Ring Theory And Algebraic Geometry

(7)

proving that is left A-linear. It is also right A-linear because ^(aOc)6

=

^d*(S)a^e(df

® c)b

i

(17)

=

(7)

=

3dt®
Notice that the dual basis for C satisfies the following equality (the proof is left to the reader):

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

Doi-Hopf Modules and Entwined Modules

15

Using this equality one computes

(31)

=

(34)

=

(7)

=

(9)

-

(18)

=

V d* * d* Z—/ J

This proves that ^ is^right C"-colinear. Conversely, given E.V2, first one needs to show that 9 = aJ~ 1 ((/>) € Vi. It is now more convenient to work with Horn. (C, A) rather than C* A. For / e Horn (C, A), 6,6' e /I, (30) can be rewritten as

(bfb')(c) = btf(c*)b'.

(35)

Take any c, d 6 C, a G /I and compute

(35)

=

is right A-linear)

=

^ is left A-linear)

=

(35)

a^4>(l ® d*) (c)

=

This proves that ^ satisfies (17). Before proving (18), we look at the right C-coaction pr o n / = c*(g>a S Horn (C, A) ^ C*(g)A Write pr(f) = /[0](8>/[i] € Horn (C,A)®C. Using (31), we find, for all c e C,

This means that for all / e Horn (C, A) )).

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

(36)

16

Brzezinski et al.

This can be used to show that 9 satisfies (18). Explicitly,

6(c(2)®d}^®c^l}

(36) (4> is right C-colinear)

=

V(c(i) <8> 0(c (2) d))

=

0(1 ® d)[0](c) ®^(1

=

<j>(\ ® d^)(c) iS> d(2)

It remains to be shown that a\ and a^1 are inverses of each other. First take 0 6 Vi. Then for all c, d <E C,

=

6(d®c).

Finally, for 4> eV2, a e A and c, d 6 C:

® c)

a Now we give an alternative description for W%.

PROPOSITION 4.8 Let C be finitely generated and protective as a k-module. Then

W = Wl ^ W2 =Romk£A(C*®A,A®C).

The isomorphism j3\ : W\ —> W2 is given by j3i(z) =

and the inverse of (3i is given by fcl(4>) = 4>(e®\}.

(37)

PROOF.- We have to show that @\(z) — is left and right ^.-linear and right Ccolinear. For all c* 6 C* and a, 6 6 A,

l (7)

=

53 a/ a^, 6*
c* (gi a)6,

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Doi-Hopf Modules and Entwined Modules

17

proving that <£ is right A-linear. The proof of left A-linearity goes as follows:

*, cl(2j)cf(l} (7)

=

=

(9)

=

(23)

=

=

Next one needs to show that <£> is right C-colinear. Using (36), one finds

E i (c*, i

(9)

=

^aza^,® (c*,C( ( 2 ) )(cf) ( 1 ) ® (cf) ( 2 )

Conversely, let e W2 and put z = 0(e ® 1) = ^^ a; ® c;. Using (30), we see that a(£l) = (e l)a = za, and z 6 Wi. Take z = '£llai®ci€Wl. Then

Finally, take £ W2, and write j3l l($) = is right A-linear, right C-colinear and left C*-linear,

i

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18

Brzeziriski et al.

and it follows that

)), as required.

D

Suppose that C is finitely generated and projective as a /c-module. From Proposition 4.7, it follows that F is separable if and only if there exists a map (f> € V^ = Horn ^ A(A <8> C, C* A) such that (/>(! C( 2 ))(c(i)) = e(c)l, for all c e C. In the Doi-Hopf case, this implies the Maschke Theorem in [12]. Now we apply the same procedure to determine when (F, G) is a Frobenius pair.

