol. 96, (2003), pp. 121{148. [4] R. Bocian and A. Skowro¶nski: "Weakly symmetric algebras of Euclidean type", Preprint, (2003). [5] R. Bocian and A. Skowro¶nski: "Socle deformations of sel¯njective algebras of Euclidean type", Preprint, (2003). [6] K. Erdmann, T. Holm, O. Iyama and J. SchrÄoer: "Radical embeddings and representation dimension", Advances Math., to appear. (arXiv:math.RT/0210362) [7] D. Happel: Triangulated categories in the representation theory of ¯nite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, 1988. [8] D. Happel and D. Vossieck: "Minimal algebras of in¯nite representation type with preprojective component", Manuscr. Math., Vol. 42, (1983), pp. 221{243. [9] T. Holm: "The representation dimension of Schur algebras: the tame case", Preprint, (2003). [10] T. Holm: "Representation dimension of some tame blocks of ¯nite groups", Algebra Colloquium, Vol. 10:3, (2003), pp. 275{284. [11] K. Igusa and G. Todorov: "On the ¯nitistic global dimension conjecture for artin algebras", Preprint, (2002). [12] O. Iyama: "Finiteness of representation dimension", Proc. Amer. Math. Soc., Vol. 131, (2003), pp. 1011{1014. [13] H. Krause: "Stable equivalence preserves representation type", Comment. Math. Helv., Vol. 72, (1997), pp. 266{284. [14] H. Krause and G. Zwara: "Stable equivalence and generic modules", Bull. London Math. Soc., Vol. 32, (2000), pp. 615-618.
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[15] H. Lenzing and A. Skowro¶nski: "On sel¯njective algebras of Euclidean type", Colloq. Math., Vol. 79, (1999), pp. 71{76. [16] J. Rickard: "Derived categories and stable equivalence", J. Pure Appl. Algebra, Vol. 61, (1989), pp. 303{317. [17] C. M. Ringel: Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer Verlag, Berlin, 1984. [18] A. Skowro¶ nski: "Sel¯njective algebras of polynomial growth", Math. Ann., Vol. 285, (1989), pp. 177{199. [19] A. Skowro¶ nski and J. WaschbÄusch: "Representation-¯nite biserial algebras", J. Reine Angew. Math., Vol. 345, (1983), pp. 172{181. [20] C. Xi: "Representation dimension and quasi-hereditary algebras", Advances Math., Vol. 168, (2002), pp. 193{212. [21] B. Zimmermann Huisgen, The ¯nitistic dimension conjectures { a tale of 3.5 decades. In: Facchini, Alberto (ed.) et al., Abelian groups and modules. Proceedings of the Padova conference, Padova, Italy, June 23-July 1, 1994. Kluwer (1995).
CEJM 2(1) (2004) 76{86
On quasi-solution to infeasible linear complementarity problem obtained by Lemke’s method L.D. Popov¤ Institute of Mathematics and Mechanics, 16 S. Kovalevskaja, 620219 Ekaterinburg, Russia
Received 5 May 2003; revised 18 November 2003 Abstract: For a linear complementarity problem with inconsistent system of constraints a notion of quasi-solution of Tschebyshev type is introduced. It’s shown that this solution can be obtained automatically by Lemke’s method if the constraint matrix of the original problem is copositive plus or belongs to the intersection of matrix classes P0 and Q0 . ® c Central European Science Journals. All rights reserved. Keywords: infeasible linear complementarity approximation MSC (2000): 65K05, 90C20, 90C49
1
problem,
Lemke’s
method,
Tschebyshev
Introduction
Let q 2 Rn be a vector and M be an (n £ n)-matrix. Consider the linear complementarity problem: ¯nd vectors w and z which satisfy the conditions w = M z + q; w ¶ 0; z ¶ 0;
(1)
w T z = 0:
(2)
Since all variables above are non-negative, one can replace (2) by wi zi = 0
¤
Let denote this problem as LCP(q; M ): E-mail:
[email protected]
for all i = 1; : : : ; n:
(3)
L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
77
The linear complementarity problem belongs to the fundamental problems and is closely connected with a theory of linear and quadratic programming and bimatrix (twoperson, nonzero-sum) games. It has been intensely studied since the 1960s. Many alternative formulations have been investigated, and e±cient algorithms have been proposed [1{5]. Yet some questions about the problems which have no solutions at all or, more widely, ill-posed (improper) optimization instances are under investigation [6{8]. One reason for the presence of unsolvable problems is the accidental corruption of the problem data. Another place where, e.g., infeasible quadratic programs can arise is in sequential quadratic programming algorithms [9{10] etc. Usually we do not know a priori whether a concrete problem is proper or not. Therefore at the 1-st stage of numerical analysis of a problem we try to solve it by means of some standard tools. If such an attempt fails then we conclude that our problem is improper and its data revision is necessary. For a linear complementarity problem it means that we have to determine (preferably small) adjustments ¢M; ¢q such that a corrected problem w = (M + ¢M )z + q + ¢q; w ¶ 0; z ¶ 0;
(4)
w T z = 0:
(5)
will be proper. This is the 2-nd stage of our analysis which may require some additional numerical techniques. When it is done we apply our standard tools again, but to the corrected problem. The solution we will obtain after this 3-rd stage may play a role of a reasonable "quasi-solution" of initial improper problem. Can one combine these three stages in an unique algorithm when one deals with a linear complementarity problem? Yes, and this unique algorithm is the well-known Lemke’s method. We intend to show that under some restrictions, applying this method to a linear complementarity problem automatically gives us an ordinary solution when this problem is proper (this fact is well-known) and its quasi-solution otherwise (this fact is new). It is true (at least) for the problems with co-positive plus matrices or matrices T from Q0 P0 ; where P0 is a class of matrices with non-negative principal minors and Q0 is a class of matrices for which the feasible linear complementarity problem always has a solution. Note, that co-positive plus matrices also belong to Q0 . But what sort of quasi-solution can Lemke’s method provide when M 2 Q0 and an initial problem is improper? It can provide a solution of (4), (5) with ¢M = 0 and ¢q 2 Arg min fk»k®;1 : K(») 6= ;g; where K(») = f(w; z) : w = M z + q + »; w ¶ 0; z ¶ 0 g; k»k®;1 = max
1·i·n
and ®1 ; : : : ; ®n are any given positive numbers.
µ
j »i j ®i
¶
(6)
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L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
It is obviously, that the problem (6) is closely connected with a linear program % = minf z0 : w = M z + q + p z0 ; w ¶ 0; z ¶ 0; z0 ¶ 0 g; where
p = (®1 ; : : : ; ®n )T :
(7) (8)
Indeed (see, e.g., [8]), any solution to (6) has the property k¢qk®;1 = %, and having % one can obtain at least one of the optimal vectors of (6), namely ¢q := % p: Using this connection we intend to prove the facts declared above. In particular, we shall use well-known assertion that triplet » = (w; ¹ z¹; z¹0 ; ) is a solution of the program (7), (8) i® w¹ = q + p z¹0 + M z¹ ¶ 0; ¼ T M µ 0; ¼ T w¹ = 0;
¼ T p µ 1;
z¹ ¶ 0;
z¹0 ¶ 0;
¼ = (¼1 ; : : : ; ¼n ) ¶ 0;
¼ T M z¹ = 0;
(¼ T p ¡
1)z0 = 0;
(9) (10) (11)
where ¼ = (¼1 ; : : : ; ¼n ) is a vector of dual prices. It is easy to see that there exists some analogy between our results and well-known properties of the simplex-method in linear programming. Namely, if initial linear program is infeasible then one can form minimal adjustments of right-hand-side components of its constraints using positive values of arti¯cial variables obtained at the 1-st phase of the simplex-algorithm. But this analogy is not comprehensive, and proofs presented below have a rather di®erent nature.
2
Brief Lemke’s method description
Let’s recall some necessary aspects of the algorithm in question. To begin let us include an arti¯cial variable z0 in the initial instance as follows w = M z + q + p z0 ;
(12)
where a column p = (p1 ; : : : ; pn )T > 0: According to standard terminology, any solution of this system is called almost complementary if zi wi = 0 for i = 1; : : : ; n and complementary if, in addition, z0 = 0: Such a solution is called basic if the columns (of extended matrix (I; q ¡ M )) corresponding to its nonzero components are linear independent (i.e., form a basis). A basic solution is called non-degenerate if the number of its nonzero components is equal to n: Lemke’s algorithm moves from one non-negative basic almost complementary solution to another in a special manner (using Lemke’s rule), aiming to meet a complementary one. At each iteration some variable enters a basis (it’s called an increasing variable) and some variable leaves it (it’s called a blocking variable). For i = 1; : : : ; n the corresponding variables zi and wi are called complementary and each is the complement of the other. According to Lemke’s rule an increasing variable is a complement of a variable leaving the basis at previous iteration. The leaving (blocking)
L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
79
variable is determined by standard primal simplex-method rule to keep all variables nonnegative. To simplify our discourses we assume that all feasible (i.e. non-negative) basic solutions of (12) are non-degenerate. Under this assumption (assumption of nondegeneracy) they determine (or correspond to) di®erent extreme points of a polyhedral set P = f(w; z; z0 ) : w = M z + q + p z0 ; w ¶ 0; z ¶ 0; z0 ¶ 0g: Therefore, each iteration of Lemke’s method corresponds to motion from some extreme point of this set to another along its edge, all points of which are almost complementary. If the edge is bounded, then an adjusted extreme point is reached which is either complementary or almost complementary. The process terminates if (i) the edge is unbounded (an alternative ray), (ii) an adjusted point coincides with a previously generated one, (iii) an adjusted point is a complementary extreme point. Note that, due to to Lemke’s rule, along an almost complementary path, the only almost complementary basic feasible solution which can reoccur is the initial one. The initial extreme point is formed as follows: z = 0;
qs z0 = ¡ = min i: qi <0 ps
µ
qi ¡ pi
¶
;
w = q + pz0 :
Here ws = 0; i.e., we have a feasible basic almost complementary solution. Moreover, this solution is the origin of a ray of P (its points are generated when z0 increases to +1). Therefore, terminate case (iii) is impossible. As for the ¯rst increasing variable, it is zs : Note that for copositive plus matrices and matrices from class Q0 with non-negative principal minors the case (i) means incompatibility of initial constraint system.
3
Copositive plus matrices
Matrix M is called copositive plus if it satisfy the conditions xT M x ¶ 0 (M + M T )x = 0
for all x ¶ 0;
for all x ¶ 0; xT M x = 0:
(13) (14)
The class of such matrices is su±ciently large and includes 1) all strictly copositive matrices, i.e., those for which xT M x > 0 for all 0 6= x ¶ 0; 2) all positive semide¯nite matrices, i.e., those for which xT M x ¶ 0 for all x: Positive matrices are obviously strictly copositive. But some examples show that matrices of the form
0
1
B 0 AC A ; where A > 0; B > 0;
M =@
BT 0
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L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
may violate relations (14). Furthermore, it is possible to build a block-diagonal matrix
0
1
B M1 0 C A
M =@
0 M2
satisfying (13), (14), out of smaller matrices M1 and M2 which also satisfy (13), (14). Moreover, if S is any skew-symmetric matrix (of suitable order) then M + S is copositive plus any time as M is so. Consequently, block matrix
0
1
T B M1 ¡ A C A
M =@
A
M2
satis¯es (13), (14) if and only if M 1 and M 2 do too. The class of all copositive plus matrices possesses an important property: if a constraint matrix of an LCP(q; M ) belongs to this class then termination of Lemke’s method on an alternative ray proves inconsistency of its constraint system [2]. The next proposition elaborates this well-known fact. Theorem 3.1. Let matrix M be copositive plus and Lemke’s method with (8) terminate on an alternative ray. Then the constraint system of the initial linear complementarity problem LCP(q; M ) is inconsistent and the origin of the alternative ray (w; z; z0 ) has the property: (w; z) is a solution of the corrected problem LCP(q¹; M ); where q¹ = q + p z0 ; z0 = %; % is the optimal value of (7): Proof. Termination in a ray means that a feasible basic solution (w; z; z0 ) and a ray direction vector (w; ¹ z¹; z¹0 ) are available such that w¹ = p¹ z0 + M z¹; (w; ¹ z¹0 ; z¹) ¶ 0;
(15)
w + ¸w¹ = q + p (z0 + ¸¹ z0 ) + M (z + ¸¹ z );
(16)
(wi + ¸w¹i ) (zi + ¸¹ zi ) = 0; i = 1; : : : ; n:
(17)
and for all ¸ ¶ 0 it holds
These relations evidently imply that the pair (w; z) solves the corrected linear complementarity problem di®ering from initial one only by the vector q¹ = q + p z0 : Next we must show that the triplet (w; z; z0 ) is a solution of (7), i.e., z0 = % > 0: To do that let us investigate the conditions (13){(17) in detail. First, from (w; ¹ z¹; z¹0 ) 6= 0 it follows that z¹ > 0: Indeed, if, on the contrary, z¹ = 0 then z¹0 > 0 and w¹ > 0 too, which implies (see (17)) z + ¸¹ z = z = 0; i.e., an alternative ray coincides with the initial one.
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81
Secondly, all points of the obtained alternative ray are almost complementary, i.e., zi wi = zi w¹i = z¹i wi = z¹i w¹i = 0; i = 1; : : : ; n:
(18)
0 = z¹T w¹ = z¹T p z¹0 + z¹T M z¹:
(19)
By (15), As matrix M is copositive plus and z¹ ¶ 0; both terms in the right side of (19) are non-negative and, consequently, are equal to zero. The scalar z¹0 = 0 because of z¹T p > 0: Thirdly, the equality z¹T M z¹ = 0 means, by assumption, that M z¹ + M T z¹ = 0
M z¹ = ¡ M T z¹:
or
But, by (15), z¹0 = 0 implies w¹ = M z¹ ¶ 0; whence M T z¹ µ 0 or, what is the same thing, z¹T M µ 0: At last, by (18), 0 = z T w¹ = z T M z¹ = z T (¡ M T z¹) = z¹T M z: Hence, z¹ 6= 0;
z¹T w = 0;
z¹T M µ 0;
z¹T M z = 0:
(20)
It makes it possible to determine dual prices ¼=
µ
1 T z¹ p
¶
z¹
which are non-negative and, by (20), together with triplet (w; z; z0 ) satisfy to optimality conditions (9){(11) for (7). The proof is complete. Note that in general, i.e., when matrix M is arbitrary, the origin (w; z; z0 ) of the terminal alternative ray in Lemke’s method provides a solution (w; z) of the corrected linear complementarity problem LCP(q + p z0 ; M ): Nevertheless, the initial problem may be solvable as well as unsolvable. Example [3] Let apply Lemke’s algorithm to LCP(q; M ) with
0
1
0
1
0 1
B0 1 C B¡ 1 C B1 C A; q = @ A; p = @ A:
M =@
00
0
1
Just the ¯rst iteration terminates on an alternative ray with z0 = 1 > 0. However,
0 1
give a solution to this problem.
0 1
B0 C B0 C w = @ A; z = @ A 0 1
82
4
L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
Matrices with non-negative principal minors
Let us consider next the matrix class P0 consisting of the matrices whose principal minors are non-negative. This class possesses many signi¯cant properties and plays a prominent role in linear complementarity theory. However, it’s too wide for our purposes, as one can see from the example above exploiting the matrix from this class. That is why we must restrict ourselves to a more narrow matrix set, e.g., by intersection P0 \Q0 . It should be noted at this point that P0 \Q0 is not contained in any of the better known matrix classes that arise within linear complementarity theory. Complicated but constructive characterizations of such matrices are given in [4]. To describe it let us introduce some notations. Let I and J be an arbitrary subsets of the index set f1; : : : ; ng. Then MIJ is a matrix obtained from (n £ n)-matrix M by retaining the rows indexed by I and the columns indexed by J: If I = f1; : : : ; ng or J = f1; : : : ; ng; we write M¢ J or MI ¢ respectively. If I or J is a singleton, say i; we simply write i: At last, if I » f1; : : : ; ng is a set of indexes, then I¹ = f1; : : : ; ng n I is the complementary index set. With this notations, a matrix M is said to have T-property if for ever nonempty index set I; the existence of a solution u to the system MII u µ 0; M II u > 0; ¹ u ¶ 0; implies there exists a nonzero vector v ¶ 0 such that v T MI¢ µ 0;
v T MII u = 0:
Let M be a constraint matrix in (1). At each iteration of Lemke’s method this matrix ¹ : changes to principal (pivotal) transform M ¹ J J = M ¡1 ; M JJ ¡1 ¹ JJ M ¹ = MJJ ¹ MJJ ;
¹ J J¹ = ¡ M ¡1 M J J¹; M JJ
¹ J¹J¹ = MJ¹J¹ ¡ M
¡1 MJJ ¹ MJ J MJ J¹;
where J is a nonempty index set of z-variables being basic. The next assertion gives the characterization of the matrices under investigation. Theorem 4.1. [4] If M 2 Q 0 \P0 then M and each of its principal (pivotal) transforms has T-property. Inversely, if M is P0 -matrix, and each of its principal transforms has T-property, then M 2 Q0 . This characterization was used in [4] to show that for any q and M 2 Q0 \P0 Lemke’s method either solves an LCP(q; M ) or proves its infeasibility (when terminates on a ray). And again, we show that in the last case some quasi-solution of (7)-type is generated. Theorem 4.2. Let M 2 Q0 \P0 and Lemke’s method with (8) terminate on an alternative ray. Then the constraint system of the initial linear complementarity problem
L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
83
LCP(q; M ) is inconsistent and the origin of the alternative ray (w; z; z0 ) has the property: (w; z) is a solution of the corrected problem LCP(q¹; M ); where q¹ = q + p z0 ; z0 = %; % is the optimal value of (7): Proof. T -property of M allows us to verify the conditions (9){(11) for the basic almost complementary solution in Lemke’s method. Nevertheless, it is easier to use well-known fact: a value of an arti¯cial variable does not increase along Lemke’s path for such matrices [3]. Let us apply Lemke’s method to initial problem assuming it is infeasible. We shall refer to this calculation as process I. Process I must terminate on a ray with arti¯cial variables z0 ¶ % in a basis. Inequality z0 < % is impossible, since, by de¯nition, the system w = M z + q + p z0 ; w ¶ 0; z ¶ 0; z0 ¶ 0; can not be consistent for z0 < %. Along with the initial problem (1), (2), let consider the corrected one: w = M z + q¹; w ¶ 0; z ¶ 0;
(21)
w T z = 0;
(22)
where q¹ = q+p z¹0 ; z¹0 is a solution of (7), i.e., z¹0 = %. As the assumption of nondegeneracy of the initial problem is valid, at least feasible almost complementary basic solutions of the corrected problem are non-degenerate too. Since conditions (21) are compatible now and M 2 P0 \Q0 ; Lemke’s algorithm applying to (21),(22), will pass through extreme points of "shifted" polyhedral set P¹ = f(w; z; z0 ) : w = M z + q + p z¹0 +p z0 ; w ¶ 0; z ¶ 0; z0 ¶ 0g
| {z }
(23)
q¹
until it terminates on a complementary solution. We shall refer to this calculation process as process II. Let show that both processes produce the same sequence of the bases. To start, consider the initial basis for process I. It consists of all w-variables, except one, say ws ; where qs qi ( µ := ) = min ; I := fi : qi < 0g: i2I pi ps Instead ws , an arti¯cial variable z0 enters the basis with a value z0 = ¡ µ ¶ z¹0 (= %) (see Section 2). If ¡ µ = z¹0 , then the proof is complete (as z0 can not increase). Otherwise we try to take the same basis as the initial one for process II. We can do that because the minima q¹i µ¹ := min ; i2I0 pi
I0 := fi : q¹i < 0g;
is attained at the same indexes i as µ: It is so, because: (i) one has q¹i =pi = qi =pi + z¹0 ; i.e., all compared fractions of processes I and II di®er from each other by a constant; (ii)
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L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
index set I0 keeps within I; (iii) qi =pi = µ implies q¹i = qi + pi z¹0 < qi ¡ pi µ = 0 as ¡ µ > z¹0 , i.e., I0 contains all indexes at which µ is attained. At the next iterations of both processes, the variables entering the basis are uniquely determined as the complement to the variables leaving it at the previous iterations. In its turn, the variable leaving the basis is determined by minimal ratio test which involves values of basic variables and positive elements of the pivotal transform column corresponding to entering variables. For both processes all the quantities coincide as far as the basis contains an arti¯cial variable. Only the values of z0 di®er from each other by the constant z¹0 . But when an arti¯cial variable leaves the basis of the corrected problem, process II terminates. Therefore, two sequences of the bases coincide till the corrected problem will be solved. To ¯nish, consider the last pivotal tableau of process II z10 : : : zs0 : : : zn0 z0 q00
m00s
w10 q10 ... ...
m01s ...
wn0 qn0
m0ns
where zs0 is the increasing variable which forces an arti¯cial variables z0 to leave the basis, i.e., 0 q00 ^ := min mis ; I 0 := fi : m0 < 0g: = µ is i2I 0 m00s pi But at this stage the pivotal tableau and the increasing variable of process I are the same z10 : : : zs0 : : : zn0 z0 q000
m00s
w10 .. .
q10 .. .
m01s .. .
wn0 qn0
m0ns
except for a value of q000 = q00 + z¹0 . That is why a new value of an arti¯cial variable in process I satis¯es z0new = q000 + m00s zsnew µ q00 + z¹0 + m00s µ^ = z¹0 (indeed, z0new = z¹0 ). Though several iterations may occur before this process terminates too, an arti¯cial variable can not increase. Consequently, one has z0 = z¹0 = % at the end of the calculation. The proof is complete.
L.D. Popov / Central European Journal of Mathematics 2(1) (2004) 76{86
5
85
Conclusions
In the article there are presented a new notion of quasi-solution to infeasible linear complementarity problem and some conditions which guarantee this solution be obtained by the classical Lemke’s algorithm. The conditions are formulated in terms of some matrix classes, in particular, classes Q0 , P0 and class of all copositive plus matrices. Among close works, in [11], using a multiple-objective framework, feasible but unsolvable linear complementarity problems are analyzed and transformed into well-posed problems by means of matrix correction, and in [12] some iterative schemes of Tikchonov’s type are applied to improper generalized monotone equations and ¯nite-dimension variational inequalities.
Acknowledgment Author would like to thank Prof.M.M.Kostreva and anonymous reviewers for helpful suggestions that have improved the presentation of these results. The work was ¯nancially supported by RFBR, grant 01-01-00563.
References [1] C.E. Lemke, J.T. Howson: "Equilibrium points of bimatrix games", SIAM Review., Vol. 12, (1964), pp. 45{78. [2] R.W. Cottle, G.B. Dantzig: "Complementarity pivote theory of mathematical programming", Linear Algebra and its Applications, Vol. 1, (1968), pp. 103{125. [3] B.C. Eaves: "The linear complementarity problem", Management Science., Vol. 17, (1971), pp. 612{634. [4] M. Aganagic, R.W. Cottle: "A constructive characterization of Q0 -matrices with nonnegative principal minors", Math. Programming., Vol. 37, (1987), pp. 223{231. [5] R.W. Cottle, J.S. Pang, R.E. Stone: Academic Press, Boston, 1992.
The Linear Complementarity Problem,
[6] O.L. Managasarian: "The ill-posed linear complementarity problem", In: M.C. Ferris, J.S. Pang: Complementarity and variational problems: State of the Art, SIAM Publications, Philadelphia, PA, 1997, pp. 226-233. [7] I.I. Eremin: Theory of Linear Optimization, Inverse and Ill-Posed Problems Series. VSP. Utrecht, Boston, Koln, Tokyo, 2002. [8] I.I. Eremin, Vl.D. Mazurov, N.N. Astaf ’ev: Improper Problems of Linear and Convex Programming, Nauka, Moscow. 1983. (in Russian) [9] Y. Fan, S. Sarkar, and L. Lasdon: "Experiments with successive quadratic programming algorithms", J. Optim. Theory Appl., VOl. 56, (1988), pp. 359{383. [10] P.E. Gill, W. Murray, A.M. Saunders: "SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization", SIAM Journal on Optimization, Vol. 12, (2002), pp. 979{1006.
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[11] G. Isac, M,M, Kostreva, M.M. Wiecek: "Multiple objective approximation of feasible but unsolvable linear complementarity problems", Journal of Optimization Theory and Applications, Vol. 86, (1995), pp. 389{405. [12] L.D. Popov: "On the approximative roots of maximal monotone mapping", Yugoslav Journal of Operations Research, Vol. 6, (1996), pp. 19{32.
CEJM 2(1) (2004) 87{102
A classi¯cation of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach Old·rich Kowalski1¤, Barbara Opozda2y , Zden·ek Vl¶a·sek1z 1
Faculty of Mathematics and Physics, Charles University, Sokolovsk¶ a 83, 186 75 Praha, Czech Republic 2 Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krak¶ow, Poland
Abstract: The aim of this paper is to classify (locally) all torsion-less locally homogeneous a¯ ne connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classi cation of all non-equivalent transitive Lie algebras of vector elds in R2 according to P.J. Olver [7]. ® c Central European Science Journals. All rights reserved. Keywords: Two-dimensional manifolds with a± ne connection, locally homogeneous connections, Lie algebras of vector ¯elds, Killing vector ¯elds MSC (2000): 53B05, 53C30
1
Introduction
z
y
¤
The ¯eld of a±ne di®erential geometry is well-established and still in quick development (see e.g. [6]). Also, many basic facts about a±ne transformation groups and a±ne Killing vector ¯elds are known from the literature (see [2], [1]). Yet, it is remarkable that a seemingly easy problem to classify all locally homogeneous torsion-less connections in the plane domains was not solved until recently. (Some partial results have been published in [3] and [4].) For dimension three it seems to be a really hard problem.
[email protected]¬.cuni.cz
[email protected] [email protected]¬.cuni.cz
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B. Opozda [8] has found a general formula expressing all torsion-less locally homogeneous a±ne connections on two-dimensional manifolds in suitable local coordinates. She proved the following Theorem 1.1. Let r be a torsion-less locally homogeneous a±ne connection on a 2dimensional manifold m . Then, either r is a Levi-Civita connection of constant curvature or, in a neighborhood u of each point m 2 m , there is a system (u; v) of local coordinates and constants p; q; c; d; e; f such that r is expressed in u by one of the following formulas: Type A: r@u @u = p@u + q@v ; r@u @v = c@u + d@v ; r@v @v = e@u + f @v :
(1)
Type B: r@ u @ u =
p@u + q@v c@u + d@v e@u + f @v ; r@ u @ v = ; r@v @v = : u u u
(2)
(Here the "Levi-Civita connection" involves also the Lorentzian case. For an application of this result see [5]). In this paper we are going to classify these a±ne connections from the group-theoretical point of view. This means that we always start with a speci¯c transitive Lie algebra of vector ¯elds from the list of Olver and we are looking for all a±ne connections for which this algebra is an a±ne Killing algebra. It may happen that the same family of a±ne connections admits di®erent a±ne Killing algebras from the Olver’s list - in such a case we are interested also in the maximal one. It also happens quite often that the given Lie algebra of vector ¯elds does not admit any invariant a±ne connection. Let us underline that this classi¯cation procedure cannot be reduced to that used by B.Opozda because, as a rule, we are using di®erent "canonical" coordinates. We try to organize our computation in (possibly) most systematic way so that the whole procedure is not excessively long. Also, because this topic is an ideal subject for a computer-aided research, we are using the software Maple V Release 4, (c) Waterloo Maple Inc., throughout this work. But we put stress on the full transparency of this procedure. According to [7], taking into account the comments on page 61, the classi¯cation of all transitive Lie algebras of vector ¯elds in R2 is given by Table 1 and Table 6 ([7], pages 472 and 476, respectively). The Olver’s tables with a slight modi¯cation are presented at the end of the article. Remark. Here = (1) = a(1) © R. In Cases 1.5 and 1.6, the functions ´1 (v); : : : ; ´k (v) satisfy a k th order constant coef¯cient homogeneous linear ordinary di®erential equation d [u] = 0. In Cases 1.5 - 1.11 we require k ¶ 1. Note, though, that if we set k = 0 in Case 1.10, and replace u by u2 , we obtain Case 1.1. Similarly, if we set k = 0 in Case 1.11, we obtain
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
89
Case 1.3. Cases 1.7 and 1.8 for k = 0 are equivalent to the Lie algebra spanf@v ; ev @u g of type 1.5. Case 1.9 for k = 0 is equivalent to the Lie algebra spanf@v ; @u ; u@u g of type 1.6. Notice that the distribution spanned by @u is invariant by the imprimitive Lie algebras from Table 1.