THEOREM 4.9 Consider an entwining structure (A, C, ijj), and assume that C is finitely generated projective as a k-module. Let F : A / t('0)5 ~* MA be the functor forgetting the C-coaction, and G — • ® C be its right adjoint. Then the following statements are equivalent: 1) (F, G) is a Frobenius pair. 2) There exist z = "^ ai ® ci £ W\ and 9 € Vj such that the maps

A®C and <j6: A ® C —> C* <S> A, given by l i

are inverses of each other. 3) C* A and A Cg> C are isomorphic as objects in AA / t(V')5PROOF.- 1) => 2). Let z e Wl and d e Vi be as in Theorem 4.4. Then <j) = (3i(z) and (f> = a i (8) are morphisms in A-Mfy)1^, and

(27) The fact that and are right A-linear and left C*-linear implies that 4>o4> = IC*®ASimilarly, for all c e C,

(18) (26)

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Doi-Hopf Modules and Entwined Modules

19

Since and <j> are left A-linear, 0 o = 2) =» 3). Obvious, since c/> and (j> are in 3) =» 1). Let (f> : C* ® A —> A <S> C be the connecting isomorphism, and put

z = (e ® 1) = E; ai ®ci e Wi, 6 = a^-1) e Vi. Applying (32) and (37), one finds e i g i l = <^)-1 ((£(£ 01)) = Evaluating this equality at c € C, one obtains (27). For all c £ C,

1 0 c = <j)(4>~l(l <£> c)) = ^az6»(cj 0 c (1) ) 0 c (2 ). i Applying e to the second factor, one finds (26). Theorem 4.4 implies that (F, G) is a Frobenius pair. D REMARK 4.10 Recently M. Takeuchi observed that entwined modules can be viewed as comodules over certain corings. This observation has been exploited in [5] to derive some properties of coring counterparts of functors F and G. It is quite clear that the procedure applied in Section 2 to extension and restriction of scalars can be adapted to functors associated to corings, leading to a generalization of the results in this Section. This will be the subject of a future publication.

5

THE FUNCTOR FORGETTING THE A-ACTION

Again, let (A,C,il>) be a right-right entwining structure. The functor

G' : forgetting the A-action has a left adjoint F' . The unit p, and the counit 77 of the adjunction are given at the end of Section 3.

LEMMA 5.1 Let M e AM(^}% N e c' M(if)}cA. C

Then F'G'(M) e A.MWOS

and

G'F' 6 .A/1 (•) ^. The left structures are given by

a(m 0 6 ) = am 0 b and pl(n 06) = /or a// a, 6 6 ^4, m £ M, n £ AT. Furthermore HM is left A-linear, and VN is left C-colinear.

Now write V = ^(G'-F', IA^ C ). W7' = MM(lc,^'G')- Following the philosophy of the previous Sections, we give more explicit descriptions of V and W . We do not give detailed proofs, however, since the arguments are dual to the ones in the previous Section. Let

V{ = {$ £ (C ® A)*

^(c (1) ®a^)c* 2) =tf(c ( 2 ) 0a)c ( 1 ) , for all c e C,aeA}. (40)

PROPOSITION 5.2 The map a : V -> V{, a(v'} =eovc is an isomorphism.

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Brzeziriski et al.

20

PROOF.- Details are left to the reader. Given $ € V, for N & M.c, the natural map v'N : N A —> IV is

n For any /c-linear map e : C —> A <8> A and c 6 C, we use the notation e(c) = e 1 (c) ®e 2 (c) (summation understood). Let W[ be the fc-submodule of Horn (C, A <8> A) consisting of maps e satisfying e 1 (c ( 1 ) )(g>e 2 (c ( 1 ) )(8)c ( 2 )

e1

=

"(I)'

(41) (42)

PROPOSITION 5.3 The map 8 : W -> W{ given by

is an isomorphism. Given e £ W[, C,1 = /3~^(e) is recovered from e as follows: for

M PROOF.- We show that 8 is well-defined, leaving other details to the reader. Consider a commutative diagram

A-

C-

•~A <8> C J^^..^ ® C ® A

3ft and right C-colinear. Write A(c) The map A = CC^A ° (7c C o,- ® a!i • Then C(i) <8> A(c( 2 )) = ^ ci(1) ig) c i(2 ) ® a, i

Applying e to the second factor, one finds C ( i ) ® e ( c ( 2 ) ) = A(c).