2
Killing vector ¯elds and some Lemmas
Choose a system (u; v) of local coordinates in a domain u » m . In u , the connection r is uniquely determined by six functions A; : : : ; F given by the formulas:
r@u @u = A@u + B@v ; r@u @v = C@u + D@v = r@v @u ; r@v @v = E@u + F @v :
(3)
The following assertion is standard: Proposition 2.1. A smooth connection r on m is locally homogeneous if and only if it admits, in a neighborhood of each point p 2 m , at least two linearly independent a±ne Killing vector ¯elds. We start with the analysis of the system of partial di®erential equations for the Killing vector ¯elds. A Killing vector ¯eld X is characterized by the equation:
[X; rY Z] ¡
rY [X; Z] ¡
r[X;Y ] Z = 0
(4)
which has to be satis¯ed for arbitrary vector ¯elds Y; Z (see [4]). It is su±cient to © ª satisfy (4) for the choices (Y; Z) 2 (@u ; @u ); (@u ; @v ); (@v ; @u ); (@v ; @v ) . Moreover, we easily check from the basic identities for the torsion and the Lie brackets, that the choice (Y; Z) = (@v ; @u ) gives the same condition as the choice (Y; Z) = (@u ; @v ). In the sequel, let us express the vector ¯eld X in the coordinate form
X = a(u; v)@u + b(u; v)@v :
(5)
If we substitute the corresponding expressions for X; Y and Z in (4), we easily see that the condition (4) reduces to six linear partial di®erential equations for the unknown functions a; b:
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1)
auu + Aau ¡
2)
buu + 2Bau + (2D ¡
3)
auv + (A ¡
4)
buv + Dau + Bav + (F ¡
5)
avv ¡
6)
bvv + 2Dav ¡
Bav + 2Cbu + Au a + Av b = 0; A)bu ¡
Bbv + Bu a + Bv b = 0;
D)av + Ebu + Cbv + Cu a + Cv b = 0;
Eau + (2C ¡
C)bu + Du a + Dv b = 0;
(6)
F )av + 2Ebv + Eu a + Ev b = 0;
Ebu + F bv + Fu a + Fv b = 0:
In the following we shall need: Lemma 2.2. If the connection r admits the Killing vector ¯eld @v , then all Christo®el symbols are functions of one variable u. If r admits Killing vector ¯elds @u and @v , then all Christo®el symbols are constant in the corresponding coordinates. Proof. Obvious from (6). Lemma 2.3. If the connection r admits the Killing vector ¯elds @u ; @v ; (pu + qv)@u + (ru + sv)@v , where p; q; r; s are constants such that ps ¡
qr 6= 0; 2p2 + 2s2 + 9qr ¡
5ps 6= 0;
(7)
then all Christo®el symbols A; B; : : : ; F with respect to the coordinates u; v are zero, i.e., r is locally °at. Proof. First we see that all A; B; : : : ; F are constants. Substituting now pu + qv and ru + sv for a(u; v) and b(u; v), respectively, into (6), we obtain the following system of 6 linear homogeneous algebraic equations for 6 unknown constants A; B; : : : ; F , namely pA ¡ ¡ rA + (2p ¡ qA + sC ¡ qB ¡
qB + 2rC = 0; s)B + 2rD = 0; qD + rE = 0;
rC + pD + rF = 0;
2qC + (2s ¡ 2qD ¡
p)E ¡
qF = 0;
rE + sF = 0:
The determinant of this system can be expressed in the form ¡ (2p2 + 2s2 + 9qr ¡ which concludes the proof.
5ps)(ps ¡
qr)2 ;
(8)
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91
Lemma 2.4. A locally °at connection (i.e., such that all Christo®el symbols vanish with respect to some local coordinates u; v) admits the 6-dimensional algebra spanf@u ; @v ; u@u ; v@u ; u@v ; v@v g of a±ne Killing vector ¯elds in corresponding coordinates. Proof. Obvious by (6) and well-known. Lemma 2.5. If r is locally °at and (u; v) is a coordinate system in which the Christo®el symbols of r vanish, then the connection does not admit a Killing vector ¯eld of the form Z = a(u; v)@u + b(u; v)@v where a(u; v) or b(u; v) is a proper quadratic polynomial of u; v. Proof. Because all coe±cients A; : : : ; F are zero, we see at once from (6) a contradiction.
3
Lie algebras containing all translations
We see ¯rst from Table 1 and Table 6 that the following Lie algebras of vector ¯elds contain the operators @u ; @v : the case 1.4, the cases 1.5 and 1.6 for ´1 (v) = const and any k ¶ 1, the cases 1.7{1.11, the cases 6.1 and 6.4{6.8. Now we formulate the ¯rst classi¯cation results: Lemma 3.1. The following Lie algebras from the Olver’s list characterize, as a±ne Killing algebras, connections with vanishing Christo®el symbols: the case 1.7 for ® 6= 0; 2; 1=2, and k = 1; 2, the case 1.7 for ® = 0, or ® = 1=2, and k = 2, the case 1.8 for k = 1, the case 1.9 for k = 1; 2, the case 6.1, 6.4, 6.5 and 6.6. Proof. From Lemma 2.2 it follows that the algebras quoted above consist of a±ne Killing vector ¯elds of the connection with constant Christo®el symbols. It remains to show that each of these algebras enforces the vanishing of all Christo®el symbols. Since all these algebras contain the operators @u ; @v , the Christo®el symbols A; : : : ; F must be constant. Now, the case 1.8 for k = 1 involves the operator (u + v)@u + v@v , i.e., one with p = q = s = 1; r = 0, which satis¯es the inequalities (7) from Lemma 2.3. The case 1.9 always involves the operator u@u + v@v , which also satis¯es the conditions (7). The case 6.1 involves the operator with p = ®; q = ¡ 1; r = 1; s = ®. We see again that (7) is satis¯ed. The cases 6.4, 6.5 and 6.6 are now obvious. It remains to discuss the case 1.7, in which we have an additional operator with p = ®; q = 0; r = 0 and s = 1. Here the values p; q; r; s satisfy (7) except the cases ® = 0; ® = 2 and ® = 1=2. Thus, for ® 2 = f0; 1=2; 2g we get our assertion. Let be now ® = 0, or ® = 1=2, and k = 2. Then we have the second additional operator v@u , i.e. one
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with p = r = s = 0; q = 1. From (8) we obtain A = B = D = 0 and, because ® 6= 2, applying (8) for the ¯rst operator we get ¯nally C = E = F = 0. This completes the proof. Lemma 3.2. The following Lie algebras of vector ¯elds do not admit any invariant a±ne connection: the case 1.4, the case 1.7 for k > 2, the case 1.8 for k ¶ 2, the case 1.9 for k > 2, the cases 1.10, 1.11 and the cases 6.7, 6.8. Proof. If there is such a connection, its Christo®el symbols must be again constant. For the case 1.7 with ® 6= 2 the result follows from Lemma 3.1 and Lemma 2.5. If ® = 2 in the case 1.7, we substitute p = 2; q = r = 0 and s = 1 into the system (8). We obtain A = B = C = D = F = 0. Now the operator v 2 @u has to be added. We put a(u; v) = v 2 ; b(u; v) = 0 in (6) and the 5th equation gives a contradiction. For the case 1.8 with k ¶ 2 we put a(u; v) = ku + v k ; b(u; v) = v in the 5th equation of (6). We obtain the following contradiction: (k 2 ¡
k)v k + (2 ¡
k)Ev 2 + (2kC ¡
kF )v k+1 = 0:
For the remaining cases, the result follows from Theorem 3.1, Lemma 2.3 and Lemma 2.5. We shall conclude this section with the following Proposition 3.3. In the case 1.7 the following holds: a) If ® = 0; k = 1, then B = C = E = F = 0 and A; D are arbitrary parameters, b) If ® = 1=2; k = 1, then A = C = D = E = F = 0 and B is an arbitrary parameter, c) If ® = 2; k = 1; 2, then A = B = C = D = F = 0 and E is an arbitrary parameter. Proof. Obvious from the equation (8) where we put p = ®; q = r = 0; s = 1 in each case, and p = r = s = 0; q = 1 in the last subcase ® = 2; k = 2. We postponed here the special cases 1.5 and 1.6 to the next section.
4
Lie algebras of types 1.5 and 1.6
In these cases we have the Killing vector ¯eld @v and thus all Christo®el symbols A; : : : ; F are functions of the variable u only. Let us check a Killing vector ¯eld of the form f (v)@u , i.e., we put a(u; v) = f (v) and b(u; v) = 0 into the system (6). We get:
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
1)
A0 (u) ¡
2)
B 0 (u) = 0;
3) 4)
¡
¢
B(u) f 0 (v)=f (v) = 0;
¢¡
¢
D(u) f 0 (v)=f (v) + C 0 (u) = 0;
A(u) ¡
¡
¡
93
¢
B(u) f 0 (v)=f (v) + D 0 (u) = 0;
¡
¢¡
5)
f 00 (v)=f (v) + 2C(u) ¡
6)
2D(u) f 0 (v)=f (v) + F 0 (u) = 0:
¡
(9)
¢
F (u) f 0 (v)=f (v) + E 0 (u) = 0;
¢
Let r be a locally homogeneous connection with a Killing vector ¯eld f (v)@u . If f 0 (v0 ) = 0 at some v0 , we see that A(u); B(u) and D(u) are constants. If some of them is nonzero, then by (9) f 0 (v) is identically zero and all Christo®el symbols are constant. Moreover, the corresponding Killing vector ¯eld is a constant multiple of @u . Assume now that f 0 (v) 6= 0 everywhere. Even here, B must be constant according to 2) of (9). Assume ¯rst B 6= 0. From 1) or 4) of (9) we see that f 0 (v)=f (v) is a nonzero constant (due to the separation of variables), i.e., f (v) = elv where l 2 R n f0g. Using the transformation of the coordinate v : v~ = lv (which does not change the form of the algebras of types 1.5 and 1.6) we can assume that l = 1 and f (v) = ev . Integrating now the system (9) we obtain the solution A(u) = C1 u + C2 ; B(u) = C1 ; D(u) = ¡ C1 u + C3 ; C(u) = ¡ C1 u2 + (C3 ¡ 3
E(u) = C1 u + (C2 ¡
C2 )u + C4 ; F (u) = C1 u2 ¡ 2
2C3 )u + (C5 ¡
2C4 ¡
2C3 u + C5 ;
(10)
1)u + C6 ;
where C1 ; : : : ; C6 are constant parameters and C1 6= 0. If now B = 0 then, by 1) and 4) of (9), A and D are constants. From 3) and 6) of (9) we see that either A = D = 0 or f 0 (v)=f (v) is a constant. In the second case we can assume again that f (v) = ev . We obtain the formula (10) with C1 = 0. If A = B = D = 0, then, by (9), one gets A = B = D = 0; C(u) = c1 ; F (u) = 2c2 ; E(u) = c3 u + c4 ;
(11)
where c1 ; c2 ; c3 ; c4 are constants. Moreover the 5th equation of (9) takes on the form f 00 (v) + 2(c1 ¡
c2 )f 0 (v) + c3 f (v) = 0:
(12)
We see that, for any choice of the parameters in (11), the corresponding equation (12) is uniquely determined, and the solutions f (v) form a uniquely determined twodimensional vector space. The generators of this vector space are the following pairs of functions, which depend on the sign of the discriminant c02 ¡ c3 , where c0 = c2 ¡ c1 : c0 2 ¡ c3 > 0: Then there is a constant c5 > 0 such that c52 = c02 ¡ c3 and we get a pair of generators fe(c0 +c5 )v ; e(c0 ¡c5 )v g, c0 2 ¡ c3 = 0: Then we have a pair of generators fec0 v ; v ec0 v g,
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c0 2 ¡ c3 < 0: Then there is a constant c5 > 0 such that c52 = c3 ¡ c02 and a pair of generators is fec0 v cos(c5 v); ec0 v sin(c5 v)g. Assume now that r admits a transitive Lie algebra of Killing vector ¯elds of type 1.5 and f (v)@u is a Killing vector ¯eld. If some of the functions A(u),B(u),D(u) is nonzero then f 0 (v)=f (v) is a uniquely determined constant by (9). Hence k = 1 and for the function ´1 (v) we can take either 1 or ev . If A = B = D = 0, then f satis¯es (12). Hence k = 2 and the corresponding Lie algebra must be generated by @v ; ´1 (v)@u and ´2 (v)@u , where the pair f´1 (v); ´2 (v)g has one of the three possible forms calculated above for the equation (12). Summing up we have got Theorem 4.1. Let r be a locally homogeneous connection whose algebra of Killing vector ¯elds is of type 1.5. Then k = 1 or 2 and we have (i) For k = 1 there are two non-equivalent algebras of Killing vector ¯elds, namely spanned by f@v ; @u g, or f@v ; ev @u g. In the ¯rst case all Christo®el symbols are arbitrary constants. In the second case, the Christo®el symbols can be expressed by the formula (10) with 6 arbitrary parameters. (ii) For k = 2 there are ¯ve non-equivalent algebras of Killing vector ¯elds spanned by the following triplets: f@v ; @u ; ev @u g; f@v ; @u ; v@u g; f@v ; e®v @u ; e¯v @u g, ® 6= ¯; ®; ¯ 6= 0; f@v ; e®v @u ; v e®v @u g, ® 6= 0; f@v ; e®v cos(¯v)@u ; e®v sin(¯v)@u g, ¯ 6= 0. The corresponding Christo®el symbols are given by the formulas (11) in which either c3 6= 0 or c3 = 0. For k > 2, an invariant a±ne connection r does not exist. Assume now that a Lie algebra of Killing vector ¯elds is of type 1.6, that is, it also contains u@u . Setting a(u; v) = u; b = 0 into (6) we get: A(u) + uA0 (u) = 0; 2B(u) + uB 0 (u) = 0; uC 0 (u) = 0;
(13)
0
D(u) + uD (u) = 0; ¡ E(u) + uE 0 (u) = 0; uF 0 (u) = 0: Comparing this with formulas (10), (11) gives A = B = D = 0. If the Christo®el symbols are constant we also have A = B = D = 0. Using (13) and previous considerations dealing with this situation we see that the Christo®el symbols are given by (11) with c4 = 0 and moreover f satis¯es (12). It follows that k = 2. Adding to each of sets of generators given by Theorem 4.1, (ii), the vector ¯eld u@u we obtain a set generating a Lie algebra. We have got: Theorem 4.2. Let r be a locally homogeneous connection whose Lie algebra of Killing vector ¯elds is of type 1.6. Then k = 2 and the Lie algebra of Killing vector ¯elds is
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95
generated by one of the following sets: f@v ; u@u ; @u ; ev @u g; f@v ; u@u ; @u ; v@u g; f@v ; u@u ; e®v @u ; e¯v @u g where ® 6= 0; ¯ 6= 0; ® 6= ¯; f@v ; u@u ; e®v @u ; v e®v @u g where ® 6= 0; f@v ; u@u ; e®v cos(¯v)@u ; e®v sin(¯v)@u g where ¯ 6= 0. The Christo®el symbols are given by (11) with c4 = 0. For k > 2, an invariant a±ne connection does not exist. Remark 4.3. In case of the ¯rst set of generators we have c3 = 0 and c1 ; c2 are arbitrary such that c0 = c2 ¡ c1 6= 0. In case of the second set of generators we have c3 = 0; c1 = c2 . Remark 4.4. The case k = 1 also makes sense here but it does not produce new locally homogeneous connections! The corresponding Lie algebras are generated by the triplets f@v ; u@u ; @u g and f@v ; u@u ; e®v @u g; ® 6= 0. For any invariant connection r, such an algebra can be extended to a 4-dimensional a±ne Killing algebra corresponding to the case k = 2.
5
Other cases
We are now left with the cases 1.1{1.3 and 6.2, 6.3. The cases 1.1{1.3 should involve the Killing vector ¯elds @v ; u@u ¡ v@v . Hence, the Christo®el symbols of r depend only on u and they satisfy the system of equations: A(u) + uA0 (u) = 0; D(u) + uD 0 (u) = 0; 3B(u) + uB 0 (u) = 0; ¡ 3E(u) + uE 0 (u) = 0;
(14)
¡ C(u) + uC 0 (u) = 0; ¡ F (u) + uF 0 (u) = 0: The general solution is of the form A(u) =
p q d ; B(u) = 3 ; C(u) = cu; D(u) = ; E(u) = eu3 ; F (u) = f u: u u u
(15)
Now, in the case 1.1, we have the third Killing vector ¯eld for which a(u; v) = 2uv, b(u; v) = ¡ v 2 . Making corresponding substitutions in (6), we get the following relations among constants: q = 0; d = 1=2; p = ¡ 1=2; f = 2c: (16) Hence we get the ¯nal solution: A(u) = ¡
1 1 ; B(u) = 0; C(u) = cu; D(u) = ; E(u) = eu3 ; F (u) = 2cu 2u 2u
with two arbitrary parameters c; e.
(17)
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In the case 1.2 we have to put a(u; v) = 2uv + 1; b(u; v) = ¡ v 2 . An easy calculation shows that here p = q = d = 0; c = ¡ 2; e = 4; f = 2. We get: A(u) = B(u) = D(u) = 0; C(u) = ¡ 2u; E(u) = 4u3 ; F (u) = 2u:
(18)
In the case 1.3 we have two additional Killing vector ¯elds, e.g. u@u and uv@u ¡ v 2 @v . An easy calculation yields the d = 1 and other constants are zero. We have: A(u) = B(u) = C(u) = E(u) = F (u) = 0; D(u) = 1=u:
(19)
The remaining cases are 6.2 and 6.3. In the case 6.2, we have again the Killing vector ¯eld @v and hence all Christo®el symbols are functions of the variable u only. Further, the occurence of the Killing vector ¯eld u@u + v@v enforces equations analogous to (14), namely ©(u) + u©0 (u) = 0 for © = A(u); B(u); : : : ; F (u):
(20)
p q c d e f ; B(u) = ; C(u) = ; D(u) = ; E(u) = ; F (u) = : u u u u u u
(21)
Hence we get: A(u) =
The last Killing vector ¯eld is characterized by a(u; v) = 2uv, b(u; v) = v 2 ¡ u2 . Substituting from this and (21) in (6), we get a system of linear equations for the constants p; q; : : : ; f and hence p = ¡ 1; q = 0; c = 0; d = ¡ 1; e = 1; f = 0:
(22)
We get the ¯nal formula A(u) = ¡
1 1 1 ; D(u) = ¡ ; E(u) = ; B(u) = C(u) = F (u) = 0: u u u
(23)
It remains the most complicated case 6.3. Here we have three Killing vector ¯elds and we write the equation (6) for each of these Killing vector ¯elds separately. As a result, we obtain a system of 18 (proper) partial di®erential equations, namely uAv ¡
vAu + B + 2C = 0;
uBv ¡
vBu ¡
A + 2D = 0;
uCv ¡
vCu ¡
A + D + E = 0;
uDv ¡
vDu ¡
B¡
uEv ¡
vEu ¡
2C + F = 0;
uFv ¡
vFu ¡
2D ¡
C + F = 0; E = 0;
(24)
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
u2 + v 2 )Av + 2uvAu + 2vA ¡
(1 ¡
2
2uB ¡
2
4uC = 0;
(1 ¡
u + v )Bv + 2uvBu + 2uA + 2vB ¡
4uD ¡
2 = 0;
(1 ¡
u2 + v 2 )Cv + 2uvCu + 2uA + 2vC ¡
2uD ¡
2uE + 2 = 0;
(1 ¡
2
2
2
2
u + v )Dv + 2uvDu + 2uB + 2uC + 2vD ¡ u + v )Ev + 2uvEu + 4uC + 2vE ¡
(1 ¡
u2 + v 2 )Fv + 2uvFu + 4uD + 2uE + 2vF + 2 = 0;
2uF = 0;
(1 + u2 ¡
v 2 )Au + 2uvAv + 2uA + 2vB + 4vC + 2 = 0;
(1 + u2 ¡
v 2 )Bu + 2uvBv ¡
2vA + 2uB + 4vD = 0;
(1 + u2 ¡
v 2 )Cu + 2uvCv ¡
2vA + 2uC + 2vD + 2vE = 0;
2
(25)
2uF = 0;
(1 ¡
2
97
(1 + u ¡
v )Du + 2uvDv ¡
2vB ¡
(1 + u2 ¡
v 2 )Eu + 2uvEv ¡
4vC + 2uE + 2vF ¡
(1 + u2 ¡
v 2 )Fu + 2uvFv ¡
4vD ¡
(26)
2vC + 2uD + 2vF + 2 = 0; 2 = 0;
2vE + 2uF = 0:
We shall solve ¯rst the system (24){(26) as a system of linear algebraic equations for the unknowns A; B; : : : ; F and their ¯rst partial derivatives. We obtain a unique solution A=
¡2u ; 1+u2 +v 2
Au =
¡2(1¡u2 +v 2 ) ; (1+u2 +v2 )2
Av =
4uv ; (1+u2 +v2 )2
B=
2v ; 1+u2 +v 2
Bu =
¡4uv ; (1+u2 +v2 )2
Bv =
2(1+u2 ¡v 2 ) ; (1+u2 +v2 )2
C=
¡2v ; 1+u2 +v2
Cu =
4uv ; (1+u2 +v 2 )2
Cv =
¡2(1+u2 ¡v 2 ) ; (1+u2 +v2 )2
D=
¡2u ; Du 1+u2 +v 2
¡2(1¡u2 +v2 ) ; (1+u2 +v 2 )2
Dv =
4uv ; (1+u2 +v 2 )2
E=
2u ; 1+u2 +v2
Eu =
2(1¡u2 +v2 ) ; (1+u2 +v 2 )2
Ev =
¡4uv ; (1+u2 +v2 )2
F =
¡2v ; 1+u2 +v2
Fu =
4uv ; (1+u2 +v2 )2
Fv =
¡2(1+u2 ¡v 2 ) : (1+u2 +v 2 )2
=
(27)
But hence we easily see that the expressions for A; B; : : : ; F are unique solutions of (24){ (26) as a system of PDE. We can write brie°y A = ¡ ½ u ; B = ½ v ; C = ¡ ½v ; D = ¡ ½u ; E = ½ u ; F = ¡ ½v ; 2
2
where ½ = log(1 + u + v ): This concludes our classi¯cation procedure.
6
Some additional geometrical properties
Let us recall the following formulas for the Ricci tensor (cf.[3], [4]):
(28)
98
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
Ric(@u ; @u ) = B v ¡
Du + D(A ¡
D) + B(F ¡
Ric(@u ; @v ) = Dv ¡
Fu + CD ¡
BE;
Ric(@v ; @u ) = Cu ¡
Av + CD ¡
BE;
Ric(@v ; @v ) = E u ¡
Cv + E(A ¡
D) + C(F ¡
C); (29) C):
We shall start with: Theorem 6.1. Exactly in the following cases the corresponding a±ne connection is locally °at: 1) All cases given by Theorem 3.1. 2) The cases b), c) from Proposition 3.3 and the case a) for D(A-D)=0. 3) The case given by formula (10) with the additional conditions C1 ¡
2C2 2 + C1 C5 ¡
C1 C4 = 0;
2C2 + C1 C6 + C2 C4 = 0; C4 2 + 2C4 ¡
C4 C5 ¡
C5 ¡
2C2 C6 + 1 = 0; C3 = ¡ C2 :
4) The case given by formula (11) for c3 ¡ c1 2 + 2c1 c2 = 0. 5) The case of constant Christo®el symbols satisfying the relations B(F ¡
C) + D(A ¡
D) = 0;
C(F ¡
C) + E(A ¡
D) = 0;
BE ¡
CD = 0;
(where the ¯rst two of them are linearly dependent). 6) The cases given by formula (17) for c = e = 0 and by formula (19). Proof. Direct check of the assumption that Ric = 0. Lemma 6.2. A connection r given by formula (10) is locally symmetric (and not locally °at in general) in the following cases: A) C1 = C2 = C3 = C5 = 0, B) C1 = C2 = C3 = 0; C4 = C5 ¡ 1, C) C1 = C2 = C3 = 0; C4 = ¡ 1, D) C1 6= 0; C2 + C3 = 0; C4 = ¡ C2 2 =C1 ; C5 = C2 2 =C1 , C6 = ¡ C2 (C1 ¡ C2 2 )=C1 2 , E) C1 6= 0; C2 + C3 = 0; C4 = ¡ (C1 + C2 2 )=C1 ; C5 = (C2 2 ¡ 2C1 )=C1 , C6 = ¡ C2 (C1 ¡ C2 2 )=C1 2 . Proof. We check the conditions for r to be Ricci-parallel.
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
99
Lemma 6.3. A connection r with constant Christo®el symbols is locally symmetric exactly in the following cases: a) E 6= 0; A = (ED ¡ CF + C 2 )=E; B = CD=E, b) E 6= 0; B = AF=E; D = CF =E, c) C 6= 0; E = F = 0; B = AD=C, d) C = E = F = 0; A = D, e) A = C = E = 0, f) F 6= 0; C = E = 0; B = D(D ¡ A)=F , g) D = E = 0; F = C. Here the non°at connections correspond to the cases b) and c). Proof. We again check the conditions for r to be Ricci-parallel. Theorem 6.4. Exactly in the following cases the Ricci tensor ¯eld Ric is a pseudoRiemannian metric and the corresponding a±ne connection r is the Riemannian connection of Ric: 1) r is given by (10) and the case D) of Lemma 6.2 holds. Here Ric is a Lorentzian metric of constant positive curvature. 2) r is given by (18). Here Ric is a Lorentzian metric Ric = ¡ 4du dv + 4u2 dv 2 of constant positive curvature. 3) r is given by (23). Here (¡ Ric) is a Riemannian metric (du2 + dv 2 )=u2 of constant negative curvature. 4) r is given by (28). Here Ric is a Riemannian metric 4(du2 + dv 2 )=(1 + u2 + v 2 )2 of constant positive curvature. In all cases, the connection r is locally symmetric. Proof. First we check step by step that the cases 1){4) are the only possibilities. As concerns the connections given by formula (10), the cases A){C) in Lemma 6.2 give Ric(@u ; @u ) = Ric(@u ; @v ) = Ric(@v ; @u ) = 0; Ric(@v ; @v ) = ¡ 1. In the case D) we have Ric(@u ; @u ) = C1 ; Ric(@u ; @v ) = Ric(@v ; @u ) = ¡ C1 u ¡ 2
2
Ric(@v ; @v ) = C1 u + 2C2 u + C2 =C1 ¡
C2 ;
(30)
1:
6 0 and the corresponding determinant of the Ricci form is equal to ¡ C1 . Here C1 = Hence the metric is Lorentzian. In the case E) we get: Ric(@u ; @u ) = Ric(@u ; @v ) = Ric(@v ; @u ) = 0 and hence Ric has the rank one or zero. The calculations in the cases 2){4) are straithforward. The last case to consider are connections with constant Christo®el symbols. According to Lemma 6.3, we have to consider only the subcases b) and c) but an immediate check shows that the Ricci form is singular, as well.
100
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
Acknowledgements · 201/02/0616 and by the research This work was partially supported by the grant GA CR project MSM 113200007.
References [1] S. Kobayashi: Transformation Groups in Di®erential Geometry, Springer-Verlag, New York, 1972. [2] S. Kobayashi, K. Nomizu: Foundations of Di®erential Geometry I, Interscience Publ., New York, 1963. [3] O. Kowalski, B. Opozda, Z. Vl¶a·sek: "Curvature homogeneity of a±ne connections on two-dimensional manifolds", Colloquium Mathematicum, Vol. 81, (1999), pp. 123{ 139. [4] O. Kowalski, B. Opozda, Z. Vl¶a·sek: "A classi¯cation of locally homogeneous a±ne connections with skew-symmetric Ricci tensor on 2-dimensional manifolds", Monatshefte fÄur Mathematik, (2000), pp. 109{125. [5] O. Kowalski, Z. Vl¶a·sek: "On the local moduli space of locally homogeneous a±ne connections in plane domains", Comment.Math.Univ.Carolinae, (2003), pp. 229{234. [6] K. Nomizu, T. Sasaki: A±ne Di®erential Geometry, Cambridge University Press, Cambridge, 1994. [7] P.J. Olver: Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, 1995. [8] B. Opozda: "Classi¯cation of locally homogeneous connections on 2-dimensional manifolds", to appear in Di®. Geom. Appl..
k+2 k+2 k+3
@v ; ² 1 (v)@u ; : : : ; ² k (v)@u
@v ; u@u ; ² 1 (v)@u ; : : : ; ² k (v)@u
@v ; v@v + ¬ u@u ; @u ; v@u ; v 2 @u ; : : : ; v k¡1 @u
@v ; v@v + (ku + v k )@u ; @u ; v@u ; v2 @u ; : : : ; v k¡1 @u
@v ; v@v ; u@u ; @u ; v@u ; v2 @u ; : : : ; vk¡1 @u
1.5.
1.6.
1.7.
1.8.
1.9.
1)uv@u ; u@u ; @u ; v@u ; v 2 @u ; : : : ; v k¡1 @u
6
4
k+4
k+3
k+1
Table 1 Transitive, Imprimitive Lie Algebras of Vector Fields in R2 .
@v ; v@v ; v 2 @v + (k ¡
1.11.
1)u@u ; v2 @v + (k ¡
@v ; 2v@v + (k ¡
1.10.
1)uv@u ; @u ; v@u ; v2 @u ; : : : ; vk¡1 @u
k+2
@v ; v@v ; v 2 @v ; @u ; u@u ; u2 @u
1.4.
uv@u
@v ; v@v ; u@u ; v 2 @v ¡
1.3.
3
(2uv + 1)@u
u@u ; v 2 @v ¡
@v ; v@v ¡
1.2.
3
2uv@u
u@u ; v 2 @v ¡
@v ; v@v ¡
Dim
1.1.
Generators
A l(2) n Rk
M l(2) n Rk
= (1) n Rk
a(1) n Rk
a(1) n Rk
R2 n Rk
R n Rk
M l(2) © M l(2)
A l(2)
M l(2)
M l(2)
Structure
O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102 101
@v ; @u ; v@v + u@u ; u@v ¡
@v ; @u ; v@v ¡
@v ; @u ; v@v ; u@v ; v@u ; u@u
6.4.
6.5.
6.6.
v2 )@u
Table 6 Primitive Lie Algebras of Vector Fields in R2 .
@v ; @u ; v@v ; u@v ; v@u ; u@u ; v 2 @v + uv@u ; uv@v + u2 @u
6.8.
u2 )@v + 2uv@u ; 2uv@v + (u2 ¡
@v ; @u ; v@v + u@u ; u@v ¡
6.7.
v@u ; (v 2 ¡
u@u ; u@v ; v@u
v@u
u@v ¡
6.3.
u2 )@v + 2uv@u ; 2uv@v + (1 ¡ v 2 + u2 )@u
8
6
6
5
4
3
3
@v ; v@v + u@u ; (v 2 ¡ u2 )@v + 2uv@u
6.2.
v@u ; (1 + v 2 ¡
3
@v ; @u ; ¬ (v@v + u@u ) + u@v ¡ v@u
Dim
6.1.
Generators
M l(3)
M o(3; 1)
a(2)
M a(2)
R2 n R2
M o(3)
M l(2)
R n R2
Structure
102 O. Kowalski et al. / Central European Journal of Mathematics 2(1) (2004) 87{102
CEJM 2(1) (2004) 103{111
Finiteness of the strong global dimension of radical square zero algebras Otto Kerner1¤, Andrzej Skowro¶ nski2y , Kunio Yamagata3z , Dan Zacharia4x 1
Mathematisches Institut, Heinrich-Heine UniversitÄat, D-40025 DÄ usseldorf, Germany 2 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toru¶ n, Poland 3 Department of Mathematics, Tokyo University of Agriculture and Technology, Nakacho 2-24-16, Koganei, Tokyo 184-8588, Japan 4 Department of Mathematics, Syracuse University, Syracuse, New York 13244, USA
Received 12 November 2003; revised 3 December 2003 Abstract: The strong global dimension of a nite dimensional algebra A is the maximum of the width of indecomposable bounded di¬erential complexes of nite dimensional projective A-modules. We prove that the strong global dimension of a nite dimensional radical square zero algebra A over an algebraically closed eld is nite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the niteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras. ® c Central European Science Journals. All rights reserved.
x
z
y
¤
Keywords: strong global dimension, repetitive algebra, local support-¯niteness, piecewise hereditary algebra, radical square zero algebra. MSC (2000): Primary 16D50, 16E10, 18E30; Secondary 16G10.
E-mail: E-mail: E-mail: E-mail:
[email protected] [email protected] [email protected] [email protected]
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1
O. Kerner et al. / Central European Journal of Mathematics 2(1) (2004) 103{111
Introduction
Throughout the paper by an algebra we mean a basic, connected ¯nite dimensional Kalgebra over a ¯xed algebraically closed ¯eld K. For an algebra A we denote by mod A the category of ¯nite dimensional (over K) right A-modules and by Db (mod A) the derived category of di®erential bounded complexes over mod A. It is known that if the global dimension, gl:dim A, of A is ¯nite, then Db (mod A) is equivalent (as a triangulated category) to the homotopy category Kb (PA ) of the category Cb (PA ) of bounded di®erential complexes over the full subcategory PA of mod A consisting of projective modules [10]. The strong global dimension of A, s:gl:dim A, is de¯ned to be the maximum of the width of indecomposable complexes in Cb (PA ) (equivalently, in Kb (PA )) [21]. In general the following inequality holds max(2; 1 + gl:dim A) µ s:gl:dim A: It seems to be rather di±cult to determine the strong global dimension of an arbitrary algebra. The following examples illustrate some phenomena: ° s:gl:dim A = 2 if and only if gl:dim A µ 1 (A is a hereditary algebra) [10], ° for a positive integer n ¶ 2 and the bound quiver algebra A(n) = KQ(n) =I (n) where KQ(n) is the path algebra of the Dynkin quiver Q(n) ®1
0 Á¡
®2
1 Á¡
2 Á¡
: : : Á¡
(n ¡
®n
1) Á¡
n
and I (n) is the ideal in KQ(n) generated by the paths ®i¡1 ®i , 2 µ i µ n, we have gl:dim A(n) = n and s:gl:dim A = n + 1, ° for the bound quiver algebra A = KQ=I, where Q is the quiver 1 ®
3
° ¯
2 and I is the ideal in KQ generated by ¯®, we have gl:dim A = 2 but s:gl:dim A = 1 [21, (4.2)]. We are concerned with the problem of deciding when, for a given algebra A, s:gl:dim A is ¯nite. This problem is also strongly related with the density of the push-down functor b ! T(A) of the trivial extension algebra associated to the canonical Galois covering F A : A T(A) = A n D(A) of an algebra A by its minimal injective cogenerator D(A) by its b (see Section 2). It is known that the strong global dimension is ¯nite repetitive algebra A for all piecewise hereditary algebras, that is, algebras A for which there exists a hereditary abelian K-category H such that the triangulated categories Db (mod A) and Db (H) are equivalent (see Section 2). The following problem has been raised by the second named author during the international conference \Frobenius Algebras and Related Topics" held in Toru¶n (September 2003).
O. Kerner et al. / Central European Journal of Mathematics 2(1) (2004) 103{111
105
Problem 1. Assume that A is an algebra of ¯nite strong global dimension. Is A then piecewise hereditary? The main aim of this note is to prove that it is true for all algebras A with rad(A)2 = 0, where rad(A) denotes the Jacobson radical of A.
2
Repetitive algebras
In the representation theory of sel¯njective (Frobenius) algebras an important role is played by the repetitive algebras, since frequently sel¯njective algebras are isomorphic to the orbit algebras of the repetitive algebras with respect to actions of admissible automorphism groups. We present in this section some results on the repetitive algebras which are relevant to the ¯niteness of the strong global dimension, and to the connection with piecewise hereditary algebras. Let A be an algebra and D(A) = HomK (A; K), the associated injective cogenerator AA-bimodule. Then the trivial extension T(A) = A n D(A) of A by D(A) is the symmetric algebra whose K-vector space is that of A © D(A) where the multiplication is de¯ned by the formula (a; f ) ¢ (b; g) = (ab; ag + f b)
b of A is the locally ¯nite for a; b 2 A and f; g 2 D(A). The repetitive algebra [15] A dimensional algebra without the identity Ab =
M
(Am © D(A)m )
m2Z
b is given by where Am = A, D(A)m = D(A) for all m 2 Z, and the multiplication in A the formula (am ; fm )(bm ; gm ) = (am bm ; am gm + fm bm¡1 ) for am ; bm 2 Am and fm ; gm 2 D(A)m . If 1A = e1 + : : : + en is a decomposition of the identity 1A of A into a sum of orthogonal primitive idempotents e1 ; : : : ; en , then b may be considered as 1Am = em;1 + : : : + em;n , em;i = ei for all m 2 Z, 1 µ i µ n, and A a locally bounded K-category with the objects class E = fem;i j (m; i) 2 Z £ f1; : : : ; ngg :
b given by the shifts º b(em;i ) = em+1;i , for The canonical automorphism ºAb : Ab ! A A b and we have the (m; i) 2 Z £ f1; : : : ; ng, is called the Nakayama automorphism of A, canonical Galois covering (in the sense of [8]) b ¡ ! A=(º b b ) = T(A) FA : A A
with in¯nite cyclic Galois group (ºAb) generated by ºAb. Moreover, we have the associated push-down functor ([4], [8])
b ¡ ! mod T(A) F¸A : mod A
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O. Kerner et al. / Central European Journal of Mathematics 2(1) (2004) 103{111
b the T(A)-module F A (M ) such that F A (M )ei which assigns to each module M from mod A ¸ ¸ L = m2Z M em;i . It is well-known (see [8]) that F¸A (M ) is exact, preserves the indecomposable modules and Auslander-Reiten sequences, and induces an injection from the set b of (ºAb)-orbits of isomorphism classes of ¯nite dimensional indecomposable A-modules into the set of isomorphism classes of indecomposable ¯nite dimensional T(A)-modules. b ! T(A) induces a Z-grading on T(A) such that mod Ab is The Galois covering F A : A isomorphic to the category modZ T(A) of Z-graded T(A)-modules, and F¸A can be interb = modZ T(A) to mod T(A) (see [9]). Hence preted as the forgetful functor from mod A the following problem arises naturally. b ! Problem 2. Let A be an algebra. When is the push-down functor F¸A : mod A mod T(A) dense?
A su±cient condition for the density of F¸A follows from a general result established in [5] (see also [6]). A repetitive algebra Ab is said to be locally support-¯nite if, for each b indecomposable projective A-module P , the set of isomorphism classes of indecompos0 b able projective A-modules P such that HomAb(P; M ) 6= 0 6= HomAb(P 0 ; M ), for some b is ¯nite. Then we have indecomposable module M in mod A,
b is locally-support ¯nite. Then Theorem 2.1 ([5]). Let A be an algebra such that A b ! mod T(A) is dense. F¸A : mod A We have also the following necessary condition for the density of the considered pushdown functor.
Theorem 2.2 ([20]). Let A be an algebra such that the push-down functor F¸A : b ! mod T(A) is dense. Then A is triangular (the Gabriel quiver of A is directed), mod A and hence gl:dim A < 1. It has been proved by D. Happel [10] that for any algebra A there is an embedding of triangulated categories b ©A : Db (mod A) ,! mod A
b is the stable category of mod Ab (modulo projectives). Moreover, ©A is an where mod A equivalence if gl:dim A < 1. The following theorem links the two problems raised in Sections 1 and 2. b is locally support-¯nite if and only Theorem 2.3 ([21]). Let A be an algebra. Then A if s:gl:dim A < 1. Two algebras A and ¤ are said to be tilting-cotilting equivalent [2] if there is a sequence of algebras A = A0 ; A1 ; : : : ; Am+1 = ¤ and a sequence of modules TAi i , 0 µ i µ m, such that Ai+1 = End(TAi ) and TAi is either a tilting or a cotilting module. Following [12], an algebra B is said to be quasitilted if gl:dim B µ 2 and every indecomposable module from
O. Kerner et al. / Central European Journal of Mathematics 2(1) (2004) 103{111
107
mod B has projective or injective dimension at most 1. The following theorem describing the repetitive algebras of arbitrary piecewise hereditary algebras is a combination of results established in [1], [2], [3], [7], [10], [11], [13], [14], [15], [16], [17]. Theorem 2.4. Let A be an algebra. The following statements are equivalent: b (1) There exists a quasitilted algebra B such that Ab ¹= B. (2) There exists an algebra C which is hereditary or canonical, such that A and C are tilting-cotilting equivalent. (3) There exists an algebra C which is hereditary or canonical such that Db (mod A) ¹= Db (mod C). (4) A is piecewise hereditary. Since the repetitive algebras of quasitilted algebras are locally support-¯nite, we obtain.
b is locally support-¯nite, Corollary 2.5. Let A be a piecewise hereditary algebra. Then A and hence s:gl:dim A < 1. b (respectively, T(A)) of tame representation type, the posiFor algebras A with A tive solutions of Problems 1 and 2 follow from the following theorem which is a direct combination of the main results of the papers [3] and [22]. Theorem 2.6. Let A be an algebra. The following statements are equivalent: b is tame and F¸A : mod Ab ! mod T(A) is dense. (1) A b is tame and locally support-¯nite. (2) A (3) A is piecewise hereditary and the Euler form ÂA is nonnegative. Recall that the Euler form ÂA of an algebra A of ¯nite global dimension is the integral quadratic form ÂA : K0 (A) ! Z on the Grothendieck group K0 (A) of A such that ÂA ([M ]) =
1 X
(¡ 1)i dimK ExtiA (M; M )
i=0
for any module M from mod A (see [18]).
3
Algebras with radical square zero
The aim of this section is to prove following theorem. Theorem 3.1. Let A be an algebra with rad(A)2 = 0. Then the following statements are equivalent: (1) s:gl:dim A < 1. b ! mod T(A) is dense. (2) F¸A : mod A (3) A is piecewise hereditary.
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b ¹= H b for a hereditary algebra H with rad(H)2 = 0. (4) A
It follows from Theorem 2.1, Theorem 2.3 and Corollary 2.5 that (3) implies (1), and b ¹= H b is locally support-¯nite (1) implies (2). Moreover, if the statement (4) holds then A (see [1, Section 3] and [7, Section 3]), and invoking Theorems 2.1 and 2.2 we conclude that gl:dim A < 1. Thus, applying Happel’s theorem [10], we obtain isomorphisms of triangulated categories
b ¹= mod H b ¹= Db(mod H): Db (mod A) ¹= mod A
Hence, (4) implies (3). Therefore, in order to prove the theorem, it remains to show that (2) implies (4). Assume that A is an algebra with rad(A)2 = 0 such that the push-down functor b ! mod T(A) is dense. Since A is a basic ¯nite dimensional algebra over the F¸A : mod A algebraically closed ¯eld K, A is isomorphic to the bound quiver algebra KQ=I where KQ is the path algebra of the ordinary quiver Q = QA of A and I is the ideal in KQ generated by all paths in Q of length 2. Moreover, the quiver Q is connected because the algebra A is assumed to be connected. Without loss of generality, we may assume that A = KQ=I. We note that in general the quiver Q may have many arrows with ¯xed source and target. In order to formulate a necessary condition for the functor F¸A to be dense, consider the quiver G = GA (graph of A in the sense of [20], [23]) obtained from Q = QA by shrinking, for any pair i, j of vertices of Q, all arrows from i to j into one arrow from i to j. We then get a new radical square zero algebra A0 = KG=J where J is the ideal in the path algebra KG generated by all paths of length 2 (called zero-relations). It has been proved in [20, Theorem 1] that the density of the push-down b ! mod T(A) forces the following clock condition: for any cycle C functor F¸A : mod A of the quiver G = GA the number of clockwise oriented zero-relations on C equals the number of counter-clockwise zero relations on C. Since rad(A0 )2 = 0, it is equivalent to say that for any cycle C of G, the number of clockwise oriented arrows in C equals the number of counter-clockwise oriented arrows in C. In particular, we conclude that A is b for a hereditary algebra triangular (Q has no oriented cycles). We will show that Ab ¹= H 2 H with rad(H) = 0. Let Q0 be the set of vertices of Q, Q1 the set of arrows of Q, and s; e : Q1 ! Q0 two maps which assign to each arrow ® of Q its source s(®) and end e(®). Consider the b whose set Q b0 of vertices equals Z £ Q0 and the set Q b1 of arrows locally ¯nite quiver Q consists of the arrows (m;®)
(m; s(®)) ¡ ¡ ¡ ! (m; e(®))
and
(m;®0 )
(m + 1; e(®)) ¡ ¡ ¡ ! (m; s(®))
b for all m 2 Z and all arrows ® 2 Q1 . Further, let Ib be the ideal in the path algebra K Q b generated by the elements of Q (m; »)(m; ´) for »; ´ 2 Q1 with s(») = e(´), (m; °)(m; ± 0 ) for °; ± 2 Q1 with ° 6= ± and s(°) = s(±), (m; » 0 )(m + 1; %) for »; % 2 Q1 with » 6= % and e(») = e(%),
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(m; ®)(m; ® 0 ) ¡ (m; ¯)(m; ¯ 0 ) for ®; ¯ 2 Q1 with e(®) = e(¯), (m ¡ 1; ®0 )(m; ®) ¡ (m ¡ 1; ¯ 0 )(m; ¯) for ®; ¯ 2 Q1 with s(®) = s(¯), (m; ®)(m; ® 0 ) ¡ (m; ¯ 0 )(m + 1; ¯) for ®; ¯ 2 Q1 with e(®) = s(¯), b Ib (see [19]). We may assume that and m 2 Z. Then it is easy to check that Ab ¹= K Q= b = K Q= b I, b and identify A = KQ=I with the path algebra of the convex subquiver f0g£Q A b consisting of the vertices (0; i), i 2 Q0 , and the arrows (0; ®), ® 2 Q1 . We note of Q b ! Ab is induced by the Nakayama quiver also that the Nakayama automorphism ºAb : A b!Q b given by the shifts º(m; i) = (m + 1; i), º(m; ®) = (m + 1; ®) automorphism ºQb : Q 0 0 and º(m; ® ) = (m + 1; ® ) for all m 2 Z, i 2 Q0 , ® 2 Q1 . b such that the path algebra H = K¢ We will de¯ne a ¯nite convex subquiver ¢ of Q 2 b ¹= H b . Clearly, the fact that rad(H)2 = 0 of ¢ is hereditary with rad(H) = 0 and A is equivalent to the fact that ¢ has sink-source orientation (there are no paths in ¢ of length ¶ 2). Since the quiver Q has no oriented cycles, we may ¯x a sink u of Q. For each vertex v of Q denote by d(v) the length of the shortest walk u = u0 ¡ ¡ u1 ¡ ¡ ¢ ¢ ¢ ¡ ¡ ur = v, with ui ¡ ¡ ui+1 being ui ! ui+1 or ui Á ui+1 , in Q connecting u and v (such a walk always exists because Q is connected). Denote by n the maximum of all d(v), v 2 Q0 . We shall de¯ne the quiver ¢ as the ¯nal quiver in an increasing chain
¢(0) » ¢(1) » : : : » ¢(n¡1) » ¢(n) = ¢
b of full convex subquivers of the quiver Q. b formed by the vertex (0; u). Further, let ¢(1) be the Let ¢(0) be the subquiver of Q b consisting of (0; u) and the vertices (0; s(®)) and the arrows (0; ®) for all subquiver of Q arrows ® of Q with e(®) = u. Clearly ¢(0) » ¢(1) . Then ¢(2) is the full subquiver of b formed by extending ¢(1) by the ends of all arrows in Q b starting from the vertices of Q ¢(1) n ¢(0) . Observe that the vertices and arrows of ¢(2) n ¢(1) are of two types: the vertices (0; e(¯)) and arrows (0; ¯), for all arrows ¯ in Q with (0; s(¯)) in ¢(1) , and the b with (0; e(°)) in ¢(1) . In vertices (¡ 1; s(°)) and arrows (¡ 1; ° 0 ), for all arrows ° of Q b obtained by extending the general, for i ¶ 2, ¢(i+1) is de¯ned as the full subquiver of Q (i) b starting from the vertices ¢(i) n ¢(i¡1) , if i is quiver ¢ by the ends of all arrows of Q b ending in the vertices of ¢(i) n ¢(i¡1) , if i is even. odd, and the sources of all arrows of Q b with sink-source orientation, We will show that ¢ = ¢(n) is a full convex subquiver of Q b Let v be a vertex containing exactly one vertex from each ºQb -orbit of the vertices of Q. b that there exists a walk in of Q di®erent from u. Then it follows from de¯nition of Q b with sink-source orientation and length d(v) connecting (0; u) with a vertex (m; v), Q for some m 2 Z. Hence ¢(d(v)) contains a vertex from the ºQb -orbit of the vertex (0; v). S Suppose now that ¢ = i¸0 ¢(i) contains two di®erent vertices from the ºQb -orbit of a vertex (0; w), w 2 Q0 . We may assume that d(w) µ d(x) for all vertices x 2 Q0 with this property. Then the fact that there are two walks with sink-source orientation in ¢ from two vertices (r; w) and (s; w), r 6= s, to the vertex (0; u) exactly means (see [20, Lemmas
110
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2 and 3]) that Q contains a subquiver of the form b1 u
a1
:::
b2
:::
br
as
w c1
c2
:::
ct
where the numbers of clockwise and counter-clockwise oriented arrows on the cycle are di®erent, a contradiction with the clock condition. This shows in particular that ¢ = S (i) b exactly once. In = ¢(n) and ¢ intersects each ºQb -orbit of the vertices in Q i¸0 ¢ particular, ¢ is a ¯nite quiver. The same argument shows also that a given vertex (m; v) b belongs to exactly one of the di®erences ¢(i) n ¢(i¡1) , 0 µ i µ n. Hence, ¢ does not of Q contain any path of length 2. Clearly then H = K¢ is a connected hereditary algebra with rad(H)2 = 0. Finally, it follows from de¯nition of the quiver ¢, that, for any arrow ® of Q, the set ¢1 of arrows of ¢ intersects each set f(m; ®) j m 2 Zg [ f(m; ®0 ); m 2 Zg exactly once. In fact, if ¢1 contains an arrow (m; ®) (respectively, (m; ®0 )) then ¢1 contains all arrows (m; ¯) (respectively, (m; ¯ 0 )) for all arrows ¯ 2 Q1 with s(®) = s(¯) and e(®) = e(¯). Applying now the construction of the bound quiver presentation of the repetitive algebra of a radical square zero algebra to the algebra H = K¢, we conclude b = ¢, b and ¯nally that Ab ¹= H. b that Q
Acknowledgements
The research was initiated during the visit of the ¯rst, third and fourth named authors in Toru¶n (March 2002), and they thank the Faculty of Mathematics and Computer Science of Nicolaus Copernicus University for the hospitality. The authors acknowledge support from the Polish Research Grant KBN No. 5 P03A 008 21.
References [1] I. Assem, J. Nehring and A. Skowro¶nski, \Domestic trivial extensions of simply connected algebras", Tsukuba J. Math., Vol. 13, (1989), pp. 31{72. [2] I. Assem and A. Skowro¶ nski, \Algebras with cycle-¯nite derived categories", Math. Ann., Vol. 280, (1988), pp. 441{463. [3] I. Assem and A. Skowro¶nski, \On tame repetitive algebras", Fund. Math., Vol. 142, (1993), pp. 59{84. [4] K. Bongartz and P. Gabriel, \Covering spaces in representations theory", Invent. Math., Vol. 65, (1982), pp. 331{378. [5] P. Dowbor and A. Skowro¶ nski, \On Galois coverings of tame algebras", Arch. Math., Vol. 44, (1985), pp. 522{529. [6] P. Dowbor and A. Skowro¶nski, \Galois coverings of representation-in¯nite algebras", Comment. Math. Helv., Vol. 62, (1987), pp. 311{337.
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[7] K. Erdmann, O. Kerner and A. Skowro¶ nski, \Self-injective algebras of wild tilted type", J. Pure Appl. Algebra, Vol. 149, (2000), pp. 127{176. [8] P. Gabriel, \The universal cover of a representation-¯nite algebra" In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer, 1981, pp. 68{105. [9] E.L. Green, \Graphs with relations, coverings and group graded algebras", Trans. Amer. Math. Soc., Vol. 279, (1983), pp. 297{310. [10] D. Happel, \On the derived category of a ¯nite dimensional algebra", Comment. Math. Helv., Vol. 62, (1987), pp. 339{389. [11] D. Happel, \A characterization of hereditary categories with tilting object", Invent. Math., Vol. 144, (2001), pp. 381{398. [12] D. Happel, I. Reiten and S.O. Smal¿, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc., Providence, Rhode Island, 1996. [13] D. Happel, I. Reiten and S.O. Smal¿, \Piecewise hereditary algebras", Arch. Math., Vol. 66, (1996), pp. 182{186. [14] D. Happel, J. Rickard and A. Scho¯eld, \Piecewise hereditary algebras", Bull. London Math. Soc., Vol. 20, (1988), pp. 23{28. [15] D. Hughes and J. WaschbÄ usch, \Trivial extensions of tilted algebras", Proc. London Math. Soc., Vol. 46, (1983), pp. 347{364. [16] H. Lenzing and A. Skowro¶nski, \Sel¯njective algebras of wild canonical type", Colloq. Math., Vol. 96, (2003), pp. 245{275. [17] J. Nehring and A. Skowro¶ nski, \Polynomial growth trivial extensions of simply connected algebras", Fund. Math., Vol. 132, (1989), pp. 117{134. [18] C.M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., Springer, Berlin, 1984. [19] J. SchrÄoer, \On the quiver with relations of a repetitive algebra", Arch. Math., Vol. 72, (1999), pp. 426{432. [20] A. Skowro¶nski, \Generalization of Yamagata’s theorem on trivial extensions", Arch. Math., Vol. 48, (1987), pp. 68{76. [21] A. Skowro¶ nski, \On algebras with ¯nite strong global dimension", Bull. Polish. Acad. Sci., Vol. 35, (1987), pp. 539{547. [22] A. Skowro¶ nski, \Tame quasi-tilted algebras", J. Algebra, Vol. 203, (1998), pp. 470{ 490. [23] K. Yamagata, \On algebras whose trivial extensions are of ¯nite representation type", In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer, 1981, pp. 364{371.