The right C-colinearity of A implies that a

(gi c *

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Doi-Hopf Modules and Entwined Modules

21

and hence proves (41). To prove (42) note that A = CA®C ° (nA <S> IG) is left right A-linear, hence e 1 (c)®e 2 (c)a

=

(IA ® £ ® /A)(Cii®c(( 1 ® c ) a ))

D

PROPOSITION 5.4 Let (A,C,ip) be a right-right entwining structure. 1 ) F ' = • ® A : Jv[c —> .M (VOS is separable if and only if there exists "9 G V[ such that for all c G C, i?(c&l)=e(c). (43)

2) G' : M^)^ —> M° is separable if and only if there exists e G W[ such that for all c<=C, e1(c)e2(c)=e(c)l. (44) 3) (F',G') is a Frobenius pair if and only if there exist •& G V{ and e & W( such that e(c)l

=

79( C ( 1 ) ®e 1 ( C ( 2 ) ))e 2 (c ( 2 ) )

(45) (46)

PROOF.- We only prove 3). If (F' , G') is a Frobenius pair, then there exist i/ G V and C' e W such that (1) and (2) hold. Take $ e V{ and e € W{ corresponding to i/ and ^' and write down (1) applied to n ® 1 with n G N e Mc , (47)

Taking N = C, n = c, and applying £c to the first factor, one obtains (45). Conversely, if -& e V{ and e e W^i satisfy (45), then (47) is satisfied for all AT e Mc , and (1) follows since v'N ® I A and C/v®/l are right A-linear. Now write down (2) applied to m e M e A1(V>)2i (48)

Take M = C ® A, m = c (g> 1, and apply £c to the first factor. This gives (46). Conversely, if $ € V{ and e e VFj satisfy (46), then application of (46) to the second and third factors in TOpi ® mli} ® 1> an<^ then EC to the second factor shows that (48) holds for all M e M(t/>)%. Finally note that (48) is equivalent to (2). D Inspired by the results in the previous Section, we ask the following question: assuming (F',G!) is a Frobenius pair, when is A finitely generated projective as a fc-module. We give a partial answer in the next Proposition. We assume that ip is bijective (cf. [2, Section 6]). In the Doi-Hopf case, this is true if the underlying Hopf algebra H has a twisted antipode. The inverse of ijj is then given by the formula

ip~l(a c) = c5(a(!)) (g> a (0 ).

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22

Brzeziriski et al.

PROPOSITION 5.5 Let (A,C,ip) be a right-right entwining structure. With notation as above, assume that (F', G') is a Frobenius pair. If there exists c & C such that e(c) = I , and if tjj is invertible, with inverse C —> C (x) A, then A is finitely generated and projective as a k-module. PROOF.- Observe first that (A, C,
that (7-10) hold, but with A and C replaced by A°p and Ccop. In particular, £(c

= e(c)a,

(49)

This can be seen as follows: rewrite (8) and (9) as commutative diagrams, reverse the arrows, and replace i/j by
a = e(c)a

(42)

= e(cv)av

=

0(c* <

Write (7®e)A(c) = Y.T=\ci®bi®ai £ C®A®A. For i = l , - - - , m , define a* e A* by (a*, a) = •&(cf <8> a^bij. Then {ctj, a* i = I , • • • , m} is a finite dual basis of A as a k-module.

D

From now on we assume that A is finitely generated and projective with finite dual basis {a,, a* | i = 1, • • • , m}. The proof of the next Lemma is straightforward, and therefore left to the reader.