CEJM 2(1) (2004) 112{122
The minimizing of the Nielsen root classes Daciberg L. Gon»calves1 ¤ , Claudemir Aniz2
y
1
Departamento de Matem¶atica - IME-USP, Caixa Postal 66281 - Ag^encia Cidade de S~ao Paulo, 05311-970 - S~ao Paulo - SP { Brasil 2 Universidade Estadual de Mato Grosso do Sul-UEMS, Rua Walter Hubacher s/n, 79750-000/Vila Beatriz Nova Andradina - MS { Brasil
Received 8 August 2003; accepted 5 November 2003 Abstract: Given a map f : X ! Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H -related to the given one for all homotopies H of the map f . We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f ] µ 1, or N R[f ] > 1 and f satis es the Wecken property. Here N R[f ] denotes the Nielsen root number. The condition \f satis es the Wecken property” is known to be equivalent to jdeg(f )j µ N R[f ]=(1 ¡ À (M2 ) ¡ À (M 1 )=(1 ¡ À (M2 )) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f ) µ N R[f ]=(1 ¡ À (M 2 ) ¡ À (M1 )=(1 ¡ À (M2 )). Also we construct, for each integer n ¶ 3, an example of a map f : Kn ! N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time. ® c Central European Science Journals. All rights reserved. Keywords: Nielsen root classes, Absolut degree, Wecken property, closed surfaces, complex MSC (2000): 55M20, 57M12, 20F99
1
Introduction
y
¤
Given a map f : M1 ! M2 consider a Nielsen root class denoted by C. For each homotopy H from f to a map f 0 , there is a well known correspondence between the Nielsen root classes of f and the Nielsen root classes of f 0 . Brooks in [Br] has de¯ned the notion of E-mail:
[email protected] E-mail:
[email protected]
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an inessential root class. In the same spirit we can de¯ne the concept of the minimal number of a root class (see section 2): De¯nition 1.1. Let M R[f; C] be the minimal cardinality among all Nielsen root classes C 0 , where C 0 is a Nielsen class of f 0 , H-related to C, for H a homotopy between f and f 0 . In many relevant cases we have that M R[f; C] is ¯nite. Then the following natural question, related to the question of minimization of number of roots in the homotopy class, arises: Given f , can we deform it to a map f 0 such that the number of points in each Nielsen root class of f 0 is the minimal number of roots of the corresponding Nielsen root class? R. Brooks has provided some positive result, see [Br, Theorem 1]. A consequence of his result is: if M1 is a complex of the same dimension as a manifold N of dimension greater or equal to 3 and all Nielsen classes are inessential, (i.e. the minimal number of roots in the class is zero) then we can deform the map to be root free. This result also gives a motivation for the above more general question and also the similar question for dim(M1 ) > dim(M2 ), which should be more subtle. In this work we consider the case where dimM1 = dimM2 . We divide into two cases, ¯rst when M1 ; M2 are closed surfaces (dimMi = 2), and the second case M1 is a complex of dimension greater than or equal to three and M2 a manifold. Here are our main results. For the case of maps between surfaces: Theorem 1.2. For maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f ] µ 1, or N R[f ] > 1 and f satis¯es the Wecken property. For the case of maps from a complex into a manifold, of the same dimension greater or equal to 3, we observe that if Kn is a manifold as result of [Sc] our question has a positive answer. So, an n-dimensionally connected complex Kn , which is not a manifold, is constructed. Let Pn be the n-dimensional real projective space. Theorem 1.3. For each n ¶ 3 there exist f : Kn ! Pn such that: (1) N R[f ] = 2 (2) The minimal number of a root class is 1 (3) M R[f; y0 ] = minf#(g ¡1 (y0 ))jg homotopic to f g > 2 It follows from [GZ], [BGZ], [GKZ] and [BGKZ1] that if f satis¯es the Wecken property then we can minimize all the Nielsen classes at the same time. Also, in light of [BGKZ2], for some pairs of surfaces Sh ; Sg all maps satisfy the Wecken property. Therefore a deformation to minimize all the classes at the same time is always possible for such pairs of surfaces. On the other hand, when the target is the torus, it is always possible to decide when the deformation is possible or not. Nielsen theory has been used to study several di®erent subjects, for example: ¯xed point theory, root theory, coincidence theory and intersection theory. The type of problem
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we consider here can be studied in any one of subjects above and the problem is related to the minimal problem in each one of the cases. This paper is organized into three sections. In section 2 we develop some generalities. Then we show that the minimal number of a Nielsen root class of a map f is independent of the class. This is Proposition 2.3. Also for maps between surfaces, the answer of our question depends only on Â(M1 ), Â(M2 ) and the index [¼1 (Sg ); f# (¼1 (S h )]. This is Theorem 2.5. In section 3 we compute the minimal number of points in a Nielsen root class for the case of maps between surfaces. These are Theorems 3.2 and 3.3. Then we show our main result for maps between surfaces which is Theorem 1.2. In section 4 we construct a family of examples fn , one for each n > 2, of maps from a complex Kn to an n-dimensional manifold N (n ¶ 3) which cannot be deformed to a map such that the number of points in each Nielsen root class is minimal. This result is Theorem 4.2. The authors would like to thank the referee for has careful reading and suggestions, improving the presentation of this work.
2
Generalities about the problem
Let f : X ! Y be an arbitrary map and y0 2 Y a given base point. We consider roots with respect to this base point, i.e. the preimage, f ¡1 (y0 ), of the base point y0 . In this section we recall some basic de¯nitions, and develop some general facts for later use. Recall that two roots x1 ; x2 2 f ¡1 (y0 ) are said to be equivalent if there is a path ¸ : [0; 1] ! X such that the class of the loop f (¸) is nullhomotopic. This is equivalent to say that for every path ¸ from x1 to x2 the homotopy class of the loop f (¸) belongs to the image f# (¼1 (X; x0 )) » ¼1 (Y; y0 ). The set of equivalence classes under this relation are called the Nielsen root classes of f . Also a homotopy H between two maps f and f 0 provides a one to one correspondence between the Nielsen root classes of f and the Nielsen root classes of f 0 . We say that two such classes under this correspondence are H related. See [Ki] for more details. Following Brooks [Br] we have the de¯nition. De¯nition 2.1. A Nielsen root class C of a map f is inessential if it is H¡ related to an empty Nielsen root class, i.e. there is a homotopy H of f such that the class C is H¡ related to an empty Nielsen root class of f 0 = H( ; 1). Otherwise it is called essential. In the same spirit of the de¯nition above, for a Nielsen root class C of f we de¯ne: De¯nition 2.2. Let M R[f; C] be the minimal cardinality among all Nielsen root classes C 0 , where C 0 is a Nielsen class of f 0 H -related to C, for H a homotopy between f and f 0 . Our main question is to know when is possible to deform a map f to some map f 0 with the property that all Nielsen root classes of f 0 have the minimal number of points. We consider only the case where the target is a manifold. Under the condition that the target is a manifold, we can show that this number
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M R[f; C] is independent of the Nielsen root class. This is relevant to our question. Proposition 2.3. If Y is a manifold then the number M R[f; C] is independent of the class C. Proof. We will show that any two root classes are H-related and the result will follow from [Br]. Consider the covering p : Y~ ! Y which corresponds to the subgroup f# (¼1 (X; x0 )) » ¼1 (Y; y0 ). Let f~ : X ! Y~ be a lift of f . Let us ¯x two points y~0 ; y~1 over y0 . Since Y~ is a manifold, there is a disk D n whose interior intersects p¡1 (y0 ) in these two points and a homeomorphism h : Y~ ! Y~ with the following properties: a) The map h is the identity in the complement of the interior of D n , b) There is a homotopy H between the map h and the identity which is relative to the complement of the interior of D n and c) h(~ y1 ) = (~ y0 ). Suppose that the Nielsen root class C of f corresponds to the preimage of y~0 and C1 is the Nielsen root class which corresponds to the preimage of the point y~1 . The composite H ¯ (f~ £ id) : X £ [0; 1] ! Y~ is a homotopy between f~ and h ¯ f~. From [Br, Lemma 1 and Lemma 2], the root classes C and (h ¯ f~)¡1 (~ y0 ) = C1 are ~ (p ¯ H ¯ (f £ id))-related and the result follows. h If we have a map f between surfaces, it follows from Kneser [Kn] (see also [Ep]) that, if deg(f ) (degree of f for the orientable case) or A(f ) (absolute degree for the nonorientable case) is zero, then the map can be deformed to be root free. So, it su±ces to consider the cases where the degree, or the absolute degree, of the map is non zero. Given X; Y two spaces, consider the group Homeo(X )£Homeo(Y ) with the operation given by (Ã1 ; Á1 ) ¤ (Ã2 ; Á2 ) = (Ã2 ¯ Ã1 ; Á1 ¯ Á2 ). De¯ne an action of this group on the set Y X , the function space of all continuous map from X to Y , by (Ã; Á):f = Á ¯ f ¯ Ã. We show that our question depends only on the orbits given by this action. Namely: Lemma 2.4. A map f : X ! Y has the property that it can be deformed to a map f 0 such that all the root classes have the minimal number of points if and only if the same is true for Á ¯ f ¯ à where à 2 Homeo(X ); Á 2 Homeo(Y ). Furthermore, the minimal number of a Nielsen root class (as de¯ned in Proposition 2.3) is the same for any two maps in the same orbit. Proof. Let H be a homotopy connecting f to f 0 such that f 0 has the property that all the root classes have the minimal number of points. Then we claim that f 0 ¯ à also has the same property. For, given a root class C of f 0 , the correspondence C ! à ¡1 (C) is a bijection between the root classes of f 0 and f 0 ¯ Ã; so, C, à ¡1 (C) have the same cardinality. Also, the homotopy H of f provides a homotopy H ¯ (à £ I) of f ¯ Ã, and the H -related classes of f corresponds to the (H ¯ (à £ I))¡ related classes of f ¯ à under the homeomorphism Ã. By contradiction, suppose that one Nielsen root class C2 of f 0 ¯ à (which has the cardinality of a root class of f 0 ) does not have the minimal number of points. Let us consider a homotopy such that the H 0 ¡ related class to C2 has less points. But this implies, by the argument above, that one of the classes of a deformation of f 0
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would have less points than a root class of f 0 , which is a contradiction. Therefore, this implies the Lemma for the maps f and f ¯ Ã. By a similar argument one can show the Lemma for f and Á ¯ f . We leave the details to the reader. Therefore, the result follows. h
For a map f : N1 ! N2 between surfaces denote by deg(f ) the degree of f in case both surfaces are orientable, and by A(f ) the absolute degree of f otherwise (see [BSc] or [Ep]). As an application of Lemma 2.4 and results of Gabai-Kazez [GK1], [GK2] we have: Theorem 2.5. Let f; g : N1 ! N2 be two maps between orientable closed surfaces (at least one of them is nonorientable) such that jdeg(f )j = jdeg(g)j (A(f ) = A(g)) and [¼1 (N2 ) : f# (¼1 (N 1 ))] = [¼1 (N2 ) : g# (¼1 (N1 ))]. The map f has the property that it can be deformed to a map f 0 such that all the root classes have the minimal number of points if and only if the same is true for g. Proof. First let us observe that if deg(f ) = 0 then the claim is true since by [Kn] if a map have degree zero it can be deformed to a root free. So let deg(f ) 6= 0. We consider ¯rst the orientable case. Let f; g : N1 ! N2 be two maps such that jdeg(f )j = jdeg(g)j. Since deg(f ) is non zero, it follows the index [¼1 (N2 ) : f# (¼1 (N 1 ))] = [¼1 (N2 ) : g# (¼1 (N1 ))] is ¯nite and denoted by `. Consider the ¯nite coverings N20 , N200 of N2 with base points s1 ; s2 , which correspond to the subgroups f# (¼1 (N1 )), g# (¼1 (N1 )), respectively. These coverings have the same number of sheets `, therefore they are homeomorphic surfaces and let Á : N20 ! N200 be a homeomorphism between them. Consider the lifts f~ : N1 ! N20 , g~ : N1 ! N200 (base point preserving) of f , g, respectively, and the map Á ¯ f . By GabaiKazez [GK1, Corollary 9.4] we have that there is a homeomorphism à : N1 ! N1 such that g~ ¯ à = Á ¯ f~. Therefore p2 ¯ g~ ¯ à = p2 ¯ Á ¯ f~ which implies g ¯ à = p2 ¯ Á ¯ f~. By the Lemma 2.4, it follows that if the result is true for g then it is true for g ¯ Ã, and consequently for p2 ¯ Á ¯ f~. Now it su±ces to compare the latter map with f . For, they have lifts Á ¯ f~, f~, respectively. The root classes of f correspond to the preimage of points in p¡1 2 (y0 ). A straightforward argument, similar to the one used in the proof of the Lemma 2.3, shows that if f~ can be deformed such that the preimage of each point of p¡1 2 (y0 ) contains the same number of points and is the minimal number of a root class, then the same is true for Á ¯ f~. Then the result follows. The nonorientable case is similar, where the results from Gabai-Kazez in [GK1] are replaced by the one’s in [GK2]. h
3
Maps between closed surfaces
In this section we show the results for maps between surfaces. Denote by N R[f ] the Nielsen root number of f , which is the number of essential Nielsen root classes. Our main result is:
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Theorem 3.1. For maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f ] µ 1, or N R[f ] > 1 and f satis¯es the Wecken property. Proof. The "if part". This is the easy part. Of course, if N R[f ] = 1 then there is only one root class and there is nothing to be proved. If N R[f ] = 0 then all classes are inessential and f satis¯es the (root) Wecken property. This follows from Kneser [Kn] which shows that f can deformed to a root free map. So, let N R[f ] > 1 and f satis¯es the Wecken property. This means that f can be deformed to a map g which has N R[f ] roots. This implies that each Nielsen root classe of g has exactly one point, since all classes are essential. Therefore the result follows. The "only if part". This is a direct consequence of Theorems 3.2 and 3.3 stated and proved below. h The result above is based in the calculation of the minimal number of a root class of a map between surfaces. This calculation seems interesting in its own right. We will answer completely this question. First we consider the orientable case. Theorem 3.2. Let f : Sh ! Sg be a map between orientable surfaces of genus h and g, respectively, ` = N R[f ] and jdeg(f )j = `d. Then, the minimal number of a root class is given by: a) If deg(f ) = 0, it is zero. b) If deg(f ) 6= 0 and ` = 1, it is maxf1,jdeg(f )j ¡ (jdeg(f )jÂ(Sg ) ¡ Â(Sh ))g. c) If 0 < jdeg(f )j µ ` ¡ Â(Sh )=(1 ¡ Â(Sg )), it is one. d) If jdeg(f )j > ` ¡ Â(Sh )=(1 ¡ Â(Sg )), ` > 1 and (jdeg(f )jÂ(Sg ) ¡ Â(Sh )) ¶ d ¡ 1, it is one. e) If jdeg(f )j > ` ¡ Â(Sh )=(1 ¡ Â(Sg )), ` > 1 and 0 µ (jdeg(f )jÂ(Sg ) ¡ Â(Sh )) < d ¡ 1, it is d ¡ (jdeg(f )jÂ(Sg ) ¡ Â(Sh )).
Proof. First we observe that the Theorem cover all maps. As result of the cases a) and b) we can consider deg(f ) 6= 0 and ` > 1. The case c) consider all maps such that 0 < jdeg(f )j µ ` ¡ Â(Sh )=(1 ¡ Â(Sg )) (independent of `). The cases d) and e) consider the remain maps where ` > 1 and jdeg(f )j > ` ¡ Â(Sh )=(1 ¡ Â(Sg )). Part a) follows from Kneser [Kn] since the map can be deformed to be root free. Part b) follows from [BGZ, Theorem 3.3]. Part c) It follows from [GZ, Theorem 3.6] that f satis¯es the Wecken property. Therefore, it can be deformed to have exactly ` roots, which is the number of root classes. Since all classes are essential it follows that each root classes has exactly one point and the result follows. Part d) Let us consider a lift f~ : Sh ! Sg0 of f , where p : Sg0 ! Sg is the cover which corresponds to the subgroup f# (¼1 (Sh )). Because ` > 1, p¡1 (y0 ) = (y1 ; :::; y` ) has more than one point. Since jdeg(f )jÂ(Sg ) ¡ Â(Sh )) ¶ d ¡ 1 it follows from [BGZ,
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Theorem 3.3] that the minimal number M R of roots satis¯es ` < M R µ (` ¡ 1)d + 1. Therefore we have ` µ M R ¡ 1 µ (` ¡ 1)d. So we can write M R as a sum of ` positive numbers where the ¯rst summand is 1. At the extreme case M R ¡ 1 = (` ¡ 1)d the other summands would be d. These summands de¯ne a partition of M R of the form m1 ; :::; m` where 0 < mi µ d and m1 = 1. There is a primitive branched covering of degree d having (y1 ; :::; y` ) as branching points and over each branching point yi has mi points. This follows from [BGZ, Proposition 5.8 ], for the case where g > 1 and, from [BGKZ3, Theorem 1.2 ] or [BGKZ4, Theorem 3.3], in the case the target is the torus. So, the map which is the composite of the constructed branched covering with the projection p is a map such that the minimal number of points of a root class is one. From Lemma 2.4, the result follows. Part e) The proof is similar to the previous case and we mention only the point where we have to adapt the argument. In this case as result of the hypothesis we have (` ¡ 1)d + 1 < M R < `d. So, we can consider a partition of this number as ` summands m1 ; :::; m` where mi = d for i = 2; :::; ` and m1 the number claimed in the Theorem. Observe that this is a partition which minimize the value of m1 . Then we continue the proof as in the previous case and the result follows. h Now we move to the case where f : M1 ! M 2 is a map between surfaces not necessarily orientable. Then we have a result similar to the previous Theorem:
Theorem 3.3. Let f : M1 ! M 2 be a map between two surfaces with nonnegative Euler characteristic, ` = N R[f ] and A(f ) = `d. Then, the minimal number of a root class is given by: a) If A(f ) = 0, it is zero. b) If A(f ) 6= 0 and ` = 1, it is maxf1, A(f ) ¡ (A(f )Â(M 2 ) ¡ Â(M1 ))g. c) If 0 < A(f ) µ ` ¡ Â(M1 )=(1 ¡ Â(M2 )), it is one. d) If A(f ) > ` ¡ Â(M 1 )=(1 ¡ Â(M2 )), ` > 1 and (A(f )Â(M2 ) ¡ Â(M1 )) ¶ d ¡ 1, it is one. e) If A(f ) > ` ¡ Â(M1 )=(1 ¡ Â(M2 )), ` > 1 and 0 µ (A(f )Â(M2 ) ¡ Â(M 1 )) < d ¡ 1, it is d ¡ (A(f )Â(M2 ) ¡ Â(M1 )).
Proof. Similar to the proof of Theorem 3.2. First we observe that the Theorem cover all maps. Part a) follows from Kneser [Kn] since the map can be deformed to be root free. Part b) follows from [BGKZ1, Theorem 1.1 ]. Part c) It follows from [GKZ,Theorem 4.6] that f satis¯es the Wecken property. Therefore, it can be deformed to have exactly ` roots, which is the number of root classes. Since all classes are essential it follows that each root classes has exactly one point and the result follows.
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Part d) Let us consider a lift f~ : M 1 ! M 20 of f , where p : M 20 ! M 2 is the cover which corresponds to the subgroup f# (¼1 (M1 )). Because ` > 1, p¡1 (y0 ) = (y1 ; :::; y` ) has more than one point. Since A(f )Â(M 2 ) ¡ Â(M1 )) ¶ d ¡ 1 it follows from [BGKZ1, Theorem 1.1] that the minimal number M R of roots satis¯es ` < M R µ (` ¡ 1)d + 1. Therefore we have ` µ M R ¡ 1 µ (` ¡ 1)d. So we can write M R as a sum of ` positive numbers where the ¯rst summand is 1. At the extreme case M R ¡ 1 = (` ¡ 1)d the other summands would be d. These summands de¯ne a partition of M R of the form m1 ; :::; m` where 0 < mi µ d and m1 = 1. There is a primitive branched covering of degree d having (y1 ; :::; y` ) as branching points and over each branching point yi has mi points. If M 2 is orientable, this is the case in the proof of Theorem 3.2. For M2 nonorientable, the existence of a primitive branched covering follows from [BGKZ1, Theorem 3.3 ], for the case where M2 has Euler characteristica negative and, from [BGKZ3, Theorem 1.2 ] or [BGKZ4, Theorem 3.3], in the case the target is the Klein bottle. So, the map which is the composite of the constructed branched covering with the projection p is a map such that the minimal number of points of a root class is one. From Lemma 2.4, the result follows. Part e) The proof is similar to the previous case and we mention only the point where we have to adapt the argument. In this case as result of the hypothesis we have (` ¡ 1)d + 1 < M R < `d. So, we can consider a partition of this number as ` summands m1 ; :::; m` where mi = d for i = 2; :::; ` and m1 the number claimed in the Theorem. Observe that this is a partition which minimize the value of m1 . Then we continue the proof as in the previous case and the result follows. h
3.1 Remark It would be interesting to explore the consequences of the above results for quadratic equations in free group. See [GZ], [BGZ] and [GKZ].
4
Maps from a complex into a manifold
In this section we construct, for each n ¶ 3, an n-dimensional complex Kn and a map f : Kn ! M which has the following property: The map f cannot be deformed to a map g such that the number of points of each root class of g is the minimum, as de¯ned in De¯nition 2.2. The complex Kn has the property that it is n-dimensionally connected, which is a property that holds for a manifold. This type of map was constructed for the ¯rst time, at least for a complex K3 of dimension 3, in [An]. Let us observe that it is not possible to construct such examples if Kn is a manifold, as result of H. Schirmer’s work [Sc]. On the otherhand if we relax the condition that Kn is n-dimensionally connected, for example allowing Kn to have cut points (i.e. the link of a vertex is not connected), then one can construct one example (as pointed out by the referee) as follows: Take the wedge of k-copies of S n and map each of them into P n by the natural covering map. If k is at least 2 it is not di±cult to show that M R[f; C] is 1,
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the Nielsen root number is 2 but the minimal number of roots is k + 1. Denote by ¢n =< v0 ; : : : ; vn > the n-dimensional simplex. De¯ne Kn = [n+1 i=1 Si , n n+1 where each Si is a copy of the sphere S , which we regard as the boundary of ¢ , i.e. Si = @ < vi0 ; : : : ; vi(n+1) > . Now we identify the faces of these complexes as follows: 1. S1 \ S2 v10 = v20 ; v11 = v21 ; : : : ; v1(n¡1) = v2(n¡1) 2. S2 \ S3 v21 = v31 ; v22 = v32 ; : : : ; v2n = v3n 3. S3 \ S4 v32 = v42 ; v33 = v43 ; : : : ; v3(n+1) = v4(n+1) 4. S4 \ S5 v43 = v53 ; v44 = v54 ; : : : ; v4(n+1) = v5(n+1) ; v40 = v50 ................ Proposition 4.1. The complex Kn de¯ned as above has the following properties: (1) Every maximal simplex is n-dimensional. (2) It is simply connected, i.e. ¼1 (Kn ) = 0. (3) It is n-dimensionally connected, i.e. any two n-dimensional simpleces can be joined by a sequence of n¡ simpleces ¢i such that the intersection of two consecutive ones is a (n ¡ 1)¡ simplex. Proof. Part 1. Given ¾ » Kn a p-simplex with p µ n ¡ 1, from the de¯nition of Kn it follows that ¾ belongs to Si , for some i = 1; : : : ; n + 1. Since Si is a manifold of dimension n, it implies that ¾ » ¿ , where ¿ is an n-simplex. Part 2. De¯ne Kp = Kp¡1 [ Sp+1, for p µ n, where K0 = S1 . We show by induction that Kp is simply connected, for all p µ n. The statement is true for p = 0, since the sphere S1 of dimension n ¶ 3 is simply connected. Suppose by induction hypothesis we have that Kp is simply connected. Since Kp+1 = Kp [ Sp+2 , and Kp \ Sp+2 is path connected, as result of the identi¯cations given above, we can use the Van Kampen Theorem (see [Ar], pg. 138) to compute ¼1 (Kp+1 ). Since the sphere Sp+2 and Kp are simply connected, the latter one as result of the induction hypothesis, we obtain that Kp+1 is also simply connected. So, the result follows. Part 3. Let ¾; ¿ » Kn , be two n-simpleces. Without loss of generality we can assume that ¾ » Sk and ¿ » Sp for some k and p such that 1 µ k µ p µ n + 1. For k = p the result is true, since the sphere Sk is a manifold and it has the property that we want to prove. Suppose that p = k + 1. Since Sk \ Sk+1 is a (n ¡ 1)-simplex s(n¡1) , let us choose two n-simplexes ¾1 , and ¿1 , belonging to Sk , and Sk+1, respectively, such that s(n¡1) is in the boundary of both n-simplexes. Because Sk ; Sk+1 are manifolds we can connect ¾ to ¾1 , and ¿1 to ¿ , therefore we can connect ¾ to ¿ , and the result follows. To ¯nish the proof we can argue by induction on the di®erence p ¡ k. Assuming the result is true for p ¡ k = l, let us show for l + 1. Given ¾ and ¿ as above, let ° be an arbitrary n-simplex
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in Sk+1 . From above we can connect ¾ to °, and by induction hypothesis we can connect ° to ¿ . Therefore we can connect ¾ to ¿ and the result follows. h Now, we show the main result of this section. Let Pn be the n-dimensional real projective space. Theorem 4.2. For each n ¶ 3 there exists f : Kn ! Pn such that: (1) N R[f ] = 2. (2) The minimal number of a root class is 1. (3) M R[f; y0 ] > 2. Proof. Two simplicial complexes K, L are homeomorphic if there is a bijection Á between the set of the vertices of K and L such that fv1 ; v2 ; : : : ; vs g is a simplex in K if and only if fÁ(v1 ); Á(v2 ); : : : ; Á(vs )g is a simplex in L (see [Ar], pg 128). Using the fact above, we can construct, for each i = 2; : : : ; n + 1, homeomorphisms hi : Si ! Si¡1 such that hi jSi \ Si¡1 is the identity. In order to de¯ne our map f : Kn ! Pn let f~1 : S1 ! S n be any homeomorphism from S 1 to the sphere S n . De¯ne f~2 = f~1 ¯ h2 : S2 ! S1 ! S n and observe that f~1 (x) = f~2 (x) for x 2 S1 \ S2 . So, from f~1 ; f~2 we obtain a well de¯ned map S1 [ S2 ! Pn which extends both maps. Inductively de¯ne f~i = fi¡1 ¯ hi : Si ! S n , which is a homeomorphism with the property f~i (x) = f~i¡1 (x), for all x 2 Si¡1 \ Si . So de¯ne f~ : Kn ! S n to be the map such that f~jSi = f~i . Let f = p ¯ f~ : Kn ! Pn be the two-fold covering map. Sn ~ f~1 ;:::; f~n+ f=(
Kn
1)
f
p
Pn
h
Part 1. Let y0 = f (v1(n¡1) ) and p¡1 (y0 ) = f~ y0 ; ¡ y~0 g. Since every map homotopic to ~ f is surjective, it follows that N R[f ] = 2 (see [Br], lemma 1 and lemma 2) . Part 2. We have v1(n¡1) 2 f~¡1 (~ y0 ) or v1(n¡1) 2 f~¡1 (¡ y~0 ). Suppose that v1(n¡1) belongs to f~¡1 (~ y0 ), so f~¡1 (~ y0 ) = fv1(n¡1) g. For, v1(n¡1) = \n+1 i=1 S i , otherwise we could ¯nd a point ~ v0 2 Kn with the property f (v0 ) = y~0 . But Kn = [n+1 i=1 Si and so v0 belong to some ~ ~ Si and f (v0 ) = f (v1(n¡1) ) , which contradicts the fact that the restriction of f~ to Si is a homeomorphism. Therefore, this class has only one point and it follows that the minimum number of a root class is 1. From Proposition 2.3 this is the case for all the other root classes and the result follows. Part 3. Since f restricted to each sphere has at least two roots, and the intersection of all the spheres has only one point, we have that M R[f; y0 ] > 2 and the result follows.
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Acknowledgements This work is part of the Projeto-Tem¶atico Topologia Alg¶ebrica, Geom¶etrica e DiferencialFapesp No 2000/05385-8.
References [An] C. Aniz: Ra¶³zes de fun»c~oes de um Complexo em uma variedade, Phd Thesis, S~ao Carlos-SP-Brazil, 2002. [Ar] M.A. Armstrong: Basic Topology. Undergraduate Texts in Mathematics, SpringerVerlag, New York Inc., 1983. [BGKZ1] S. Bogatyi, D.L. Gon»calves, E.A. Kudryavtseva and H. Zieschang: \Minimal number of roots of surface mappings\, Matem. Zametki, (to appear). [BGKZ2] S. Bogatyi, D.L. Gon»calves, E.A. Kudryavtseva and H. Zieschang: \On the Wecken property for the root problem of mappings between surfaces\, Moscow Math. Journal, (to appear). [BGKZ3] S. Bogatyi, D.L. Gon»calves, E.A. Kudryavtseva and H. Zieschang: \Realization of primitive branched coverings over surfaces following the Hurwitz approach\, Central Europ. J. of Mathematics, Vol. 2, (2003), pp. 184{197. [BGKZ4] S. Bogatyi, D.L. Gon»calves, E.A. Kudryavtseva and H. Zieschang: "Realization of primitive branched coverings over closed surfaces", Kluwer, Preprint, (to appear). [BGZ] S. Bogatyi, D.L. Gon»calves and H. Zieschang: \The minimal number of roots of surface mappings and quadratic equations in free products\, Math. Z., Vol. 236, (2001), pp. 419{452. [Br] R.B.S. Brooks: \On the sharpness of the ¢2 and ¢1 Nielsen numbers\, J. Reine Angew. Math., Vol. 259, (1973), pp. 101{109. [BSc] R. Brown and H. Schirmer: \Nielsen root theory and Hopf degree theory\, Paci¯c J. Math., Vol. 198(1), (2001), pp. 49{80. [Ep] D.B.A. Epstein: \The degree of a map\, Proc. London Math. Soc., Vol. 3(16), (1966), pp. 369{383. [GZ1] D. Gabai and W.H. Kazez: \The classi¯cation of maps of surfaces\, Invent. Math., Vol. 90, (1987), pp. 219{242. [GZ2] D. Gabai and W.H. Kazez: \The classi¯cation of maps of nonorientable surfaces\, Math. Ann., Vol. 281, (1988), pp. 687{702. [GKZ] D.L. Gon»calves, E.A. Kudryavtseva and H. Zieschang: \Roots of mappings on nonorientable surfaces and equations in free groups\, Manuscr. Math., Vol. 107, (2002), pp. 311{341. [GZ] D.L. Gon»calves and H. Zieschang: \Equations in free groups and coincidence of mappings on surfaces\, Math. Z., Vol. 237, (2001), pp. 1{29. [Kn] H. Kneser: \Die kleinste Bedeckungszahl innerhalb einer Klasse von FlÄachen abbildungen\, Math. Ann., Vol. 103, (1930), pp. 347{358. [Ki] T.H. Kiang: The theory of ¯xed point classes, Springer-Verlag, Berlin-HeidelbergNew York, 1989. [Sc1] H. Schirmer: \ Mindestzahlen von Koinzidenzpunkten\, J. Reine Angew. Math., Vol. 194(1-4), (1955), pp. 21{39.
CEJM 2(1) 2004 123{142
Cartan matrices of sel¯njective algebras of tubular type Jerzy BiaÃlkowski
¤
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toru¶ n, Poland
Received 4 August 2003; accepted 23 September 2003 Abstract: The Cartan matrix of a nite dimensional algebra A is an important combinatorial invariant re®ecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of nite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame sel njective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class b of sel njective algebras of tubular type, that is, the orbit algebras B=G of the repetitive algebras b B of tubular algebras B with respect to the actions of admissible groups G of automorphisms of b The aim of the paper is to describe the determinants of the Cartan matrices of sel njective B. algebras of tubular type and derive some consequences. ® c Central European Science Journals. All rights reserved. Keywords: Cartan matrix, determinant, sel¯njective algebra, repetitive algebra, tubular algebra MSC (2000): Primary: 15A15, 16D50, 16G60; Secondary 16G20, 16G70
1
Introduction and the main results
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Throughout the paper K will denote a ¯xed algebraically closed ¯eld. By an algebra we mean a ¯nite dimensional K-algebra with an identity, which we shall assume (without loss of generality) to be basic and connected. For an algebra A, we denote by mod A the category of ¯nite dimensional right A-modules and by D the standard duality HomK (¡ ; K) on mod A. By Drozd’s remarkable Tame and Wild Theorem [6] the class of algebras may be divided into two disjoint classes. One class consists of tame algebras for which the inE-mail:
[email protected]
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decomposable modules occur in each dimension d, in a ¯nite number of discrete and a ¯nite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all ¯nite dimensional K-algebras. Accordingly we may realistically hope to classify the indecomposable ¯nite dimensional modules only for tame algebras. An algebra A is called sel¯njective if A ¹= D(A) in mod A, that is, the projective A-modules are injective. Further, A is called symmetric if A and D(A) are isomorphic as A-bimodules. For a sel¯njective algebra A, we denote by ¡sA the stable AuslanderReiten quiver of A, obtained from the Auslander-Reiten quiver ¡A of A by removing all projective modules and arrows attached to them. A component of ¡sA of the form ZA1 =(¿ r ), r ¶ 1, is called a stable tube of rank r. It is known [12] that a component in ¡sA is a stable tube (of rank r) if and only if it consists of indecomposable periodic modules (of period r) with respect to the action of the Auslander-Reiten translation ¿A = D Tr. We also note that ¿A = 2A ¯ N A , where A is the Heller syzygy operator and N A : mod A ! mod A is an equivalence induced by the Nakayama automorphism ºA of A. In particular, ¿A = 2A if A is symmetric. Recall also that the Cartan matrix CA of A is the matrix (dim K HomA (Pi ; Pj ))1·i;j·n for a complete family P1 ; : : : ; Pn of pairwise nonisomorphic indecomposable projective A-modules. An important class of sel¯njective b b is the repetitive algebra [14] algebras is formed by the algebras of the form B=G where B (locally ¯nite dimensional, without identity)
b= B
M
(Bm © Qm )
m2Z
b of an algebra B, where Bm = B and Qm = D(B) for all m 2 Z, the multiplication in B is de¯ned by (am ; fm )m ¢ (bm ; gm )m = (am bm ; am gm + fm bm¡1 )m2Z b for am ; bm 2 Bm , fm ; gm 2 Qm , and G is an admissible group of K-automorphisms of B. b!B b is the Nakayama automorphism of B b given by the identity In particular, if ºBb : B shifts Bm ! Bm+1 and Qm ! Qm+1 , then the in¯nite cyclic group (ºBb ) generated by ºBb b b ) is the trivial extension T (B) = B n D(B) of B by D(B), and is admissible and B=(º B is a symmetric algebra. Following [17] by a tubular algebra we mean a tubular extension (equivalently, coextension) B of a tame concealed algebra C of tubular type (2; 2; 2; 2), (3; 3; 3), (2; 4; 4), or (2; 3; 6). Then the rank n of the Grothendieck group K0 (B) of B is equal to 6, 8, 9, or 10, respectively. Let B be a tubular algebra and e1 ; : : : ; en a complete set of primitive orthogonal idempotents of B such that 1 = e1 + ¢ ¢ ¢ + en . Then we have the canonical set E = fem;i j 1 µ i µ n; m 2 Zg of primitive orthogonal idempotents of the repetitive algebra b such that em;1 + ¢ ¢ ¢ + em;n is the identity of the diagonal algebra Bm = B of B. b B b is called admissible if G acts freely A group G of automorphisms of the K -algebra B b on the set E and has ¯nitely many orbits. Then the orbit algebra B=G is de¯ned [11] and is a (¯nite dimensional) sel¯njective algebra, called a sel¯njective algebra of tubular
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b on the set E is given type [3]. The action of the Nakayama automorphism ºBb of B by ºBb (em;i ) = em+1;i for (m; i) 2 Z £ f1; : : : ; ng, and the in¯nite cyclic group (ºBb ) is b is said to be rigid if for any (m; i) 2 Z £ f1; : : : ; ng admissible. An automorphism % of B there exists j 2 f1; : : : ; ng such that %(em;i ) = em;j . Following [18] a tubular algebra b such that ’d = %º b for B is said to be exceptional if there is an automorphism ’ of B B d b ¶ some d 2 and a rigid automorphism % of B. Moreover, if ’ = ºBb then ’ is called b is said to be positive if, for each a d-root of ºBb . Further, an automorphism à of B (m; i) 2 Z £ f1; : : : ; ng, we have Ã(em;i ) = ep;j for some p ¶ m and j 2 f1; : : : ; ng. Hence b is a shift of B b in the same direction as º b . Finally, a a positive automorphism of B B b positive but not rigid automorphism of B is said to be strictly positive. Let B be a tubular algebra and e1 ; : : : ; en a complete set of its primitive orthogonal idempotents. An automorphism % of B is said to be rigid if for any i 2 f1; : : : ; ng there exists j 2 f1; : : : ; ng such that %(ei ) = ej . Clearly, each rigid automorphism of b Similarly, we de¯ne the rigid B corresponds to exactly one rigid automorphism of B. automorphism of a tame concealed algebra. Moreover, if B is a tubular extension of a tame concealed algebra A then any rigid automorphism % of B induces exactly one rigid automorphism of A. We are concerned with the properties of tame sel¯njective algebras whose stable Auslander-Reiten quiver consists only of stable tubes. A large class of such algebras is provided by the sel¯njective algebras of tubular type. This is the class of all nondomestic polynomial growth algebras having simply connected Galois coverings [18]. Moreover, it has been recently shown [3] that a sel¯njective algebra A is of tubular type if and only if A is tame, admits a simply connected Galois covering, and ¡sA consists only of stable tubes. On the other hand, in the process of classifying tame blocks of group algebras of ¯nite groups, K. Erdmann discovered various families of tame symmetric algebras (of quaternion type) having at most three simple modules, nonsingular Cartan matrix, and the stable Auslander-Reiten quiver consisting of stable tubes of ranks at most 2, but only very few of them admit simply connected Galois coverings (see [7], [8], [9]). The program of a classi¯cation of all sel¯njective algebras of tubular type, initiated in [13], [16] and [18], has been recently completed in [3], [4] and [15]. Moreover, all sel¯njective algebras socle equivalent to the sel¯njective algebras of tubular type have been described in [5]. It is conjectured by A. Skowro¶ nski that every tame sel¯njective algebra whose stable Auslander-Reiten quiver consists only of stable tubes is of quaternion type or socle equivalent to a sel¯njective algebra of tubular type. By a classical Brauer’s theorem the determinant of the Cartan matrix of the group algebra KG of a ¯nite group G over a ¯eld K of characteristic p > 0 is a power of p (see [2, (5.7.2)]). In particular, all blocks of group algebras have nonsingular Cartan matrices. On the other hand, the trivial extensions T (B) of all tubular algebras have singular Cartan matrices [4]. The aim of this paper is to describe the determinants of the Cartan matrices of arbitrary sel¯njective algebras of tubular type. If has been proved in [18] that every sel¯njective algebra ¤ of tubular type is of the b form ¤ = B=G, where B is a tubular algebra and G is an in¯nite cyclic group generated
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b For a tubular algebra B and a strictly by a strictly positive automorphism à of B. b we de¯ne (in Section 2) a natural number d (B; Ã) 2 positive automorphism à of B, f0; 4; 6; 12; 16; 256; 1024g. The main result of the paper is the following theorem. b and Theorem. Let B be a tubular algebra, à a strictly positive automorphism of B, b A = B=(Ã) the associated sel¯njective algebra of tubular type. Then d (B; Ã) is the determinant of the Cartan matrix CA of A. As a consequence of the theorem we obtain the following periodic behaviour of the determinants of the Cartan matrices of sel¯njective algebras of tubular type.
b Corollary 1. Let B be a tubular algebra and à a strictly positive automorphism of B. 6 b b Then the determinants of the Cartan matrices of the algebras B=(Ã) and B=(ú b ) are B equal. The following proposition (compare with [10, Corollary 8.3] in the case when ½ is the identity) shows that there are many sel¯njective algebras of tubular type having singular Cartan matrices.
b Then the Proposition 2. Let B be a tubular algebra and % a rigid automorphism of B. m b Cartan matrices of the algebras B=(%º b ), m ¶ 1, are singular. B
In the classi¯cation of sel¯njective algebras of tubular type a distinguished role is played by the exceptional tubular algebras [18], that is, the tubular algebra B for which b such that ’m = %º b for some rigid there exists a strictly positive automorphism ’ of B B b and m ¶ 2. For an exceptional tubular algebra B we de¯ne pB to automorphism % of B be: 2 if B is of tubular type (2; 2; 2; 2) or (2; 4; 4), 3 if B is of tubular type (3; 3; 3), and 6 if B is of tubular type (2; 3; 6). The following corollary shows that there are also many sel¯njective algebras of tubular type having nonsingular Cartan matrices. Corollary 3. Let B be an exceptional tubular algebra. Then the following statements hold: b such that the Cartan matrix (i) There exists a strictly positive automorphism à of B b of the algebra B=(Ã) is nonsingular. (ii) If ’ is a nontrivial root of the Nakayama automorphism ºBb then the Cartan b matrix of the algebra B=(’) is nonsingular. b the determinants of the Cartan (iii) For any strictly positive automorphism à of B, pB b b matrices of the algebras B=(Ã) and B=(úBb ) are equal.
2
The invariant d (B; ’)
Consider the following family of bound quiver algebras (where a dotted line means that the sum of paths indicated by this line is zero if it indicates exactly three parallel paths,
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127
the commutativity of paths if it indicates exactly two parallel paths, and the zero path if it indicates only one path) 5
6
Á °
5
à ¾
3
3
®
4
¯
1 ®°Á = ¯¾Á
1 2 ®» = °´; ®³ = °!
®°Ã = ¸¯¾Ã
¾» = ¯´; ¾³ = ¸¯!
B1 (¸)
7
¾ °
!
®
¯
B2 (¸)
¶ 2 K n f0; 1g 8
6
»
4
2
´ ³
¶ 2 K n f0; 1g
6
5 2
3 1
4
B3
B4
B5
B6
B7
B8
B9
B10
B11 8
9 6
3
B12
10 7
4 1
5 2
B13 B14 \ \ b b14 form a complete The repetitive algebras B1 (¸), B2 (¸), ¸ 2 K n f0; 1g, B3 ; : : : ; B family of exceptional repetitive algebras of tame tubular type (see [4], [15] and [18]). b we will denote For a tubular algebra B and a strictly positive automorphism ’ of B, by d (B; ’) the integer de¯ned as follows: b and i is a (1) If B is an arbitrary tubular algebra, ½ is a rigid automorphism of B,
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positive integer then d (B; ½º i ) = 0. (2) If B is an exceptional algebra of tubular type (2; 2; 2; 2) isomorphic to the algebra b given by the B1 (¸), for some parameter ¸ 2 k n f0; 1g, ’ is an automorphism of B shifts: ’(em;i ) = em;i+3 , ’(em;i+3 ) = em+1;i , for m 2 Z and i 2 f1; 2; 3g, b corresponding to the rigid automorphism of B and ³ is a rigid automorphism of B which ¯xes the vertices 1 and 4 and exchanges the vertices 2 with 3 and 5 with 6, then, for all positive integers m, we put d (B; ’m ) = d (B; ³’m ) =
8 > < 16 > :0
if
m ² 1 (mod 2)
if
m ² 0 (mod 2)
:
b corresponding to the rigid auFurther, if there exists a rigid automorphism ¾ of B tomorphism of B which ¯xes exactly four vertices of B and exchanges the remaining pair (either 2 and 3, or 5 and 6) of vertices of B (char K 6= 2, ¸ = ¡ 1) then we set d (B; ¾’m ) = 0 for all positive integers m. (3) If B is an exceptional algebra of tubular type (2; 2; 2; 2) isomorphic to the algebra b given by B2 (¸) for some parameter ¸ 2 k n f0; 1g and ’ is an automorphism of B the shifts: ’(em;i ) = em;i+2 , for m 2 Z, i 2 f1; 2; 3; 4g and ’(em;i ) = em+1;i¡4 , for m 2 Z, i 2 f5; 6g, b corresponding to the rigid automorphism of B and ³ is a rigid automorphism of B which exchanges exactly two pairs of vertices then, for all positive integers m, we put 8 d (B; ’m ) = d (B; ³’m ) =
> < 12 > :0
if
m ² §1 (mod 6)
if
m 6² §1 (mod 6)
:
b which exchanges exactly Furthermore, if there exists a rigid automorphism ¾ of B one pair of vertices of B (1 and 2, 3 and 4 or 5 and 6) or all three pairs (char K 6= 2, ¸ = ¡ 1) then we put d (B; ¾’m ) = 0 for all integers m. (4) If B is an exceptional algebra of tubular type (3; 3; 3), then: b has no nontrivial rigid automorphism (in that case Bb is isomorphic to one (a) if B b4; : : : ; B b8), ’ is a 2-root of º b , and i 2 Z, then of the algebras B B d (B; ’i ) =
8 > < 12 > :0
if
i ² §1 (mod 6)
if
i 6² §1 (mod 6)
;
b has no nontrivial rigid automorphism, Á is a 4-root of º b (in that case B b is (b) if B B b4 ; B b6 ), and i is an odd integer then isomorphic to one of the algebras B d (B; Ái ) =
8 > <6 > :0
if
i ² §1 (mod 6)
if
i ² 3 (mod 6)
;
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129
b is isomorphic to the algebra B b3 (in that case B b has a nontrivial rigid twist, (c) if B b is determined by the shifts and ºBb has no root of order 4), automorphism ’ of B ’(em;1 ) = em;5 , ’(em;2 ) = em;8 , ’(em;3 ) = em;7 , ’(em;4 ) = em;7 , ’(em;5 ) = em+1;1 , ’(em;6 ) = em+1;4 , ’(em;7 ) = em+1;3 , ’(em;8 ) = em+1;2 , for m 2 Z, b3 , ¾ is a rigid automorphism of B b of order 2 and ³ is a rigid automorphism in B b of order 3 then, for all positive integers i, we put of B d (B; ’i ) =
8 > > 4 > > <
256 > > > > : 0
d (B; ¾’i ) =
8 > < 12
if
i ² §1 (mod 6)
if
i ² 3 (mod 6)
if
i ² 0 (mod 2)
if
i ² §1 (mod 6)
> : 0 if i 6² §1 (mod 6) 8 > < 16 if i ² 1 (mod 2) i d (B; ³’ ) = : > : ² 0
if
i
;
;
0 (mod 2)
b is isomorphic (5) If B is an exceptional algebra of tubular type (2; 4; 4) (in that case B b9; : : : ; B b13 ) and ’ is a 3-root of º b then, for all positive to one of the algebras B B integers i, we set 8 i
d (B; ’ ) =
> <6 > :0
if
i ² §1 (mod 6)
if
i 6² §1 (mod 6)
:
b is isomorphic (6) If B is an exceptional algebra of tubular type (2; 3; 6) (in that case B b14 ), ’a is the automorphism of B b given by the shifts to the algebra B ’a (em;1 ) = em;7 , ’a (em;2 ) = em;6 , ’a (em;3 ) = em;10 , ’a(em;4 ) = em;9 , ’a (em;5 ) = em;8 , ’a (em;6 ) = em+1;1 , ’a (em;7 ) = em+1;2 , ’a (em;8 ) = em+1;3 , ’a (em;9 ) = em+1;4 , ’a(em;10 ) = em+1;5 , for m 2 Z, d b in B 14 and ’b is the automorphism of B given by the shifts ’b (em;1 ) = em;6 , ’b (em;2 ) = em;7 , ’b (em;3 ) = em;8 , ’b (em;4 ) = em;9 , ’b (em;5 ) = em;10 , ’b (em;6 ) = em+1;2 , ’b (em;7 ) = em+1;1 , ’b (em;8 ) = em+1;5 , ’b (em;9 ) = em+1;4 , ’b (em;10 ) = em+1;3 , for m 2 Z, d in B 14 , then, for all positive integers i, we put d (B; ’ia ) =
8 > > 16 > > <
1024 > > > > : 0
if
i ² §1 (mod 6)
if
i ² 3 (mod 6)
if
i ² 0 (mod 2)
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and d (B; ’ia ’b ) = 0 : (Note that ’b ’a = ºBb = ’a ’b and ’2b = %ºBb = ’2a , where % is a nontrivial rigid b automorphism of B.) It follows from the main results of [3] and [15] that d (B; ’) is well and uniquely de¯ned.
3
Proof of the Proposition 2
Let A be a tame concealed algebra. Denote by ©A the Coxeter matrix ¡ CAt CA¡1 of A. We will identify the elements of Grothendieck group K0 (A) with the vectors from Zn (as coe±cients in the canonical base e1 ; : : : ; en ). We denote by qA the Euler form qA : Zn ! Z of A de¯ned by qA (x) = xt (CAt )¡1 x, for x 2 Zn (see [17, (3.5),(4.3)]). The radical of qA is of the form rad qA = ZhA for some positive vector hA with at least one coordinate equal to 1. It follows from [17, (2.4)] that ©A hA = hA . Moreover, there exists a Z-linear map @A : Zn ! Z called the defect such that ©dA (x) = x + @A (x)hA for all vectors x 2 Zn . Finally, we note that @A (x) = @A (© A (x)) for all x 2 Zn . We will identify the defect @A (1) (n) (1) (n) with the 1 £ n matrix @A = [@A ¢ ¢ ¢ @A ] such that [@A ¢ ¢ ¢ @A ]x = [@A (x)] for all x 2 Zn . Denote by In the n £ n identity matrix. Since @A In = @A ©A , we obtain the following equalities
¡
¢
¡
@A CA + CAt = @A (In ¡
¢
(¡ CAt CA¡1 )CA = @A ((In ¡
¡
©A )CA ) = @A (In ¡
Now, we may prove the following lemma.
¢
©A ) CA = 0:
Lemma 3.1. Let B be a tubular algebra which is an extension of a tame concealed algebra A. Let @e be the one-row matrix obtained from the matrix @A by taking the corresponding elements from @A for the idempotents ei from A and 0 for the remaining coordinates. Then ¡ ¢ e @ CBt + CB = 0 : Proof. We may assume (if necessarily, after reordering the primitive idempotents of B) that the Cartan matrix CB of B is of the form
2
Then
·
@e = @A 0
6 CA 0 7 5:
CB = 4
¸
·
3
C1 C2
¸
= @A(1) ¢ ¢ ¢ @A(nA ) 0 ¢ ¢ ¢ 0 :
We know that for every row V of the matrix C1 vector V t is either the zero vector or is the dimension vector of some module M from the mouth of a stable tube in the AuslanderReiten quiver of A. In the second case, the module M is regular, and consequently, by
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131
[17, (4.3)], @A V t = @A (dimM ) = 0. Thus we obtain @A C1t = 0. Therefore, we obtain
·
¸
02
3
2
31
· ¸ t t ¡ t ¢ B6 CA C1 7 6 CA 0 7C e @ CB + CB = @A 0 @4 5+4 5A = @A (CAt + CA ) @A C1t = 0: 0 C2t
C1 C2
This shows the claim.
For a rigid automorphism ½ of an algebra A with K0 (A) of rank n, we denote by ½ the permutation of the set f1; : : : ; ng such that ½(ei ) = e½(i) for i 2 f1; : : : ; ng. Moreover, we denote by ½e the matrix of ½. Using this notation we have the following lemma. Lemma 3.2. Let A be a hereditary algebra of Euclidean type and ½ be a rigid automorphism of A. Then @A (½ex) = @A (x) for all vectors x 2 Zn .
Proof. For l 2 f1; : : : ; ng, denote by Sl the matrix (xi;j )ni;j=1 with
xi;j =
8 > > 1 if i = j 6= l > > > > > > > < 1 if i = l and there exists an arrow
from i to j or from j to i in QA :
> > > > > > ¡ 1 if i = j = l > > > : 0 in other cases
Matrices Sl , l 2 f1; : : : ; ng, are the matrices of the re°ections in the sense of GelfandPonomarev (see [1]). We may assume that 1; : : : ; n is an admissible sequence of sinks (see [1]). Then the sequence ½¡1 (1); : : : ; ½¡1 (n) is also an admissible sequence of sinks. Therefore, following [1, VII.4.7] we have Sn : : : S1 = ©A = S½¡ 1 (n) : : : S½¡ 1 (1) : We note also that for l 2 f1; : : : ; ng we have the equality of matrices Sl = ½eS½¡ 1 (l) ½e¡1 :
Therefore we obtain
¡
¢
½e©A ½e¡1 = ½e (Sn : : : S1 ) ½e¡1 = ½e S½¡ 1 (n) : : : S½¡ 1 (1) ½e¡1 =
¡
¢
¡
¢
= ½eS½¡ 1 (n) ½e¡1 : : : ½eS½¡ 1 (1) ½e¡1 = Sn : : : S1 = ©A :
Hence ½e©A = ©A ½e. Let x 2 Zn be an arbitrary vector. From the above formula we obtain ½ex + @A (x)½ehA = ½e (x + @A (x)hA ) = ©nA ½ex = ½ex + @A (½ex) hA ;
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so @A (x)½ehA = @A (½ex) hA . Let Q be the quiver of A and Q0 be the set of vertices of Q. Denote by (hA )v for v 2 Q0 the coordinate of hA corresponding to the vertex v. We have the following equalities @A (½ex)
X
(hA )v =
v2Q0
= @A (x)
X
v2Q0
X
v2Q0
(@A ½e(x)hA )v =
(½ehA )v = @A (x)
X
v2Q0
X
(@A (x)½ehA )v =
(hA )v :
v2Q0
By the de¯nition, hA is a nonzero, nonnegative vector, and hence we obtain the required claim @A (½ex) = @A (x) : Consider the following algebras:
B15
B16
B17
By analysis of re°ection sequences of tubular algebras we obtain the following proposition.
b has a nontrivial rigid autoProposition 3.3. Let B be the tubular algebra such that B morphism. Then one of the following conditions is satis¯ed: b ¹= B c0 for some tubular algebra B 0 , which is a tubular extension of some hered(i) B e n or D e n, itary algebra of Euclidean type A b ¹= B b3 , (ii) B is of tubular type (3; 3; 3) and B b is isomorphic to one of (iii) B is of tubular type (2; 4; 4) and the repetitive algebra B op b17 , B b17 , the algebras B b is isomorphic to one of (iv) B is of tubular type (2; 3; 6) and the repetitive algebra B b14 , B b15 , B b16 . the algebras B Proof. Each tubular algebra is one of the tubular types (2; 2; 2; 2), (3; 3; 3), (2; 4; 4), (2; 3; 6). Let B be a tubular algebra. All re°ection sequences of the tubular algebras of type (2; 2; 2; 2) were presented in [18]. In each of them, there is a tubular algebra which e n or D en . is a tubular extension of a hereditary algebra of one of the Euclidean types A If B is of tubular type (2; 3; 6) then the claim follows from [15, Theorem 5.3]. Namely, the algebras B14 , B15 , B16 are equal respectively to the algebras B35 , B33 , B34 from [15, Section 5], and the algebras B1 ; : : : ; B32 from [15, Section 5] are the tubular extensions e n or D e n . The lists of the representatives of of hereditary algebras of Euclidean types A
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133
tubular algebras from the sequences of tubular algebras of the types (3; 3; 3) and (2; 4; 4) having a nontrivial rigid automorphism were presented, respectively in [3, Section 5] and [3, Section 6]. The algebra B3 is equal to the algebra B24 from [3, Section 5] and the algebra B17 is equal to the algebra B21 from [3, Section 6]. Direct checking shows that all those re°ection sequences contain either an algebra which is a tubular extension e n or D e n or an algebra which is of a hereditary algebra of one of the Euclidean types A op isomorphic to one of the algebras B3 , B17 , B17 . See Appendix 1 for details. From Lemma 3.2 and Proposition 3.3 we obtain the following corollary. Corollary 3.4. For each tubular algebra B there exists a tame concealed algebra A such that the following conditions are satis¯ed: b ¹= B b0 for some tubular extension B 0 of A, (i) B (ii) for any rigid automorphism ½ of the algebra A and for any vector x 2 Zn (where n is the rank of the Grothendieck group K0 (A)) the following equality is satis¯ed
@A (½ex) = @A (x):
b has no nontrivial rigid automorphisms, then the claims follow trivially. AsProof. If B b has a nontrivial rigid automorphism. sume that B By the Proposition 3.3 we may choose an algebra B 0 in such way that A is either a hereditary algebra of Euclidean type or one of the bound quiver algebras listed bellow:
. In the ¯rst case, the claim is satis¯ed by Lemma 3.2. In the second case, it is enough to show that the equality @A ½e = @A
is satis¯ed for any rigid automorphism ½ of A. The coe±cients of the matrix @A corre-
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sponding to the vertices of quivers of the above algebras A are as follows 1
1
1
1 0
0
0 0 0
0
0
¡ 1
0
0 ¡ 1
1
0 ¡ 1
¡ 1
1
1
0 ¡ 1
1
0¡ 10
¡ 1 2 ¡ 1:
1¡ 11
¡ 10¡ 1
We know also that any rigid automorphism ½ of those algebras ¯xes or exchanges pairwise idempotents corresponding to the vertices. Therefore we may assume that ½ is of order 2. Then we obtain the equality @A (½(i)) = @A (i) for all i 2 f1; : : : ; ng. Hence @A ½e = @A .
m b Proof (of Proposition 2). Let ¤ be the Cartan matrix of the algebra B=(%º b ) and ½ B
b Then be the rigid automorphism of B corresponding to the rigid automorphism %¡1 of B. 2
6 CB 6 6 Ct 6 B 6 6 6 0 ¤=6 6 .. 6 . 6 6 6 0 6 4 0
0
0 ¢¢¢ 0
CB 0 ¢ ¢ ¢ 0 CBt .. .