LEMMA 5.6 Let (A, C, i]j) be a right-right entwining structure, and assume that A is finitely generated and projective as a k-module. Then A* (g> C € cM.($)CA. The structure is given by the formulae pr(a*®c]

=

I/ * ,~~ \ _ p \Q> 09 Cj ^

a* <S>c (1) <8>C( 2 ),

(52)

/ * \ y^ /o * /o\ T^ \d , &iih )C( -\\ 09 a^ Qv C / r j \ .

iooi

/CCQ°\

We now give alternative descriptions of V and W.

PROPOSITION 5.7 Let (A,C,il>) be a right-right entwining structure, and assume that A is finitely generated and projective as a k-module. Then there is an isomorphism Pi '• W[ —> W!} = Horn kA(A* ®C,C® A),

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

Doi-Hopf Modules and Entwined Modules

23

Pie = f l , with

The inverse of f3\ is given by J3l l (fi) = e with i

PROOF.- We first prove that (3i is well-defined. a) 0i(e) = ft is right A-linear: for all a* e A*, c e C and 6 6 A, we have

\^/

(42)

—

\

5 ^"ibib

=

(a*,e 1 (c ( 2 ) )*}cf 1 ) ® e 2 (c ( 2 ) )fe

b) /?i(e) = fi is right C-colinear: for all a* 6 A* and c e C1, we have

(9) (41)

, =

(

^^,

( l )

(3)$

(a*,e 1 (c ( 2 ) )^)cj' 1 ' ) ®e 2 (c ( 2 ) )®c ( 3 ) ) <8>C( 2 ).

c

e

) /?i( ) = fi is left C-colinear: for all a* e A* and c e C, we have

(9)

The proof that /3f ^Jl) = e satisfies (41) and (42) is left to the reader. The maps /?i and /9f 1 are inverses of each other since

(a*,e 1 (c))a l (g)e 2 (c) = el(c) ® e 2 (c),

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24

Brzeziriski et al.

) ((a*, (ai

At the last step, we used that for all c 6 C and a £ A, ®a) = c®a. D

PROPOSITION 5.8 Let (A,C,^) be a right-right entwining structure If A is a finitely generated projective k -module, then the map

<*i • V{ -» V2' - Horn g£(C7 ® A, A* ® C1), defined by a\ ($} = SI, with fi(c ® a) = (i9, C(!) (g) a^ai)a* (g) c*2) is on isomorphism. The inverse of ai is given by aj~ 1 (fi) = i? with

PROOF.- We first show that ai is well-defined. Take •& e V{, and let ai(i?) a) fi is right A-linear since for all a, b G A and c € C,

(2)

b) 0 is right C*-colinear since for all a G A and c G C, /3 r (Q(c®a)) (9)

= = =

c) f2 is left C'-colinear since for all a € A and c € C1,

(9)

=

*

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Doi-Hopf Modules and Entwined Modules \ ^ f\ f j

!

„

25 \ li)' ^ „

*

)a -

(7) (40)

Conversely, given fi, we have to show that a^^fi) — $ satisfies (40). Take any c ig> a e C ® ^. and write Q(c ® a) = ^ 6* ®dL £ A* ®C. Since O is right and left C-colinear, we have

Therefore we can compute a),

Thus (40) follows. Finally, we show that c*i and aj"1 are inverses of each other. c

},

1

We know that ai(a^ 1 (O)) is right A-linear. Hence suffices it to show that

for all c € C. From (51), we compute ((a* <S> c)b, IA 0 £c) = (a*, 6}e(c) = (a* g) c, 6 ® e c ). Now write fi(c ® 1) = X^r- a* ® cr

anc

^ compute

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26

Brzezinski et al.

THEOREM 5.9 Let (A,C,t{j) be a right-right entwining structure, and assume that A is finitely generated and protective as a k-module. With notation as above, we have the following properties: 1) F' is separable if and only if there exists SI 6 V^ such that for all c £ C, A®£c)

=£c(c).