CB ¢ ¢ ¢ 0 .. . . .. . . .
½eCBt 0 0 .. .
0
0 ¢ ¢ ¢ CB
0
0 ¢ ¢ ¢ CBt CB
0
3
7 7 7 7 7 7 7 7: 7 7 7 7 7 7 5
We note that we have the equalities ½et = ½e¡1 = g ½¡1 and (½eCB )t = CBt ½et = CBt g ½¡1 . By Corollary 3.4 we may assume that B is a tubular extension of tame concealed algebra A such that for any rigid automorphism ³ of A the equality @A ³e = @A is satis¯ed. We may also assume that 2 3
6 CA C1 7 5;
CB = 4
0 C2
as in the proof of Lemma 3.1. Then ½e is of the form
2
3
A 07 6 ½f 5;
½e = 4
0 ¤
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 123{142
135
for some rigid automorphism ½A of A. Therefore, it follows from the assumptions on A that @A CA = @A (½f A CA ) :
Let @e be the matrix de¯ned as in Lemma 3.1. Then we obtain
0 2 · ¸ ³ ´t t A B 6 ½f (½eCB )t @e = e @ ½eCB = @ @A 0 4
0 2 31t · ¸ 07 C B 6 CA 0 7C 5 CB A = @ @A ½f 5A = A 0 4
0 ¤
=
µ·
@A ½f A CA 0
¸¶t
=
µ·
@A CA 0
3
¸¶t
1t
C1 C2
0 2 31t · ¸ t B 6 CA 0 7C = @ @A 0 4 @: 5A = CBt e C1 C2
Further, applying Lemma 3.1 again we have also the equalities
¡
CBt
+ CB
¢
³ ¡ ¢´t t t e e @ = @ CB + CB = 0:
Hence, if we multiply the matrix ¤ by the vector
t
·
b @ = e @ e @ ¢¢¢ e @
¸t
;
t then we obtain a zero-vector. On the other hand, @b is a nonzero vector, because @ is nonzero map. Therefore ¤ is singular.
4
Periodicity of Cartan matrices
For a tubular algebra B we denote by nB the number of pairwise nonisomorphic simple modules and by dB the integer equal to 2 if B is of type (2; 2; 2; 2), to 3 if B is of type (3; 3; 3), to 4 if B is of type (2; 4; 4), and to 6 if B is of type (2; 3; 6). Further, we denote by ¤dBB +1 the following matrix of degree nB (dB + 1):
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2
¤dBB +1
6 CB 6 6 Ct 6 B 6 6 6 0 =6 6 .. 6 . 6 6 6 0 6 4 0
0
0 ¢¢¢ 0
CB 0 ¢ ¢ ¢ 0 CBt
CB ¢ ¢ ¢ 0 .. .. . . .. . . . .
3
0 7
7
0 7 7 0 .. .
0
0 ¢ ¢ ¢ CB 0
0
0 ¢ ¢ ¢ CBt CB
7 7 7 7: 7 7 7 7 7 7 5
For a set S we will denote by jSj the number of the elements of S. Now we present two conditions playing an important role in our considerations. De¯nition. We will say that a tubular algebra B satis¯es the conditions (i) and (ii) if, respectively: (i) for all sets zr ; zc » f1; : : : ; nB g, jzr j = jzc j < nB , the determinants of the matrix obtained from CB by deleting rows indexed by the numbers from zr and columns indexed by the numbers from the set zc and the matrix obtained from ¤dBB +1 by deleting rows indexed by the numbers from zr and columns indexed by the numbers from the set zc + dB nB = fk + dB nB ; k 2 zc g are equal; b such that the Cartan matrix ¤ of (ii) for any positive rigid automorphisms à of B b the algebra B=(Ã) is of degree less than (dB + 3)nB , the equality det ¤ = d (B; Ã)
holds. Proposition 4.1. The algebras B1 (¸), B2 (¸), ¸ 2 K n f0; 1g, B3 ; : : : ; B 14 (listed in Section 2) satisfy the conditions (i) and (ii). Proof. This proposition was proved using a computer program. The condition (ii) needs only few computations and is easy to check (see Appendix 2.2 for the exemplary program). We note also that some of the determinants of these Cartan matrices were previously computed (see for example [4, Section 3]). Computations for the condition (i) are more complicated, but even \brutal force" algorithm computing all necessarily minors (see Appendix 2.1) gives the (positive) answer in a reasonable time. We will need also a theorem which will allow us to use the conditions introduced above. Theorem 4.2. Let B be a tubular algebra satisfying the condition (ii), à a positive b Then the determinants of the Cartan matrices of the algebras automorphism of B. 2n dB +2nB B b b B=(ú ) are equal. b ) and B=(úB b B
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 123{142
137
2nB 0 b Proof. Let ¤ = (aij )m i;j=1 be the Cartan matrix of the algebra B=(úB b ), and let ¤ be
2nB +dB b the Cartan matrix of the algebra B=(ú ). Then the both matrices ¤ and ¤0 are of b B the form 2 3 CB 0 0 ¢ ¢ ¢ 0 C1 C2
6 6 6 Ct 6 B 6 6 0 6 6 6 .. 6 . 6 6 6 0 6 6 6 6 0 4
CB 0 ¢ ¢ ¢ 0
0
0
CBt .. .
0 .. .
0 .. .
0
0 ¢ ¢ ¢ CB 0
0
0
0 ¢ ¢ ¢ CBt CB 0
0
0 ¢ ¢ ¢ 0 C3 C4
0
CB ¢ ¢ ¢ 0 .. . . .. . . .
7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 5
where C4 is a square matrix of degree less than nB . Moreover, if m is divisible by nB , then the matrices C2 , C3 , C4 degenerate to matrices of degree 0. Note also that ¤ is a square matrix of degree m and ¤0 is a square matrix of degree m + 2nB . We will not use the structure of the matrices C1 , C2 , C3 , C4 . Anyway we may see that the matrix C3 can be obtained from the matrix CBt by removing some rows, the square matrix C4 can be obtained from the matrix C3t by removing the rows with the same indices, and the matrix [C1 C2 ] can be obtained from the matrix CBt by exchanging some columns and inserting some zero-columns. We recall that, if m is divisible by nB , then by Proposition 2 the matrices ¤ and ¤0 are singular, so we obtain the required claim. In order to simplify the notation, we will write n instead of nB and d instead of dB . Denote by S(m) the set of permutations of the set f1; : : : ; mg and by S(m; ¤) the following subset of S(m): S(m; ¤) = Then det ¤ =
X
(
) m ¯ Y ¯ ¾ 2 S(m) ¯ ai ¾(i) 6= 0 :
sign ¾
¾2S(m)
For a ¯xed ¾ 2 S(m; ¤), denote
i=1
m Y
ai ¾(i) =
i=1
X
sign ¾
¾2S(m;¤)
m Y
ai ¾(i) :
i=1
bk = j¾ (fkn + 1; : : : ; (k + 1)ng) \ fkn + 1; : : : ; (k + 1)ngj ; for k = 0; : : : ;
and
¥m ¦ n
¡ b00
1,
¯ n³j m k ´ o¯ ¯ ¯ ¡ 1 n + 1; : : : ; m ¯ ; = ¯¾ (f1; : : : ; ng) \ n
b0k = j¾ (fkn + 1; : : : ; (k + 1)ng) \ f(k ¡
1)n + 1; : : : ; kngj ;
(1)
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J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 123{142
¥ ¦
¡ 1. for k = 1; : : : ; m n ¥ ¦ ¡ 1, is the number of elements of the form ai¾(i) ; i 2 Note that bk , for k = 0; : : : ; m n f1; : : : ; mg from the (k + 1)-th copy of the matrix CB (ordering from the left side) in the matrix ¤, b00 is the number of elements of that form in the matrices C1 and C2 , ¥ ¦ ¡ 1, is the number of elements of that form in the k-th and b0k , for k = 1; : : : ; m n t copy of the matrix CB (again ordering from the left side) in the matrix ¤. Clearly, in each column (respectively, in each row) of ¤ we have exactly one element of the form ai¾(i) ; i 2 f1; : : : ; mg. From the assumption, the elements ai¾(i); i 2 f1; : : : ; mg, are nonzero, hence we obtain the following inclusions: n³j m k ´ o ¡ 1 n + 1; : : : ; m ; ¾ (f1; : : : ; ng) ³ f1; : : : ; ng [ (2) n ¾ (fkn + 1; : : : ; (k + 1)ng) ³ f(k ¡ 1)n + 1; : : : ; (k + 1)ng ; (3) for 1 µ k µ
¥m ¦
¡
n
1 and
¾ ¡1 (fkn + 1; : : : ; (k + 1)ng) ³ fkn + 1; : : : ; (k + 2)ng ;
¥m ¦
(4)
for 0 µ k µ n ¡ 2. ¥ ¦ ¡ 2, we obtain from the inclusion (4) the equalities Note also that, for k = 0; : : : ; m n¥ ¦ m 0 bk + bk+1 = n, and, for k = 0; : : : ; n ¡ 1, from the inclusions (2) and (3) we obtain equalities bk + b0k = n. Indeed, we have equalities b0 + b00 = j¾ (f1; : : : ; ng) \ f1; : : : ; ngj
¯ ¡
+ ¯¾ f1; : : : ; ng \
¡©¥ m ¦ n
¡
¢
1 n + 1; : : : ; m
ª¢¯ ¯
¯ ¡ ©¡¥ m ¦ ¢ ª¢¯ ¯ ¡ = ¯¾ (f1; : : : ; ng) \ f1; : : : ; ng [ 1 n + 1; : : : ; m n
(2)
= j¾ ¡1 (fkn + 1; : : : ; (k + 1)ng)j = n;
bk + b0k = j¾ (fkn + 1; : : : ; (k + 1)ng \ fkn + 1; : : : ; (k + 1)ng)j + j¾ (fkn + 1; : : : ; (k + 1)ng \ f(k ¡ = j¾ (fkn + 1; : : : ; (k + 1)ng) \ f(k ¡
1)n + 1; : : : ; kng)j 1)n + 1; : : : ; (k + 1)ngj
(3)
= j¾ ¡1 (fkn + 1; : : : ; (k + 1)ng)j = n;
for k = 1; : : : ;
¥m ¦ n
¡
1, and
bk + b0k+1 = j¾ (fkn + 1; : : : ; (k + 1)ng \ fkn + 1; : : : ; (k + 1)ng)j + j¾ (f(k + 1)n + 1; : : : ; (k + 2)ng \ fkn + 1; : : : ; (k + 1)ng)j = j(fkn + 1; : : : ; (k + 1)ng \ ¾ ¡1 fkn + 1; : : : ; (k + 1)ng)j + j(f(k + 1)n + 1; : : : ; (k + 2)ng \ ¾ ¡1 fkn + 1; : : : ; (k + 1)ng)j = j¾ ¡1 (fkn + 1; : : : ; (k + 1)ng) \ fkn + 1; : : : ; (k + 2)ngj (4)
= j¾ ¡1 (fkn + 1; : : : ; (k + 1)ng)j = n
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 123{142
139
¥ ¦
¡ 2. Therefore, we obtain the equalities b0 = b1 = ¢ ¢ ¢ = bb m c¡1 and for k = 0; : : : ; m n n b00 = b01 = ¢ ¢ ¢ = b0 m ¡1 . bnc 0 0 m+dn Let ¤ = (aij )i;j=1 and S(m+dn) be the set of permutations of the set f1; : : : ; m+dng. We denote by S(m + dn; ¤0 ) the following subset of f1; : : : ; m + dng: S(m + dn; ¤0 ) =
(
) ¯ m+dn Y ¯ ¾ 2 S(m + dn) ¯ a0i ¾(i) 6= 0 : i=1
Then, for a ¯xed ¾ 2 S(m + dn; ¤0 ), denote
bk = j¾ (fkn + 1; : : : ; (k + 1)ng) \ fkn + 1; : : : ; (k + 1)ngj ; for k = 0; : : : ;
¥m ¦ n
0 b0
and
+d¡
1,
¯ n³j m k ´ o¯ ¯ ¯ = ¯¾ (f1; : : : ; ng) \ + d ¡ 1 n + 1; : : : ; m + dn ¯ ; n
0
bk = j¾ (fkn + 1; : : : ; (k + 1)ng) \ f(k ¡
1)n + 1; : : : ; kngj ;
¥m¦
0
0
+ d ¡ 1. Then the equalities b0 = b1 = ¢ ¢ ¢ = bb m c+d¡1 , b0 = b1 = ¢ ¢ ¢ = n 0 0 bb m c+d¡1 and b0 + b0 = n are satis¯ed. n Denote Z(j; n) = fz ³ f1; : : : ; ng j j = jzjg, for j 2 f0; : : : ; ng, ze = f1; : : : ; ng n z for all z 2 Z(j; n), j 2 f0; : : : ; ng, and s(z) = jf(x;³y) 2 z £ zejx´< ygj for ³ z 2 Z(j; n). ´ for k = 1; : : : ;
n
(1)
(j)
Further, for all j 2 f0; : : : ; ng, zr ; zr 2 Z(j; n), zr = zr ; : : : ; zr
(1)
(j)
, zc = zc ; : : : ; zc
,
³ ´ ³ ´ (1) (n¡j) (1) (n¡j) zer = zer ; : : : ; zer , zcc = zc ; : : : ; zc , denote by B (¤; zr ; zc ) the matrix obtained (1)
(j)
from the matrix ¤ by removing the rows indexed by 1; : : : ; n; n + zr ; : : : ; n + zr ; 2n + (1) (j) 1; 2n + 2; : : : ; m and the columns indexed by 1; : : : ; n; n + zc ; : : : ; n + zc ; 2n + 1; 2n + 2; : : : ; m, and by C (¤; zr ; zc ) the matrix obtained from the matrix ¤ by removing the (1) (n¡j) (1) (n¡j) rows indexed by n + zer ; : : : ; n + zer , the columns indexed by n + zec ; : : : ; n + zec , 2 and replacing the elements with indices from the set fn + 1; : : : ; 2n ¡ jg by zeros. We note that det C (¤; zr ; zc ) is a complementary minor to the minor det B (¤; zr ; zc ) of the matrix ¤. If j = n, then B (¤; zr ; zc ) is degenerated to the empty matrix. In that case we will assume that det B (¤; zr ; zc ) = 1. In the formula (1) on the determinant of the matrix ¤ we put together the summands for which the sets ¾ ¡1 (f1; : : : ; ng) \ fn + 1; : : : ; 2ng £ ¾ (f2n + 1; : : : ; 3ng) \ fn + 1; : : : ; 2ng are equal. Then, following (1) we see that each such group can be presented as a product of the determinants of pair matrices multiplied by some (determined by the number of inverses) power of number ¡ 1. Therefore we have the equality det ¤ =
n X
X
j=0 zr ;zc 2Z(j;n)
(¡ 1)s(zr )+s(zec )+j¤(n¡j) det B (¤; zr ; zc ) det C (¤; zr ; zc ) :
(5)
140
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 123{142
Indeed, from the previous considerations, we obtain
¯ ¡1 ¯ ¯¾ (f1; : : : ; ng) \ fn + 1; : : : ; 2ng¯ = b0 = b0 = j¾ (f2n + 1; : : : ; 3ng) \ fn + 1; : : : ; 2ngj : 1 2 In the formula (5), for a ¯xed permutation ¾ 2 S(m; ¤), we have j = b01 (= b02 ),
©
zr = z 2 f1; : : : ; ng j z + n 2 ¾ ¡1 (f1; : : : ; ng) and
ª
zc = fz 2 f1; : : : ; ng j z + n 2 ¾ (f2n + 1; : : : ; 3ng)g : We note that the matrix B (¤; zr ; zc ), can be also obtained from the matrix ¤0 by (1) (j) removing the rows indexed by 1; : : : ; n; n+zr ; : : : ; n+zr ; (d+2)n+1; (d+2)n+2; : : : ; m (1) (j) and the columns indexed by 1; : : : ; (d + 1)n; (d + 1)n + zc ; : : : ; (d + 1)n + zc ; (d + 2)n + 1; (d + 2)n + 2; : : : ; m. Denote by C 0 (¤0 ; zr ; zc ) the matrix obtained from the matrix ¤0 (1) (n¡j) by removing the rows indexed by n + zer ; : : : ; n + zer ; 2n + 1; : : : ; (d + 2)n and the (1) (n¡j) e columns indexed by n + 1; : : : ; (d + 1)n; (d + 1)n + zc ; : : : ; (d + 1)n + zec , and then 2 fn ¡ replacing the elements with indices from the set + 1; : : : ; 2n jg by zeros. 0 0 Then det C (¤ ; zr ; zc ) is a complementary minor to the minor det B (¤0 ; zr ; zc ) of the matrix ¤0 . By the formula (1) applied to the matrix ¤0 , in the same manner as to the matrix ¤, we obtain the equality 0
det ¤ =
n X
X
(¡ 1)s(zr )+s(zec )+(j+dB n)¤(n¡j) det B (¤; zr ; zc ) det C
0
(¤0 ; zr ; zc ) : (6)
j=0 zr ;zc 2Z(j;n)
In this case we put together the summands for which the sets
©
zr = z 2 f1; : : : ; ng j z + n 2 ¾ ¡1 (f1; : : : ; ng) and
ª
zc = fz 2 f1; : : : ; ng j z + dn 2 ¾ (f2n + 1; : : : ; 3ng)g 0
0
are equal. Then we have the equalities jzr j = b1 = bd+1 = jzc j (in the formula (6) that value is equal to j). Expanding the formula (1) applied to the matrix C 0 (¤0 ; zr ; zc ) we also use the equalities jzer j = b1 = bd = jzec j. It follows from the condition (i) that we have det C (¤; zr ; zc ) = det C 0 (¤0 ; zr ; zc ). Furthermore, it follows from the assumptions that dn is an even number, because d and n can not be both odd. Therefore, we obtain the required equation det ¤ = det ¤0 . As a consequence of Proposition 4.1 and Theorem 4.2 we obtain the following corollary. Corollary 4.3. Let B be an exceptional tubular algebra and à be a strictly positive b Then the determinant of the Cartan matrix of B=(Ã) b automorphism of B. is equal to d (B; Ã).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 123{142
5
141
Proof of the main theorem
The main theorem is now a consequence of Proposition 2 and Corollary 4.3.
6
Appendix
Appendix is available in here.
Acknowledgments The author would like to thank Andrzej Skowro¶ nski for suggesting the problem and for many helpful discussions. The author also acknowledges support from Polish Scienti¯c KBN Grant No. 5 P03A 008 21.
References [1] I. Assem, D. Simson and A. Skowro¶ nski: \Elements of Representation Theory of Associative Algebras. I: Techniques of Representation Theory\, In: London Mathematical Society Student Texts, Cambrige University Press, in press. [2] D. Benson: \Representations and Cohomology I", In: Cambridge Studies in Advanced Mathematics, Vol. 30, Cambridge, 1991. [3] J. BiaÃlkowski and A. Skowro¶ nski: \Sel¯njective algebras of tubular type\, Colloq. Math., Vol. 94, (2002), pp. 175{194. [4] J. BiaÃlkowski and A. Skowro¶ nski: \On tame weakly symmetric algebras having only periodic modules\, Archiv. Math., Vol. 81, (2003), pp. 142{154. [5] J. BiaÃlkowski and A. Skowro¶ nski: \Socle deformations of sel¯njective algebras of tubular type\, J. Math. Soc. Japan, in press. [6] Yu.A. Drozd: \Tame and wild matrix problems\, In: Representation Theory II, Lecture Notes in Math., Vol. 832, Springer, Berlin-Heidelberg-New York, 1980, pp. 242{258. [7] K. Erdmann: \Algebras and quaternion defect groups I\, Math. Ann., Vol. 281, (1988), pp. 545{560. [8] K. Erdmann: \Algebras and quaternion defect groups II\, Math. Ann., Vol. 281, (1988), pp. 561{582. [9] K. Erdmann: Blocks of tame representation type and related algebras, Lecture Notes in Math., Vol. 1428, Springer, Berlin-Heidelberg-New York, 1990. [10] K. Erdmann, O. Kerner and A. Skowro¶nski: \Self-injective algebras of wild tilted type\, J. Pure Appl. Algebra, Vol. 149, (2000), pp. 127{176. [11] P. Gabriel: \The universal cover of a representation-¯nite algebra\, In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer, BerlinHeidelberg-New York, 1981, pp. 68{105. [12] D. Happel, U. Preiser and C.M. Ringel: \Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to D Tr-periodic modules,"
142
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In: Representation Theory II., Lecture Notes in Math., Vol. 832, Springer, BerlinHeidelberg-New York, 1980, pp. 280{294. [13] D. Happel and C.M. Ringel: \The derived category of a tubular algebra," In: Representation Theory I, Lecture Notes in Math., Vol. 1177, Springer, BerlinHeidelberg-New York, 1986, pp. 156{180. [14] D. Hughes and J. WaschbÄ usch: \Trivial extensions of tilted algebras\, Proc. London Math. Soc., Vol. 46, (1983), pp. 347{364. [15] H. Lenzing and A. Skowro¶nski: \Roots of Nakayama and Auslander-Reiten translations\, Colloq. Math., Vol. 86, (2000), pp. 209{230. [16] J. Nehring and A. Skowro¶ nski: \Polynomial growth trivial extensions of simply connected algebras\, Fund. Math., Vol. 132, (1989), pp. 117{134. [17] C.M. Ringel: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., Vol. 1099, Springer, Berlin-Heidelberg-New York, 1984. [18] A. Skowro¶ nski: \Sel¯njective algebras of polynomial growth\, Math. Annalen, Vol. 285, (1989), pp. 177{199.
CEJM 2(1) 2004 143{176
Computational details { appendix to the article \Cartan matrices of sel¯njective algebras of tubular type" Jerzy BiaÃlkowski
1
Re° ection sequences of tubular algebras.
In this section we present full lists of re°ection sequences of tubular algebras of the types (3; 3; 3) and (2; 4; 4), having a nontrivial rigid automorphism. The lists were generated by a computer program. First, we created the ¯les: and ( downloadable from http://www.mat.uni.torun.pl/~jb/en/research/tubular/, 80+551 pages, 1.1+15 MB) containing the full list of tubular algebras of the types (3; 3; 3) and (2; 4; 4) respectively. The output generated by the program includes: ° the full list of all tubular algebras of the types (3; 3; 3) and (2; 4; 4) respectively (the program uses the Bongartz-Happel-Vossieck list, [2],[3], of tame concealed algebras and tubular extensions of such algebras in the sense of [4]), ° the re°ection equivalence classes of tubular algebras of the type (3; 3; 3) (respectively, of the type (2; 4; 4)), ° nontrivial rigid automorphisms of the repetitive algebras from the pairwise nonequivalent re°ection classes of tubular algebras of the type (3; 3; 3) (respectively, of the type (2; 4; 4)). The main purpose of presenting those lists is to complete the proof of [1, Proposition 3.3]. This is done by observing that: ° All the sequences, excluding for the type (3; 3; 3) and and (being opposite to each other) for the type (2; 4; 4), contain an algebra which is a tubular e n or D e n. extension of some hereditary algebra of Euclidean type A ° The algebra in the sequence for the type (3; 3; 3) is isomorphic to B3 . Likewise, the algebras and from the sequences and for the op type (2; 4; 4), are respectively isomorphic to the algebras B17 and B17 .
144
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 5
6
7
7
3
4 0
2
2
1
1
7
6
5
3
4 2
1
4
0
3
4
3
1
2
2
1
0
3
4
K 99 (seq 1)
K 3 (seq 1)
7
6
7 6
6
3
5
5 2
0
5
5 1
1
3 0
2
0
K 2 (seq 2)
6
7
1
1
7
1
3 4
1 K 10 (seq 0)
5
2
6
0
2
4
7
4
4
0
2
7
4
7
3
K 15 (seq 1)
6
3
6
0
7
0
0
5
6 3
K 137 (seq 0)
7
3
K 120 (seq 1)
5
5
6
5
1
6
0
5
5
6
K 132 (seq 0)
K 1 (seq 0)
2
7
1
0
3
4
K 122 (seq 2)
K 80 (seq 2)
2
4
2
K 43 (seq 2)
4
K 165 (seq 2) 7
7
5
6
7
7
6
7
6 6
3
3 4
1
2
5
K 58 (seq 3)
4
1
1
3
0
1
2
3
4
4
1
2
5
4
3
0
K 204 (seq 4)
7
6 1
5
2 3
4
1 K 7 (seq 5)
6
6
4
2
0
K 157 (seq 4)
7
7
0 4
5
2
K 57 (seq 4)
6
1
4
7
1
3
3
7
7
5
0
K 126 (seq 3)
5
4
5
2
K 20 (seq 3)
6
K 21 (seq 4)
7
K 42 (seq 5)
2
6
5
K 6 (seq 4)
2
5
1
K 159 (seq 3)
2
1
3
0
4
7
0
6
2
3
0 2
3
7
6
4
0
1 K 11 (seq 3)
7
3
5
0
2
6
3
5
1
0
0
5
6
4
3
0
5
K 139 (seq 5)
6
4
1
2 0
5
0
1
2
3
4
K 202 (seq 5)
K 79 (seq 5)
Fig. 1 Re®ection sequences of the tubular algebras having a nontrivial rigid authomorphism of type (333).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 5
6
6
7
3
7
0
1
1
0
K 8 (seq 6)
6
5
5
2
0
4
6
4
2
4
3 2
3
3 5
4
1
2
0
0
K 73 (seq 7)
K 67 (seq 7)
2
1
4
4
5
7
5
2
0
4
0
5 2
1
0
3
4
K 136 (seq 8)
2
4 K 34 (seq 8)
6
5
1
3 4
2
4
2
1
0
3
4
2
3
4
4
0
6
1
7
6
3
5
5 2
0
5 2
0
K 205 (seq 10)
K 23 (seq 10)
1
0
3
4
K 117 (seq 10)
K 64 (seq 10)
7 6
7
6
4 K 25 (seq 10)
7
7 6
3
5
5 0
0
1
2
3
K 203 (seq 11)
4
3
1
2
4 K 36 (seq 11)
6
5 0
5
1
2 K 26 (seq 11)
3 4
4
2
1
1
2
7
6
5
K 62 (seq 9)
5 1
2
4
7
7
1
1
K 101 (seq 9)
K 24 (seq 9)
6
0
1
7
3
2
5
6
5
7
5 0
1
7
3
6
K 90 (seq 7)
7
7
6
2
3
K 12 (seq 9)
K 91 (seq 9)
4
6
4
2
0
3
6
0
6
2
7
7
3
5
5
K 76 (seq 8)
6
1
6
0
K 93 (seq 8)
3
6
1
6
K 4 (seq 8)
1
5
5
0
5
7
6 1
2
K 63 (seq 6)
6
2
1
3
1
K 69 (seq 6)
6
7
0
3
4
0
3
2
5
7
4
0
3
7
1
K 9 (seq 7)
7
6
7
0
0
6
0
7
4
K 52 (seq 7)
7
1
6
7
3
4
K 96 (seq 6)
6
1
3
1
4
5
2
5
3
K 49 (seq 6)
7
3
5
3
2
7 7
2
4
145
0
3
K 105 (seq 11)
4
1
2
5
0
K 74 (seq 11)
Fig. 2 Re®ection sequences of the tubular algebras having a nontrivial rigid authomorphism of type (333).
146
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
7 7
6
7
6
7
7
7 6
3
5 0
3
1
6
6
5
5
2
5
5 5
1
2
0
3
4
1
1
3 0
2
2
4
1
0
3
4 2
1
K 113 (seq 12)
6
K 27 (seq 12)
0
3
4
0
K 130 (seq 12)
4
2
K 53 (seq 12)
K 94 (seq 12)
4
K 164 (seq 12)
7
6
7
6
7
7
7
7
6 3
1
2
3
5
0
2
5 0
4
5
6
6
5
5
5 1
1
3
3
1
0
2
4
6
2 2
K 39 (seq 13)
K 92 (seq 13)
7
4
3
5
0
0
4
1
3
1
5
2
0
4
2
1
4
2
1
2
5
4
3
5
6
K 148 (seq 14)
0
K 190 (seq 14)
0
6
K 61 (seq 14)
5
6
6
K 89 (seq 14)
3
7
2
3
1
5 1
2 3
1
2
2
K 188 (seq 15)
6
5 0
1
7
7
7
3
0
4
4
4
3
5
6
K 154 (seq 15)
6
K 38 (seq 15)
0
7
6 3
1
0
3
1
2
5
4 0
K 134 (seq 16)
K 29 (seq 16)
0
1
2
K 189 (seq 16)
K 70 (seq 16)
7
7
6
7
4
5
7
6
6
3 5
4
6
5 0
K 107 (seq 17)
6
3 4
2
4
3
5 3
5
1
0
5
6
5
2
2
7 4
6
1
K 75 (seq 15)
7
7
4
0
K 95 (seq 15)
1
3
7
7
2
0
K 124 (seq 13)
7
3
K 37 (seq 14)
0
1
3
2
0
4 2
7
6
1
4
3
K 103 (seq 13)
6
5 0
1
4
K 50 (seq 13)
K 162 (seq 13)
7 6
0
4
3
3
1
2
4 K 41 (seq 17)
0
1 K 191 (seq 17)
2
4
1
2
5
0
K 68 (seq 17)
Fig. 3 Re®ection sequences of the tubular algebras having a nontrivial rigid authomorphism of type (333).
4
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
5
6 3
7
7
5
6
4
6 3
5
2
0
1
2
3
4
2
K 206 (seq 18)
K 0 (seq 18)
7
7 4
0
1
2
3
4
2
K 207 (seq 19)
K 5 (seq 19)
7
7
4 2
1 K 13 (seq 19)
6
1
7
0 1
5
4
4
0 2
3
5
K 167 (seq 18)
6 3
5
0
1
5
6
6
2
K 14 (seq 18)
6 3
1
0 1
5
7
7 4
0
147
3
5
7
2
4
6
7
6
6
1
5
3 2
0
1
0
3
5
0
4
2
K 169 (seq 20)
K 119 (seq 20)
1
0
3
4
K 131 (seq 20)
K 16 (seq 20)
5
6
4
7
7
3
7
5
6
7
4 1
1
2
2
2
1
2
3 4
3
0
5
6
0
K 147 (seq 21)
0
4
1
3
0
5
6
K 153 (seq 21)
K 173 (seq 21)
K 17 (seq 21)
5
6
7
7
2
4
6
4 1
5
1
3 0
2
6
5
6
6
3
3
0
K 18 (seq 22)
7
7
0
1
2
K 161 (seq 22)
7
3
5
7
2
4
6
7
6
4 2 1
2
0
6
1
5
3 1
0
4
K 163 (seq 22)
7
5
3
0
2
K 182 (seq 22)
4
5
1
3
K 123 (seq 23)
4
0 K 170 (seq 23)
5
2
1
0
3
4
K 114 (seq 23)
K 54 (seq 23)
Fig. 4 Re®ection sequences of the tubular algebras having a nontrivial rigid authomorphism of type (333).
148
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
6
7
7
4
6
5 4
5
1
3
3
1
0
0
0
2
K 56 (seq 24)
6
6
3
5
2
7
7
5 1
1
0
2
K 183 (seq 24)
4
2
K 166 (seq 24)
7
4
K 160 (seq 24)
7
6
3
3
5
7
2
4
6
7
6
4
5
1
5
3 1
6
2
2
1
0
0
3
5
0
4
2
1
K 171 (seq 25)
K 129 (seq 25)
0
3
4
K 104 (seq 25)
K 97 (seq 25)
4
7
7
3
7
5
6
7
6 4
5
1
2
2
1
2
3 1
2
4
3
0
5
6
0
K 158 (seq 26)
0
4
1
3
0
5
6
K 140 (seq 26)
K 175 (seq 26)
K 98 (seq 26) 7
2
1
7
6
6
5
5
0
3
4 2
1
K 100 (seq 27)
0
3
5
7
2
4
6
1 3
0
4
K 172 (seq 27)
K 138 (seq 27)
7
7
4
7
7
3
6 6
5 2
5 5
2
1
0
5
3
4
0
1
2
3
4
2
K 200 (seq 28)
K 102 (seq 28)
1
0
3
0
4
7
6 6
7
4
6 3
5
0
5 6
5
1
1
K 178 (seq 28)
K 135 (seq 28)
7
7
2
6
6
5
3
K 106 (seq 29)
4
0
1
2
3
K 201 (seq 29)
4
2
1
0
3
K 118 (seq 29)
4
0
1
2
K 181 (seq 29)
Fig. 5 Re®ection sequences of the tubular algebras having a nontrivial rigid authomorphism of type (333).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
7
5
7
4
6
4
0
3 3
4
0
1
2
4
7
2
1
7
3
5
3
6
2
K 19 (seq 32)
1
2
7 4
5
6
1
2
3
5
1
2
K 176 (seq 31)
7
1
0
3
K 125 (seq 31)
4
0
K 186 (seq 31)
7 1
3 0
0
K 187 (seq 30)
6
0
4
4 1
4
7
2
K 121 (seq 31)
5
3
6
5
0
0
K 133 (seq 30)
6
1
6
3
5
K 179 (seq 30)
K 108 (seq 30)
2
5
6
5
1
7
7
6
2
149
2
6 0
3
5 4
K 168 (seq 32)
Fig. 6 Re®ection sequences of the tubular algebras having a nontrivial rigid authomorphism of type (333).
150
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 7
5
6
7
7
8
8
8
8
3
6 6
4 0
5 5
0 1
2
2
3
1
4
5
6
1
3
7
0 2
K 921 (seq 0)
1
0
3
4
K 719 (seq 0)
K 1 (seq 0)
2
4
K 820 (seq 0) 7
5
6
7
8 7
8
8
8 6
3
6
4
0
5 1
0 1
2
3
4
5
6
2
1
7
8
3
2
1 4
1
2
3
4
5
6
7
4
7
8
0
0
2
6
7
4
1
3
0
6
1
2
3
4
5
6
7
K 847 (seq 3)
7
8
3 5
7
4
5
6
5
7
0 1
2
0
1
K 2035 (seq 4)
2
3
1
2
K 1473 (seq 4)
4 K 200 (seq 4)
5
6
7
8
6
7
8
8
8
3 4
5
4
3
5
7
0 1
0
2
2
K 2032 (seq 5)
6
7
8
3
4
3
7
7
8
4
6 0 1
2
5 6
6
2
7
6
2
1
8
6 0
7
5
3 3
4
5 4
K 291 (seq 7)
2
1 K 59 (seq 7)
2
1
0
K 709 (seq 7)
4
8
6
4
3
7
8
0 2
0
K 622 (seq 6)
K 1336 (seq 6)
1
5
5
2
0
8
4
5
K 475 (seq 6)
7
3
5 3
1
K 376 (seq 6)
8
7
3
5
0
1
0
8
8
6
K 17 (seq 6)
4
2
6
K 81 (seq 5)
4
1
5
K 1482 (seq 5)
6 3
7
1
8
1
0 2
0
4
7
7
4
1
2
1 K 45 (seq 5)
3
5
K 1400 (seq 3)
K 6 (seq 4)
5
0
8
4
0
5
8
K 726 (seq 3)
6
1
0
3
2
3
5
4
3
6
8
1
0
6
5
8
8
2
7
4
2
3
6
K 1427 (seq 2)
1
6
2
K 798 (seq 2)
8
3
1
2
3
K 1039 (seq 2)
K 63 (seq 3)
7
1
4
0 2
6
8
7
K 337 (seq 3)
4
0
4
5
0
5
6 3
6 3
7
0
2
5
8
3
5
K 345 (seq 2)
K 3 (seq 2)
7
4
0
1
8
6 3
0
K 602 (seq 1)
8
8
1
0
1
4
K 802 (seq 1)
7
4
2
2
K 61 (seq 1)
6
5
0
K 993 (seq 1)
5
3
7
3
4
2
6
K 1342 (seq 7)
1
2
5
0
K 511 (seq 7)
Fig. 7 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 5
6
7
7
8 8
3
7
8
8 8
8 6
4
3 0
1
1
2
3
4
1
5
6
7
1
2
K 914 (seq 8)
3
4
2
5
6
0
2
1
0
3
4
3
K 679 (seq 8)
6
K 580 (seq 8)
5
6
7
7
8
8
8
7
8 8
3
6
4
0 1
0
2
6
4
5
6
2
1
3
5
6
2
5
1
0
1 0
6
3
4
2
2
1
7
8
4
5
3
4
5
6
7
0
K 956 (seq 10)
1
2
3
6
0
1
2
1
0
3
4
0
1
5
6
2
3
4
5
0
6
7
1
4
5
2
3
6
2
3
7
7
4
5
6
0
1
2
3
3
6
8
0
4
5
2
3
6
5
6
7
4
5
4
7
K 932 (seq 11)
8 8
6
8
5
7
0 1
0
3
4
2
K 673 (seq 12)
2
1
3
4
5
6
7
0
K 1022 (seq 12)
1
0
K 1480 (seq 12)
1
2
3
4
K 2375 (seq 12)
K 29 (seq 12)
7
3
7
8
8
6
7
8
8
8
6
4
5
4
5
2
3
6
7
0
5
0 2
2
1
1
0
3
4
0
5
6
1
2
3
0
4
7
1
7
8
1
2
K 1471 (seq 13)
K 2380 (seq 13)
K 687 (seq 13)
K 28 (seq 13)
3
4
5
6
7
6
7
K 893 (seq 13)
8
8
8
8
3
6
4 1
0
2
4
5
6
1
3
0
2
5
1
7
8
2
4
3
4
5
K 930 (seq 14)
8 8
6 0 0
1
0
2
6
7
5
0
4 1
1 K 18 (seq 15)
3
8
4
2
0
K 601 (seq 14)
K 1397 (seq 14)
8
3
7
1
2 K 50 (seq 14)
7
6
4
7
K 1040 (seq 14)
6
0
5
0
5
6
8
K 1485 (seq 11)
K 2403 (seq 11)
K 609 (seq 11)
8
3
5
K 2401 (seq 10)
8
7
1
7
2
4
7
7
K 1467 (seq 10)
8
K 24 (seq 11)
1
3
8
8
5
2
6
2
K 883 (seq 9)
6
6
0
5
1
4
8
4
1
3
K 26 (seq 10)
7
3
2
4
0
K 717 (seq 10)
5
0
4 0
1
2
8
3
5
2
5
K 586 (seq 9)
7
8 6
1
K 646 (seq 9)
K 1409 (seq 9)
K 54 (seq 9) 8
3
0
4
7
K 1001 (seq 9)
7
7 5
0 3
5
K 1418 (seq 8)
8
2
7
4
4
7
K 1027 (seq 8)
K 20 (seq 8)
1
6
5 2
1
0
5
0
0 2
151
2
3
4
5
K 848 (seq 15)
6
7
1
2
3
4
5
K 969 (seq 15)
6
7 2
1
0
3
K 718 (seq 15)
4
3
5
K 1430 (seq 15)
Fig. 8 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
152
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
7
5
8
7
8
1
4
2
3
4
5
6
7
K 955 (seq 16)
8
8
6 3
4
0 5
5
0
1
1
2
2
3
4
5
6
7 2
K 931 (seq 17)
K 344 (seq 17)
K 13 (seq 17)
7
3
K 621 (seq 16)
1
0
6
0
6
4
5
1
7
8
2
2
1 K 57 (seq 16)
7
0
5
2
K 336 (seq 16)
3
6
0 2
6
8
4
5
0
5
7
8
3
6 4
7
8
1 3
6
1
0
3
7
8 8
4
K 661 (seq 17) 7
8
8
8 6
3
6
4 0
6
3
4
5
7
5 5
0 1
2
2
1
3
4
5
6
7
0
K 1029 (seq 18)
1
1
3
2
0 2
K 2033 (seq 18)
1
0
3
4
K 697 (seq 18)
K 2 (seq 18)
2
4
K 808 (seq 18) 7
5
6
7
8 7
8
8
8
8 6
3 0
2
3
4
5
6
7
7
8
3
4
2
7
4
6
3
3
2
8
4
4 2
5
7
1
1
2
7 3
3
5
3
0
5
4
5 4
6 2
K 781 (seq 21)
K 1412 (seq 21)
6
8
7
3
4
5
0
1
2
7
4
5
6
1
4
K 2034 (seq 22)
K 2407 (seq 22)
2
5
K 532 (seq 21)
8
8
1 6
1 0
K 1333 (seq 21)
6
2
2
4
8
3
7
3
5
6
1
2
3
0 6
7
1
6
8
0
8
0
7
K 1415 (seq 20)
4
2
K 82 (seq 22)
3
0
5
0
5
6
K 796 (seq 22)
3
4
K 824 (seq 22)
6
8
7
7
8
8 7
3
7
1
5
8
5
K 2404 (seq 23)
6
1
2
2
4 K 198 (seq 23)
0
2
6
3
4
5
0
1
2
7
5 4
3
8
6 1
0 4
0
K 757 (seq 20)
1
3
3
6
K 1345 (seq 20)
4
5
6 4
8
2
0
8
2
7
K 65 (seq 21)
2
1
2
5
8
4
5
1
5
0
1
2
2
1
K 454 (seq 20)
7
0 7
7
K 292 (seq 21)
1
8
0
2
5
0
0
8
1
K 377 (seq 20)
6
0
2
3
5
0
1
3
1
6
8
7
0
7
K 2036 (seq 19)
7
6
K 5 (seq 20)
6
4
5
8
4
4
3
4
8
1
1
3
0
K 613 (seq 19)
8
3
7
1
4
3
K 814 (seq 19)
7
0 2
6
5
2
1 K 62 (seq 19)
6
3
0
2
K 885 (seq 19)
5
5
1
0 1
6
4
4
3
0
5
K 724 (seq 23)
6
K 2037 (seq 23)
K 825 (seq 23)
Fig. 9 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
6
7
7
8
3
7
5
1
4
2
1
0
3
0
4
1
2
3
6
6
7
8
3
1
0 3
4
K 712 (seq 25)
2
4
6
7
3
0
0
1
2
3
4
K 2367 (seq 25)
8 8
0
5
1
2
6
7
4
4
0
1
2
3
4
4
1
2
5 3
0
K 2384 (seq 26)
K 174 (seq 26)
6
8
7
3
4
7
6 0 1
2
7
4 4
2
4
3
K 175 (seq 27)
1
5
2
K 519 (seq 27) 7
7
7
8
8
8 6
6
5
1
0
6
1
3
4
5
0
K 1410 (seq 27)
8
2
6
3
1
8
3
8
8
5 0
K 2382 (seq 27)
7
8
3
5
5
K 1419 (seq 26)
K 471 (seq 26)
7
0
6
8
1
2
5
K 723 (seq 25)
3
2
6
4
6
5
1
5
2
7
8
3
0
6
1
K 107 (seq 25)
7
6
5
6
8
5
5
0
0
K 795 (seq 24)
8
7
6
3
7
8
0
4
4
K 2369 (seq 24)
K 606 (seq 24)
K 106 (seq 24)
1
2
5
2
2
8
6
1
7
8 6
5 0
7
8
153
1
2
3
4
5
6
7
K 915 (seq 28)
4
5 5
5
0
2
3 0
2
1
K 374 (seq 28)
K 83 (seq 28)
1
0
3
4
K 715 (seq 28)
2
4
K 811 (seq 28) 7
6
7
8 7
8
3
6
5 0
2
3
8
6
0
1
8
7
8
4
5
6
7
K 1002 (seq 29)
1
1
2
5
1 3
3
0
4
2 2
K 199 (seq 29)
6
5
1
0
3
4
5
0
2
K 607 (seq 29)
4
4
K 300 (seq 29)
K 805 (seq 29) 7 7
7
8
7
8
8
8
8 6 6
1
6 3
4
1
0
6
5
0
3 2
1
2
3
4
5
6
7
K 1023 (seq 30)
K 282 (seq 30)
4
5 5
5
0
2
2
1
K 373 (seq 30)
0
1
3
3 0
4
K 664 (seq 30)
2
4
K 804 (seq 30) 7
8 7
7
8
8
8
7
8
6 6
1
0
6 3
1
2
3
4
5
K 894 (seq 31)
6
7
4
0
3
5
2 2
4
1
0
6 3
0
2 K 369 (seq 31)
1
5 1
5
3
K 619 (seq 31)
4
4
5
0
2 K 299 (seq 31)
K 822 (seq 31)
Fig. 10 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
154
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 7
8
1
7
6 3
4
3
5
0
8
8
1
2
6
8
0
5
0
1
7 3
4
4
0
4
2
5 3
6
6
8
5
7
2
5 1
2 K 314 (seq 32)
7
6
0
1
K 1500 (seq 32)
2
3
4
K 2373 (seq 32)
K 1343 (seq 32)
K 594 (seq 32)
7
7
8
3
1
2
1
5
0
8
6
7
6 3
4
4
8
0
5
0
4
5
0
0
2
1
2
3
3
3
4
5
6 2
K 1499 (seq 33)
K 2381 (seq 33)
7
5
1
4
1
7
2
K 317 (seq 33)
6
8
8
6
K 1338 (seq 33)
K 576 (seq 33) 7
7
8
7
8
8
6
7
8
6
1
6 3
4
3
0
1
5
5
5 1
5
2
0
3
4 0
2 K 333 (seq 34)
6
0
2
K 571 (seq 34)
1
2
3
K 810 (seq 34) 7
7
7
8
4
K 2377 (seq 34)
4
8
6
7
8
8 6
3
1
2
1
5
0
6 3
4
4
5
0
0
2
1
2
3
2
K 589 (seq 35) 8
7
8 8
6
4
6 6
5
5 2
2
1
K 335 (seq 36)
7
0
3
4
2
1
K 675 (seq 36)
0
7
5
1
2
4
3
4
2
3
4
5
6
4
7
1
K 928 (seq 36)
3
0
4
2
8
8 6 6
3
4
5
0
1
2
5
1
2
3
4
5
6
7
K 967 (seq 37)
7
8
8
3
0
4
7
1
2
2
5
1
2
7
6
7 2
1
0
3
4
K 681 (seq 38)
7
8
8 6
6
5
5
6 0
5 2
K 334 (seq 39)
4
5 5
8
6
0
4
K 490 (seq 38)
8
8
1
3
K 859 (seq 38)
0
K 342 (seq 38)
7
0
3
5 4
K 711 (seq 38)
3
6
6 4
0
K 667 (seq 37)
8
6
1
1
8
6
3
2
K 2027 (seq 37)
K 469 (seq 37)
7
5
7 5
2 0
K 343 (seq 37)
5
K 517 (seq 36)
3 1
2 0
7
0
5
7
4
1
8
8
3
0
K 2042 (seq 36)
6
K 685 (seq 37)
0
0
7
8
4
1
5
6
1
7
3
8
6
0
4
3
K 703 (seq 36)
8
1
3
5
7
2
8
8
6
6
0
2
7
8
1 3
4
K 816 (seq 35) 7
7
3 0
4
K 2370 (seq 35)
K 341 (seq 35)
6
5 1
5
3 1
2
1
0
3
K 625 (seq 39)
4
2
1
0
3
K 651 (seq 39)
4
2
3
4
5
6
K 1036 (seq 39)
7
4
1
2
5
0
K 496 (seq 39)
Fig. 11 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 7 7
155
8
8 8
7
8 8
8
6 6 0 1
2
6
7
1
2
1
5
2
0
4
6
1 5
0 3
0
5 2
K 1426 (seq 40)
K 472 (seq 40)
3
4
5
0
1
2
7
4
5
2
3
4
0
1
K 2043 (seq 40)
6
K 799 (seq 40)
7
K 1483 (seq 40)
K 588 (seq 40) 7
7
8
6
3
4 4
3 5
3
8
8 7 8
8
8
6
6 1
2
0
6
7
4
3 4 3
4
1
2
5
5
1
2
6
7
6
3
3
4
0
K 1465 (seq 41)
0
K 1399 (seq 41)
5
1
1
5
2
0
4
7
1
2
K 2028 (seq 41)
6
2
K 582 (seq 41)
8
8
5
7
4
6
8
5
3
0
K 508 (seq 41) 7
7
0
5
3
8
4
K 817 (seq 41)
8
6 3
1
5
2
0
4
3
4
5
2
3
6
7
0
5
2
1
0
3
0
4
1
2
0
1
1
2
K 1481 (seq 42)
K 2253 (seq 42)
K 668 (seq 42)
6
3
4
5
6
7
K 919 (seq 42)
K 574 (seq 42) 7
8 7
8 8
8
5
7
4
6
8
1
2
6 3
1
5
2
0
4
0
4
5
2
3
6
3
7
5 1 2
1
0
3
3
4
5
6
7
0
1
K 991 (seq 43)
6
K 592 (seq 43)
7
2
4
K 704 (seq 43)
8 8
4
7
3
6
2
5
0
K 1472 (seq 43)
8
K 2250 (seq 43)
8
6 0
4
1
5
3
1
1
7
3
4
2 2
0
0
5
0
6
1
2
K 1346 (seq 44)
3
4
5
6
7
K 913 (seq 44)
K 2146 (seq 44)
K 595 (seq 44)
7
3
8
8
46
7
2
8
5
8
6 0 1
7 3
4
4
5
1
0
3
5
1 2
2
1
2
6
K 1335 (seq 45)
3
4
5
6
0
7
K 1000 (seq 45)
0
K 2151 (seq 45)
K 598 (seq 45) 7
8 8
8
5
7
4
6
8
1
2
6 0
4
5
2
3
6
3
7
5 1 2
1
0
3
4
2
3
4
5
6
7
0
1
K 1038 (seq 46)
K 615 (seq 46)
7
0
K 1484 (seq 46)
8
8
5
7
4
6
K 2252 (seq 46)
8
8
6
3
0
4
5
2
3
6
7
5 1
2
3
4
5
K 861 (seq 47)
6
7 2
1
0
3
K 702 (seq 47)
4
0
1 K 2255 (seq 47)
2
0
1
K 1466 (seq 47)
Fig. 12 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
156
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
7
7
8
4
8
8
5
6
7
8
6 6
3
7 5
1
5
3
2
4
0
2 0
1
2
1
K 2129 (seq 48)
7
0
3
4
1
2
4
3
0
5
6
K 756 (seq 48)
K 2260 (seq 48)
K 682 (seq 48)
8 7
8
4
8 7
8
5
6
1
2
6 6
7
5
1
3
2
5 3
4 2
2
1
0
3
4
4
0
K 672 (seq 49)
3
1
4
5
7
2
2
3
4
5
6
6
1
0
7
K 892 (seq 50)
5
8
4
7
3
6
1
5 0
6
6
7
1
2
1
1
2
5
6
7
3
3
4
5
6
7
0
1
K 1021 (seq 51)
K 2186 (seq 51)
8
0 1
4
2
5
6
7
0
1
1
2
3
4
5
6
3
8
7
4
7
0
1
2
3
4
5
6
7
6
2
7
8
6
6
3
0
3 2
4
2
1
1 7
8
0
1
1
K 442 8 (seq 54)
5
K 451 (seq 54)
7
7
8
6
6
5 5
1
2 0
6
0 K 31 (seq 55)
4 4
8
4
2
3
5
K 529 (seq 54)
7
3
2 0
K 30 (seq 54)
5
K 372 (seq 53)
8
5
4
0
K 2406 (seq 53)
8
4
6 3
2
8
8
1
7
5
K 298 (seq 53)
K 21 (seq 53)
7
4
7
8
0
1
3
1
6 3
2
6
6
K 1337 (seq 51)
4
8
1
0
5
2
K 60 (seq 52)
7
6
5
8
2
K 2408 (seq 52)
K 0 (seq 52)
5
4
0
2
6
4
7
3
7
0
5
3
K 2361 (seq 51)
8
4
8
8
3
0
3
K 2359 (seq 50)
5
0
8
5
K 1344 (seq 50)
8
7
7
4
2
K 2191 (seq 50)
2
K 2258 (seq 49)
0
3 1
0
6
8
6
0
5
8
3
8
0
K 780 (seq 49)
K 2128 (seq 49)
2
1
0
5
3
K 603 (seq 55)
4
0
1
2
3
K 2365 (seq 55)
4
2
1
0
3
4
K 720 (seq 55)
Fig. 13 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
8
5
6
7
3
7
8
8
6
4
6
5 5
2
2
1
1
7
3
5
0
3
4
0
1
8
7
2
3
4
2
1
K 2366 (seq 56)
K 612 (seq 56)
K 32 (seq 56)
6
7
6
0
5
7
8
0
3
4
K 700 (seq 56)
8
7
8
8
4
6
6 6
7 5
3
0 4
2
1
6 3
0
8
7
4
2 0
K 498 (seq 57)
7
8
5
8
6
6
3
3 2
2
1
2
1
4 4
0
K 466 (seq 58)
7
3
5
0
K 34 (seq 58)
5
1
7
6
4
6
1
K 2130 (seq 57)
8
0
5
4
2
K 492 (seq 57)
7
3
4
5 0
1 K 33 (seq 57)
5
3 2
1
8
5
7
4
6
5
0
K 438 (seq 58)
8
2
1
K 514 (seq 58)
8
8
4 0 1
1
2
2
6
3 3
0
5
1
2
6
3
K 2212 (seq 59)
K 1417 (seq 59)
7
2
7
4
K 66 (seq 59)
6
1
4
0 5
0
3
7
8 8
4
7
3
6
2
5
5
K 1408 (seq 59)
8
8
4 1
0
1
2
1
2
3
4
5
6
0
7
K 968 (seq 60)
0
4
K 71 (seq 60)
5
0
3
6
7
8 8
1
2
3
4
5
6
7
6
7
K 929 (seq 60)
K 2137 (seq 60)
7
3
6
2
5
8
8
4 1
0
1
2
0
3 1
2
3
4
5
6
0
7
K 1028 (seq 61)
0
1
2
3
4
5
K 884 (seq 61)
K 2142 (seq 61)
K 75 (seq 61)
5
6
7
8 8
6
7
8
8
4 0 1
1
2
2
6
3
0
5
7
1
2
6
4 3
0
7
4
0
5
1
2
3
4
3
K 2356 (seq 62)
K 1401 (seq 62)
5
K 1428 (seq 62)
K 78 (seq 62)
5
6
7
8 8
4
5
6
7
8
8
4 0 1
1
2
2
3 0
6
0
3
7
1
2
4 3
6
7
4 5
K 1411 (seq 63)
0
1 K 2207 (seq 63)
2
3
5
K 1414 (seq 63)
K 80 (seq 63)
Fig. 14 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
157
158
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 6
7
7
8
3
8
6
5 0
2
2
4
8
3 1
K 477 (seq 64)
8
7 3
4
4
1 3
6
6
7
5
2
K 176 (seq 65)
7
0
1
4
5
0
0
5
0
2
1
8
1
2
2
1
K 441 (seq 64)
8
3
3
4 4
0
7
5 0
3
K 513 (seq 64)
7
6
5
0
K 105 (seq 64)
8
6
2
1
7
8
5
3
1 4
6
7
4
5
2
K 1334 (seq 65)
8
7
1
6 3
4
8
0
4
2
5
6 3
4
6
2
0
4
6
K 1347 (seq 65)
7
8
3 0
2
1
2
K 2247 (seq 66)
K 375 (seq 66)
7
7
5
5
0
K 290 (seq 66)
1
K 577 (seq 65)
1
5
3
6
K 583 (seq 65)
7
8
8
7
8
8
8
8 8 6
1 3
4
1
2
0 6
7
0
4 2
4
2
3
K 311 (seq 67)
1
2
5
5
1
7
3
7
2
2
1
K 320 (seq 68)
0
3
7
1
4
1
5
2
0
2
0
4
2
1
8
6
4
0
5
1
3
4
7
8
6 6
3 5
3
5
4 2
1
1
7
0
3
4
0
7
8
1
2
2
1
K 2256 (seq 70)
K 649 (seq 70)
K 439 (seq 70)
7 4
5
0
3
4
K 678 (seq 70)
8
8
7
8
6
6
6
6
3 5
3
5
4 2
0
2
8
6
2
8
K 2362 (seq 69)
7
8
6
5
4
5
2 K 346 (seq 69)
4
3
5
0
K 443 (seq 69)
0
7
6 3
8
0
5
K 666 (seq 68)
1 4
1
7
3
7
6
3
K 338 (seq 69)
2
K 2368 (seq 68)
8
5
6
2
3
K 1416 (seq 67)
5
0
K 618 (seq 68)
8
5
5
6
5
0
7
2 0
5
5
4
1
8
6
6
3
7
6
1
7
4
6
K 487 (seq 67)
7
8
6 4
K 493 8 (seq 67)
8
4
5
K 1498 (seq 67)
0
K 1413 (seq 67)
2
3
4
0
3
1
7
3
5
7
6
0
6
1
K 440 (seq 71)
1
0
3
K 705 (seq 71)
4
0
1 K 2257 (seq 71)
2
2
1
0
3
4
K 669 (seq 71)
Fig. 15 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176 7
8
7
8
1
6 1
0
2
6
1
2
5 3
0
6
0
5
3
K 1429 (seq 72)
K 456 (seq 72)
7
2
8
7
4
6
6
7
7 3
4 3
4
5
0
2
5
4
1
2
5
5
0
0
K 1396 (seq 72)
K 823 (seq 72)
8
8
8
6 1
4
7
8
7
3 4
8
159
1
2
3
4
5
6
K 2405 (seq 72)
K 534 (seq 72)
8
8
6 0
4
1
5
2
3
6
0
3
7
1
2
6
4
1
2 3
0
0
5
1
2
7
7
8
4 6
6
5
5
5
K 1398 (seq 73)
8
K 600 (seq 73) 8
3
K 2214 (seq 73)
K 1431 (seq 73)
7
7
4
5
6
1
2
3 2
1
0
3
4 2
1
K 627 (seq 74)
7
8
1
4
3
5
6 4
3
4
5
6
7
1
2
3
4
5
6
7
4
5
6
7
4
7
3
6
2
5
3
4
5
6
7
5
7
0
1
6
2
3
4
5
6
5
0
1
2
7
0
7
K 2149 (seq 76)
7
5
7
6
4
6
8
1
2
8
3
0
1
1
2
3
4
5
6
0
7
K 1037 (seq 77)
K 2221 (seq 77)
5
8
4
7
5
7
3
6
4
6
8
1
2
8
3
0
1
5
8
4
7
3
6
0
4
8
3
0
3
K 2026 (seq 75)
K 2383 (seq 75)
4
0
8
1
1
2
3
4
5
6
0
7
K 992 (seq 78)
K 2219 (seq 78)
8
8
2
K 933 (seq 79)
8
8
0
3
2
4
K 2182 (seq 78)
8
2
1
2
K 920 (seq 78)
1
0
7
5
0
3
5
K 2041 (seq 75)
K 2177 (seq 77)
8
2
4
2
K 860 (seq 77)
1
6
K 1041 (seq 76)
0
2
5
3
0
8
1
6
8
K 849 (seq 76)
6
8
2
0
0
3
K 2259 (seq 74)
K 783 (seq 75)
8
2
0
4
8
1
K 759 (seq 75)
1
3
7
2
0
0
K 708 (seq 74)
7
0
1
K 2189 (seq 79)
1
2
3
4
5
K 957 (seq 79)
6
7
0
1
2
3
4
5
6
K 2400 (seq 79)
Fig. 16 Re®ection sequences of the tubular algebras having a nontrivial of tubular type (244).