2) G' is separable if and only if there exists 51 £ W^ such that for all c 6 C, ]T ai(ec ® lA)tt(a*i <E> c) = ec(c}l. i

3) The following assertions are equivalent: a) (F',G') is a Frobenius pair. b) There exist e £ W[, •& € V{ such that Q, = /?i(e) and O = cti(i)) are inverses of each other. c) A* C and C ® A are isomorphic objects in

PROOF.- We only prove a) => 6) in 3). First we show that £7 is a left inverse of fi. Since fi o Q is right ^4-linear, it suffices to show that

(40)

=

^(c ( 2 ) ®e 1 (c( 3 ))^)cj' 1 ) ®e 2 (c ( 3 ) )

(45)

=

c®l.

To show that f2 is a right inverse of O we use that fl o Jl is right (7-colinear and conclude that it suffices to show that for all c e C and a* G A* , (I A' ® e c )(n(n(a* ® c))) = £ C (c)a*. Both sides of the equation are in A*, so the proof is completed if we show that both sides are equal when evaluated at an arbitrary a & A. Observe that ))(i) ® e 2 (c (2) )*a l )a* ® (cf 1} )f 2)

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Doi-Hopf Modules and Entwined Modules

27

hence (I A- <8>e c )(n(n(a*®c)))(a) (42)

=

(a

(7)

-

{^a^e^VKcJf ®e 2 (c* 2) ))

(9)

- (a*,a^ 1 (( C *) ( 2 ) V)^((^)f 1 ' ) ® e 2 ((^) ( 2 ) ))

(46)

=

(a* ) a^)e( C *) =
as required.

6

D

THE SMASH PRODUCT

Let (B, A, R) be a factorization structure (sometimes also called a smash or twisted tensor product structure, cf. [27] [22, pp. 299-300] [16]). This means that A and B are /c-algebras and that R : A®B—+B(><)Aisa fc-linear map such that for all

a,c<E A, b, de B, =

bRr®arcR,

(54)

R(a®bd)

=

bRdr®aRr}

(55)

R(a®lB) R(lA®b)

- IB® a, = b®lA.

(56) (57)

We use the notation R(a ® b) = bR aR. B$RA is the fc-module B A with new multiplication = bdR#aRc. is an associative algebra with unit IB#^A if and only if (54-57) hold. In this Section we want to examine when B#RA/A and B$RA/B are separable or Frobenius. This will be a direct application of the results in the second part of Section 2. In Section 2, take R = A, S = B#RA. For v € Vi = Horn #^(5, R), define K : B -> A by Then 17 can be recovered form K, since z/(6#a) = «(6)a. Furthermore

a«(6) = al/(6#l) = v(bR#aR) = K,(bR)aR and we find that

V ^ Vl =* V3 = {K : B-+A o/e(6) = K,(bR)aR}.

(58)

Now we simplify the description of W = Wi C (B#RA) ®A (B#RA). Note that there is a /c-module isomorphism

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28

Brzeziriski et al.

defined by

Let W3 = j(Wi) C B ® B ® A. Take e = bl <8> 62 ® a2 & B ® B ® A (summation implicitely understood). Then e e W$ if and only if (4) holds, for all s = 6#1 and s = l#a with 6 e B and a e A, if and only if bb1 ®b2 ®a2 1

2

(6 )R ® (6 2 ) r (8) aRra

=

b1 626B ® a^, 1

2

2

= b ® 6 a a,

(59)

(60)

for all a € yl, fe € S. This implies isomorphisms W7 ^ M/i ^ VK3 = {e = b1 62 ® a2 e B ® B & A | (59) and (60) hold}.

(61)

Using these descriptions of V and W, we find immediately that Theorem 2.2 takes the following form.

THEOREM 6.1 Let (B, A, R) be a factorization structure over a commutative ring k. 1) B#RA/A is separable (i.e. the restriction of scalars functor G : A/ls#RA -^ MA is separable) if and only if there exists e = b1 ® 62 <S> a2 G Ws such that

6 1 6 2
(62)

is split (i.e. the induction functor F : MA ~* MB#RA *s separable)

if and only if there exists K 6 V-j suc/i i/ia^

K(!B) = IA-

(63)

5j B^nA/A is Frobenius (i.e. (F,G) is Frobenius pair) if and only if there exist K e Vs, e e W3 such that

(b2)R ® K^^jja 2 = 61 ® «(6 2 )a 2 = 1B ® 1A.