160
2
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
Determinants of Cartan matrices.
In this section we present computer programs verifying the conditions (i) and (ii) from the [1, Proposition 4.1]. The programs consist of a set of procedures written in Maple. In the sequel, we assume that the ¯les , , , and contain the Cartan matrices of exceptional tubular algebras. Further, each Cartan matrix of a tubular algebra B is given in the form
2
0 6 C1 0 6 6 C 0 6 2 C1 6 6 6 C3 C2 C1 6 6 .. .. .. 6 . . . 6 6 6C 6 r¡1 Cr¡2 Cr¡3 4
3
¢¢¢ 0 0 7
7
¢¢¢ 0 0 7 7 ¢¢¢ .. .
0 0 .. .. . .
¢ ¢ ¢ C1 0
Cr Cr¡1 Cr¡2 ¢ ¢ ¢ C2 C1
7 7 7 7; 7 7 7 7 7 7 5
where r is the maximal order of the roots of the Nakayama automorphism ºBb . Moreover, the Cartan matrix of the unique exceptional tubular algebra of the type (2; 3; 6) is given as 2 3
6 C1 0 7 4 5: C2 C1
The above assumptions on the form of Cartan matrices aim at simplifying the procedures used to verify the condition (ii). In the ¯le we have de¯ned the following table of Cartan matrices of algebras B1 (¸), B2 (¸), ¸ 2 K n f0; 1g. Exceptional2222 :=
22 66 66 66 66 66 66 66 66 66 44
1
0
0
0
0
1
1
0
0
0
1
1
1
1
0
3 2
0 7 6 1 6 0 7 7 6 0
0
0
0
0
1
0
0
0
1
1
1
1
0
7 1 0 1 0 0 0 7 7; 7 2 1 1 1 0 0 7 7 1 1 1 1 1 0 5
0 77 7 0 7 77
6 6 1 1 1 0 0 0 6 6 1 1 0 1 0 0 6 6 4 1 1 1 1 1 0 1
33
1
77 77 77 77 77 77 55
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
In the ¯le
161
we have de¯ned the table of Cartan matrices of algebras B3 ; : : : ; B8 .
Exceptional333 :=
22
3 2
3
66 66 66 66 66 66 66 66 66 66 66 66 66 66 44
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
1
0
0
0
6 0 7 7 6 0 7 6 6 0 7 7 6 1
0
0
1
0
0
0
0 7 7
1
1
1
1
0
0
1
1
0
0
1
0
1
1
1
0
1
0
0
1
0
1
1
1
0
0
0 7
6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
1
0
0
1
1
1
1
1
0
1
0
0
0
1
0
1
0
0
1
6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
1
0
1
0
0
0
0
2
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
1
1
0
1
1
0
1
0
1
1
1
1
1
1
0
1
2
2
7 6
7 6 1 ;6 6 6 1 2 1 1 1 1 0 0 6 7 6 1 0 1 1 1 1 0 0 7 6 1 7 6 1 1 0 1 1 0 1 0 5 4 1 0 7 7 0 7 7
3 2
1
7 7
0 7 7
7
0 7; 7 0 7 7
7 7 1 1 1 1 1 1 0 5 1
3
7 7 7 7 7 7 7 7; 7 7 7 7 7 7 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
1
1
0
1
0
0
0
0 7 7
1
1
0
1
1
0
0
1
1
1
1
0
1
0
1
1
1
1
1
1
0
0 7
7 7 7 7 7 7 7 7; 7 7 7 7 7 7 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
1
0
0
1
0
0
0
2
0
1
1
1
0
0
7 0 7 77 77 7 0 7 77
0
0
0
1
0
1
0
1
1
1
1
1
0
0
3 2
7 7
0 7 7
0 7 7; 7 0 7 7
7 7 0 1 0 1 1 0 1 0 5 1
33 77
7 0 7 77 77 7 0 7 77
77 77 1 0 0 1 1 0 1 0 55 0 77 1
162
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
In the ¯le B9 ; : : : ; B13 .
we have de¯ned the following table of the Cartan matrices of algebras
22
66 66 66 66 66 66 66 66 66 Exceptional244 := 6 6 66 66 66 66 66 66 66 44 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
1
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
1
0
1
0
3
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
1
1
1
0
0
0
0
0 7 7 0 7
1
0
1
0
1
0
0
0
0 7 7 0
1
1
1
1
0
1
0
0
0 7 0
1
1
1
1
1
1
1
0
0
1
0
1
1
0
1
0
1
0
1
1
1
1
1
1
1
0
1
3 2
7
7 7 7; 7 7 7 7 7 7 7 5
3
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0 7 6 1 7 6 0 7 6 0 1
0
1
1
0
0
0
0
0 7 7 0 7
0
0
0
1
0
1
0
1
0
0
0
0 7 7 0
1
0
0
0
1
1
1
1
0
1
0
0
0 7 0
0
0
1
0
0
1
1
1
1
0
1
1
0
0
1
1
1
0
1
0
0
0
1
1
0
1
0
1
0
0
1
0
0
0
1
1
1
1
1
1
1
1
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
0
0
0
1
1
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
1
1
1
0
0 0
7 7 7 7 7 7 7; 7 7 7 7 7 7 7 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
3 2
7
7 7 7; 7 7 7 7 7 7 7 5 33
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0 7 6 1 7 6 0 7 6 0 0
1
0
1
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
1
0
0
0
0
1
0
1
0
1
0
1
1
1
0
1
0
1
1
1
0
1
1
0
1
7 0 7 77 77 0 77
0
1
0
0
0
1
0
1
0
1
1
0
0
0
1
0 0
7 7 7 7 7 7 7; 7 7 7 7 7 7 7 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0 77 77 0 77
77
7 0 7 77 77 0 77 77 77
7 0 7 77 77 0 55
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In the ¯le of the algebra B14 .
163
we have de¯ned a (singleton) table containing the Cartan matrix
Exceptional236 :=
22 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 44
33
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
7 0 7 77 77 7 0 7 77
1
1
1
1
0
1
0
0
0
1
1
0
1
1
0
1
0
0
0
1
0
1
0
1
0
1
0
1
0
0
1
0
0
1
0
0
77
7 0 7 77 77 7 0 7 77 77
0 77 77 7 0 7 77
77 77 1 1 1 1 1 1 1 0 1 0 55 0 77 1
164
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
2.1 Condition (i). All procedures used to verify the condition (i) in [1, Section 4], as de¯ned in the ¯le , are presented below. The procedure constructs the matrix ¤dBB +1 from the matrix ¤dBB , as in the de¯nition of the condition (i) (see [1, Section 4]). The parameter is equal to the dB matrix ¤B , the parameter equals dB +1, and the variable corresponds to the matrix dB +1 ¤B . maketransl := proc(B1; s) local n, l, i, j, k, B2; n :=coldim(B1); l := s*n; B2 := matrix(l; l); for i to l do for j to l do B2 [i, j ] := 0 end do end do; for k from 0 to s ¡ 1 do for i to n do for j to n do B2 [i + k*n, j + k*n] := B1 [i, j ] end do end do end do; for k to s ¡ 1 do for i to n do for j to n do B2 [i + k*n, j + (k - 1 )*n] := B1 [j, i ] end do end do end do; B2 end proc;
The procedure makes the main computation. The parameters and are matrices, and the parameter is equal to the di®erence of their respective dimensions. (i.e. the number of rows of minus the number of rows of , which is also equal to the number of columns of minus the number of columns of ). The behaviour of this procedure depends on the value of the parameter . This
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165
parameter admits values 1, 2 and 3. (1) For = 1 the procedure compares the determinants of the matrices and . (2) For = 2 the procedure recursively removes in all possible manners the number of rows (out of requested ), indexed by the numbers equal or greater than . Upon each such sequence of removals the procedure performs a call to itself with set to 1. (3) For = 3 the procedure recursively removes in all possible manners the number of columns (out of requested ), indexed by the numbers equal or greater than (for the matrix ) or + (for the matrix ). Upon each such sequence of removals the procedure calls itself with set to 2. comput := proc(B1, B2, ¯rst, toDel, numbDel, di®er, step) local i, n, results; if toDel = 0 then if step = 1 then results := det(B1 )= det(B2 ) else results := comput(B1, B2, 1, numbDel, numbDel, di®er, step - 1 ) end if elif step = 2 then n := rowdim(B1 ); results := true; for i from ¯rst to n + 1 - toDel do results := results and comput(delrows(B1, i .. i),delrows(B2, i .. i ), i, toDel - 1, numbDel, di®er, step) end do else n := coldim(B1 ); results := true; for i from ¯rst to n + 1 -toDel do results := results and comput(delcols(B1, i .. i ), delcols(B2, i + di®er .. i + di®er), i, toDel - 1, numbDel, di®er, step) end do end if; results end proc
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Hence, to sum up, the procedure checks for equality of all corresponding pairs dB of minors of the matrices := ¤B and := ¤dBB +1 , with ¤dBB and ¤dBB +1 de¯ned as in [1, Section 4]. The procedure veri¯es if the tubular algebra B having the given Cartan dB matrix (denoted by ¤B in [1, Section 4]) satisfy the condition (i). As B is of one of the tubular types (2; 2; 2; 2), (3; 3; 3), (2; 4; 4), (2; 3; 6), the dimension of is 6, 8, 9 or 10. The procedure calls the procedure , to construct a second matrix (denoted by ¤dBB +1 ibidem) as well as the procedure to verify if those matrices have the property described in the condition (i).
checkalg := proc(A) local n, s, B, results, i ; n := coldim(A); if n = 6 then s:= 2 elif n = 8 then s:= 3 elif n = 9 then s:= 4 elif n = 10 then s := 6 else ERROR(`Wrong dimension of the matrix`, n) end if; B := maketransl(A, s+1 ); results := true; for i from 0 to n - 1 do results := results and comput(A, B, 0, 0, i, n*s, 3) end do; results end proc
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167
The procedure calls for each element of the table the procedure and declares whether the corresponding tubular algebra satis¯es the condition (i) or not. The procedure also displays the information whether all algebras corresponding to the Cartan matrices from satisfy the condition (i).
checkalgebras := proc(matrlist) local i, results; results :=true; for i to nops(matrlist) do if not checkalg(matrlist[i ]) then print (`A contradiction found`,matrlist [i ]); results := false else print(`Matrix satisfy the condition`, matrlist[i ]) end if end do; if results then print(`All matrices satisfy the condition`) end if ; results end proc
We can use the afore-mentioned procedures in the following way: [> [> [> [> [> [> [> [> [> [>
with(linalg): read "./cond_i.txt": read "./exc2222.txt": checkalgebras(Exceptional2222); read "./exc333.txt": checkalgebras(Exceptional333); read "./exc244.txt": checkalgebras(Exceptional244); read "./exc236.txt": checkalgebras(Exceptional236);
2.2 Condition (ii). First, we describe the procedures , , de¯ned in the ¯le . Let B be an exceptional tubular algebra B having no nontrivial rigid automorphisms and let be the Cartan matrix of B of the form described at the beginning of this section. Further, let be the maximal order of the roots of the Nakayama automorphism ºBb . b are of the form (’m ), where ’ is We note that in this case all admissible groups of B the (unique nontrivial) root of ºBb of maximal order, and m is a positive integer. The
168
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procedure computes the determinants of the Cartan matrices of the algebras m b B=(’ ) for the appropriate values of m (as speci¯ed in the de¯nition of the condition (ii) in [1, Section 4]). determ1 := proc(B1, st ) local d, m, n, i, j, s, B2, B3, B, determ; n := coldim(B1 ); s := n/st ; if n = 6 then d := 2 elif n = 8 then d := 3 elif n = 9 then d := 4 elif n = 10 then d := 6 else ERROR(`Wrong dimension of the matrix`, n, eval(n)) end if; B2 := transpose(B1 ); B3 := matrix(n, 2*n); for i to n do for j to n do B3 [i, j ] := B1 [j, i ]; B3 [i, n + j ] := B1 [i, j ] end do end do; for i to n do for j to n + 1 do B3 [i, j ] := B3 [i, j + i - 1] end do; for j from n + 2 to 2*n do B3 [i, j ] := 0 end do end do; determ := determinants; for m from n by s to (d + 1)*n do B := matrix(m, m); for i to m do for j to m do B [i, j ] := 0 end do end do;
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for i to n do for j to n do B [i, j ] := B1 [i, j ] end do end do; for i to n do for j to n do B [i, m - n + j ]:= B [i, m - n + j ] + B2 [i, j ] end do end do; for i from n + 1 to m do for j from i - n to i do B [i, j ] := B3 [((i - 1) modn) + 1, n + 1 + j - i ] end do end do; determ := determ, det(B ) end do; print(determ) end proc
The procedure should be used for an exceptional tubular algebra B for which there exists a nontrivial root ’ of the Nakayama automorphism ºBb such that all positive b are of the form %’m for some rigid automorphism % of B b and some automorphisms of B positive integer m. This procedure computes determinants of the Cartan matrices of m b algebras B=(%’ ) for a given algebra B, a rigid automorphism % and for values of m speci¯ed in the de¯nition of the condition (ii) in [1, Section 4]. The parameter should be equal to the order of ’, should be the Cartan matrix of B in an appropriate form and should be the matrix corresponding to the rigid automorphism ’. determ2 := proc(B1, st, B1r ) local d, m, n, i, j, s, B2, B3, B, determ; n := coldim(B1 ); s := n/st ; if n = 6 then d := 2 elif n = 8 then d := 3 elif n = 9 then d := 4 elif n = 10 then d := 6 else ERROR(`Wrong dimension of the matrix`, n,eval(n)) end if;
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B2 := transpose(multiply(B1r, B1 )); B3 := matrix(n, 2*n ); for i to n do for j to n do B3 [i, j ] :=B1 [j, i ]; B3 [i, n + j ] :=B1 [i, j ] end do end do; for i to n do for j to n + 1 do B3 [i, j ] := B3 [i, j + i - 1] end do; for j from n + 2 to 2*n do B3 [i, j ] := 0 end do end do; determ := determinants; for m from n by sto (d + 1)*n do B := matrix(m, m); for i to m do for j to m do B [i, j ] := 0 end do end do; for i to n do for j to n do B [i, j ] :=B1 [i, j ] end do end do; for i to n do for j to n do B [i, m - n + j ] := B [i, m - n + j ] + B2 [i, j ] end do end do; for i from n + 1 to mdo for j from i - n to i do B [i, j ] := B3 [((i - 1) modn) + 1, n + 1 + j - i] end do end do; determ := determ, det(B ) end do; print(determ) end proc
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171
The procedure should be used only for the exceptional tubular algebras of the tubular type (2; 3; 6). We know that there is only one such algebra B14 (up to re°ection sequence) and that ºBb14 has no nontrivial root. Moreover we know that admissible groups b14 are of the form (’m+1 ) or (’b ’m ) for the automorphisms of the automorphisms of B a a ’a and ’b described in [1, Section 2] and for some positive integer m. The parameter should be the Cartan matrix of B14 , given in an appropriate form, while the parameter should be equal to 2 (note that s = 2 is such that ’sa = %ºBb14 = ’sb for the (unique) rigid b14 ). The last parameter automorphism % of B is the table of matrices describing m+1 m the actions of ’a or ’b ’a (we omit the detailed description of its form). Procedure computes the determinants of appropriate algebras of a tubular type. determ3 := proc(B1, st, tabB2 ) local d, m, n, i, j, s, B2, B3, B, determ; n := coldim(B1 ); s := n/st ; if n = 6 then d := 2 elif n = 8 then d := 3 elif n = 9 then d := 4 elif n = 10 then d := 6 else ERROR(`Wrong dimension of the matrix`, n,eval(n)) end if; B3 := matrix(n, 2*n ); for i to n do for j to n do B3 [i, j ] := B1 [j, i ]; B3 [i, n + j ] := B1 [i, j ] end do end do; for i to n do for j to n + 1 do B3 [i, j ] := B3 [i, j + i - 1] end do; for j from n + 2 to 2*n do B3 [i, j ] := 0 end do end do; determ := determinants;
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for m from n by s to (d + 1)*n do B2 := transpose(multiply(tabB2 [((m/s - 1) mod nops(tabB2 )) + 1], B1 )); B := matrix(m, m); for i to m do for j to m do B [i, j ] := 0 end do end do; for i to n do for j to n do B [i, j ] := B1 [i, j ] end do end do; for i to n do for j to n do B [i, m - n + j ] := B [i, m - n + j ] + B2 [i, j ] end do end do; for i from n + 1 to m do for j from i - n to i do B [i, j ] := B3 [((i - 1) mod n) + 1, n + 1 + j - i ] end do end do; determ := determ, det(B ) end do; print(determ) end proc
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Clearly, using the procedures , , , we may easily check that the tubular algebras B1 (¸), B2 (¸), ¸ 2 K n f0; 1g, B3 ; : : : ; B14 do satisfy the condition (ii). We present this veri¯cation below. [> [> [> [> d> j b [> d> j b [> [> d> j b d> b d> b d> b d> b [>
with(linalg): read "./cond_ii.txt": read "./exc2222.txt": aut11:=array(sparse,1..6,1..6): aut11[1,1]:=1: aut11[2,2]:=1: aut11[3,3]:=1: aut11[4,4]:=1: aut11[5,6]:=1: aut11[6,5]:=1: aut12:=array(sparse,1..6,1..6): aut12[1,1]:=1: aut12[2,3]:=1: aut12[3,2]:=1: aut12[4,4]:=1: aut12[5,5]:=1: aut12[6,6]:=1: aut1i:=array(identity,1..6,1..6 ): aut1r:=array(sparse,1..6,1..6): aut1r[1,1]:=1: aut1r[2,3]:=1: aut1r[3,2]:=1: aut1r[4,4]:=1: aut1r[5,6]:=1: aut1r[6,5]:=1: determ2(Exceptional2222[1],2,aut 1i); determinants ,0,16,0,16,0 determ2(Exceptional2222[1],2,aut 1r); determinants ,0,16,0,16,0 determ2(Exceptional2222[1],2,aut 11); determinants ,0,0,0,0,0 determ2(Exceptional2222[1],2,aut 12); determinants ,0,0,0,0,0
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[> [> [> [> d> j b [> d> j b [> [> d> j b d> b d> b d> b d> b [>
with(linalg): read "./cond_ii.txt": read "./exc2222.txt": aut21:=array(sparse,1..6,1..6): aut21[1,1]:=1: aut21[2,2]:=1: aut21[3,3]:=1: aut21[4,4]:=1: aut21[5,6]:=1: aut21[6,5]:=1: aut22:=array(sparse,1..6,1..6): aut22[1,2]:=1: aut22[2,1]:=1: aut22[3,4]:=1: aut22[4,3]:=1: aut22[5,6]:=1: aut22[6,5]:=1: aut2i:=array(identity,1..6,1..6 ): aut2r:=array(sparse,1..6,1..6): aut2r[1,1]:=1: aut2r[2,2]:=1: aut2r[3,4]:=1: aut2r[4,3]:=1: aut2r[5,6]:=1: aut2r[6,5]:=1: determ2(Exceptional2222[2],3,aut 2i); determinants ,0,0,12,0,12,0,0 determ2(Exceptional2222[2],3,aut 2r); determinants ,0,0,12,0,12,0,0 determ2(Exceptional2222[2],3,aut 21); determinants ,0,0,0,0,0,0,0 determ2(Exceptional2222[2],3,aut 22); determinants ,0,0,0,0,0,0,0
[> [> [> [> [> d> j j b [> d> j j b d> b d> b d> b [>
with(linalg): read "./cond_ii.txt": read "./exc333.txt": aut31:=array(identity,1..8,1..8 ): aut31:=array(sparse,1..8,1..8): aut31[1,1]:=1: aut31[2,3]:=1: aut31[3,2]:=1: aut31[4,4]:=1: aut31[5,5]:=1: aut31[6,7]:=1: aut31[7,6]:=1: aut31[8,8]:=1: aut32:=array(sparse,1..8,1..8): aut32[1,1]:=1: aut32[2,3]:=1: aut32[3,4]:=1: aut32[4,2]:=1: aut32[5,5]:=1: aut32[6,7]:=1: aut32[7,8]:=1: aut32[8,6]:=1: determ2(Exceptional333[1],2,aut3 i); determinants ,0,256,0,4,0,4,0 determ2(Exceptional333[1],2,aut3 1); determinants ,0,0,0,12,0,12,0 determ2(Exceptional333[1],2,aut3 2); determinants ,0,16,0,16,0,16,0
J. BiaÃlkowski / Central European Journal of Mathematics 2(1) 2004 143{176
[> [> [> [> d> b d> b d> b d> b d> b d> b d> b d> b d> b d> b [>
with(linalg): read "./cond_ii.txt": read "./exc333.txt": read "./exc244.txt": determ1(Exceptional333[2],4); determinants ,0,6,0,6,0,0,12,6,0,6,12,0,0 determ1(Exceptional333[3],4); determinants ,0,6,0,6,0,0,12,6,0,6,12,0,0 determ1(Exceptional333[4],2); determinants ,0,0,0,12,0,12,0 determ1(Exceptional333[5],2); determinants ,0,0,0,12,0,12,0 determ1(Exceptional333[6],2); determinants ,0,0,0,12,0,12,0 determ1(Exceptional244[1],3); determinants ,0,0,6,0,6,0,0,0,6,0,6,0,0 determ1(Exceptional244[2],3); determinants ,0,0,6,0,6,0,0,0,6,0,6,0,0 determ1(Exceptional244[3],3); determinants ,0,0,6,0,6,0,0,0,6,0,6,0,0 determ1(Exceptional244[4],3); determinants ,0,0,6,0,6,0,0,0,6,0,6,0,0 determ1(Exceptional244[5],3); determinants ,0,0,6,0,6,0,0,0,6,0,6,0,0
[> [> [> [> d> j j b [> d> j j b [> [> d> j j b d> b d> b [>
with(linalg): read "./cond_ii.txt": read "./exc236.txt": auta:=array(sparse,1..10,1..10) : auta[1,2]:=1: auta[2,1]:=1: auta[3,5]:=1: auta[4,4]:=1: auta[5,3]:=1: auta[6,6]:=1: auta[7,7]:=1: auta[8,8]:=1: auta[9,9]:=1: auta[10,10]:=1: autb:=array(sparse,1..10,1..10) : autb[1,1]:=1: autb[2,2]:=1: autb[3,2]:=1: autb[4,4]:=1: autb[5,5]:=1: autb[6,7]:=1: autb[7,6]:=1: autb[8,10]:=1: autb[9,9]:=1: autb[10,8]:=1: autid:=array(identity,1..10,1.. 10): autr:=array(sparse,1..10,1..10) : autr[1,2]:=1: autr[2,1]:=1: autr[3,5]:=1: autr[4,4]:=1: autr[5,3]:=1: autr[6,7]:=1: autr[7,6]:=1: autr[8,10]:=1: autr[9,9]:=1: autr[10,8]:=1: determ3(Exceptional236[1],2,[au ta,autr,autb,autid]); determinants ,0,1024,0,16,0,16,0,1024,0,16 ,0,16,0 determ3(Exceptional236[1],2,[au tb,autid,auta,autr]); determinants ,0,0,0,0,0,0,0,0,0,0,0,0,0
175
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References [1] J. BiaÃlkowski: Cartan matrices of sel¯njective algebras of tubular type, CEJM, Vol. 2, (2004), pp. 123{142. [2] K. Bongartz: Critical simply connected algebras, Manuscr. Math. Vol. 46, (1984), pp. 117{136. [3] D. Happel and D. Vossieck: Minimal algebras of in¯nite representation type with preprojective component, Manuscr. Math., Vol. 42, (1983), pp. 221{243. [4] C. M. Ringel: \Tame Algebras and Integral Quadratic Forms," Lecture Notes in Math., Vol. 1099, Springer, 1984.