(64)

Theorem 2.7 can be reformulated in the same style. Notice that

Hom fl (5',^) = RomA(B#RA,A') ^Eom(B,A). Rom(B,A) has the following (A,B#rtA)-l>imodule structure (cf. (6)):

(cf(b#a))(d)=cf(db)a, for all a,c G C and b, d e B. From Proposition 2.5, we deduce that

V ^ V2 - 1/4 = Horn ^^(SfeA, Horn (£,>!)).

(65)

If B is finitely generated and projective as a /c-module, then we find using Proposition 2.6

W ^ W2 <* W4 = KomA>B#RA(Rom(B,A),B#RA). Theorem 2.7 now takes the following form:

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

(66)

Doi-Hopf Modules and Entwined Modules

THEOREM 6.2 Let (B,A,R) be a ring k, and assume that B is finitely {bi, b* | i — 1, • • • , 771} be a finite dual 1) B^RA/A is separable if and only

29

factorization structure over a commutative generated and projective as a k-module. Let basis for B. if there exists an (A, B#RA}-bimodule map

(j>: Horn (B, A) ^ B* A -> B#RA such that

2) B^RA/A is split if and only if there exists an (A, B^RA)-bimodule map B#RA -» Horn (B, A) such that

3) B#RA/A is Frobenius if and only if B* ® A and B#RA are isomorphic as (A,B#RA)-bimodules.

This is also equivalent to the existence of K € ¥3, e =

b1 ig> b2 <8> a2 £ W% such that the maps

and

!>: B#RA^Kom(B,A),

$(b#a)(d) = K(bdR)aR

are inverses of each other. The same method can be applied to the extension B#RA/B. There are two ways to proceed: as above, but applying the left-handed version of Theorem 2.7 (left and right separable (resp. Frobenius) extension coincide). Another possibility is to use "op" -arguments. If R : A®B —> B®A makes (B, A, R) into a factorization structure, then

R: Bop® A°p -> A°p <S> B°p

makes (A°P,B°P,R) into a factorization structure. It is not hard to see that there is an algebra isomorphism

(A0v#ABop)op * B#RA. Using the left-right symmetry again, we find that B#RA/ B is Frobenius if and only if (A 0 P#^B°P)°P/B is Frobenius if and only if (Aop#RBop)/Bop is Frobenius, and we can apply Theorems 6.1 and 6.2. We invite the reader to write down explicit results. Our final aim is to link the results in this Section to the ones in Section 4, at least in the case of finitely generated, projective B. Let (A,C,if)) be a right-right entwining structure, with C finitely generated and projective, and put B = (C*)°p. Let {ci,c* i = 1, . . . , n} be a dual basis for C. There is a bijective correspondence between right-right entwining structures (A, C, tj}} and smash product structures (C*op, A,R). R and ip can be recovered from each other using the formulae

J?(a®c*) =

(c*,cf)cJ(gia V ) ,

tp(c® a) =

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

((c*)fl,c) Ci aR.

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Moreover, there are isomorphisms of categories

and In particular, B^^A can be made into an object of ^.A/f ('!/') 5, and this explains the structure on C* ® A used in Section 4. Combining Theorems 4.9 and 6.2, we find that the forgetful functor M^}1^ —» MA and its adjoint form a Frobenius pair if and only if C* ® A and A <8> C are isomorphic as (A, (C*) op #fl J 4)-bimodules, if and only if the extension (C*)op#RA/A is Frobenius.

REFERENCES [1] K.I. Beidar, Y. Fong and A. Stolin, On Frobenius algebras and the Yang-Baxter equation, Trans. Amer. Math. Soc. 349 (1997), 3823-3836.

[2] T. Brzezinski, On modules associated to coalgebra-Galois extensions, J. Algebra 215 (1999), 290-317. [3] T. Brzezinski, Frobenius properties and Maschke-type theorems for entwined modules, Proc. Amer. Math. Soc., to appear. [4] T. Brzeziriski, Coalgebra-Galois extensions from the extension point of view, in "Hopf algebras and quantum groups", S. Caenepeel and F. Van Oystaeyen (Eds.), Lee. Notes Pure Appl. Math. 209, Marcel Dekker, New York, 2000.

[5] T. Brzezinski, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, preprint math.RA/0002105. [6] T. Brzeziriski and P. M. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), 1347-1367.

[7] T. Brzeziriski and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467-492. [8] S. Caenepeel, B. Ion and G. Militaru, The structure of Frobenius algebras and separable algebras, K-theory, to appear. [9] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Smash biproducts of algebras and coalgebras, Algebras and Representation Theory 3 (2000), 19-42. [10] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules, Applications, Adv. Math. 145 (1999), 239-290. [11] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules II, in "Hopf algebras and quantum groups" , S. Caenepeel and F. Van Oystaeyen (Eds.), Lect. Notes Puere Appl. Math. 209 Marcel Dekker, New York, 2000.

[12] S. Caenepeel, G. Militaru, and S. Zhu, A Maschke type theorem for Doi-Hopf modules, J. Algebra 187 (1997), 388-412. [13] S. Caenepeel, G. Militaru, and S. Zhu, Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997), 4311-4342.

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[14] S. Caenepeel, G. Militaru, and S. Zhu, Crossed modules and Doi-Hopf modules, Israel J. Math. 100 (1997), 221-247. [15] S. Caenepeel and §. Raianu, Induction functors for the Doi-Koppinen unified

Hopf modules, in "Abelian groups and Modules", A. Facchini and C. Menini (Eds.), Kluwer Academic Publishers, Dordrecht, 1995, p. 73-94.

[16] A. Cap, H. Schichl and J. Vanzura. On twisted tensor product of algebras. Common. Algebra 23 (1995), 4701-4735.

[17] Y. Doi, Unifying Hopf modules, J. Algebra 153 (1992), 373-385. [18] L. Kadison, The Jones polynomial and certain separable Frobenius extensions, J. Algebra 186 (1996), 461-475.

[19] L. Kadison, Separability and the twisted Frobenius bimodules, Algebras and Representation Theory 2 (1999), 397-414.

[20] L. Kadison, "New examples of Frobenius extensions", University Led. Series 14, Amer. Math. Soc., Providence, 1999. [21] M. Koppinen, Variations on the smash product with applications to groupgraded rings, J. Pure Appl. Algebra 104 (1995), 61-80.

[22] S. Majid. Foundation of Quantum Group Theory. Cambridge University Press 1995. [23] C. Nastasescu, M. van den Bergh, and F. van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397-413.

[24] R. Pierce, Associative algebras, Grad. Text in Math. 88, Springer Verlag, Berlin, 1982. [25] M. D. Rafael, Separable functors revisited, Comm. in Algebra 18 (1990), 14451459. [26] A. del Rio, Categorical methods in graded ring theory, Publ. Math. 72 (1990), 489-531.

[27] D. Tambara. The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 37 (1990), 425-456.

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Computing the Gelfand-Kirillov Dimension II J. L. BUESO, J. GOMEZ-TORRECILLAS and F.J. LOBILLO, Departamento de Algebra, Universidad de Granada. E18071-Granada. Spain. E-mail: [email protected]

1

INTRODUCTION

This paper has a twofold goal and a mixed nature. We propose an algorithm to compute the Gelfand-Kirillov dimension for finitely generated modules over solvable polynomial algebras and, from the theoretical point of view, we characterize these algebras within the class of all filtered algebras. When looking for a notion of dimension for modules over a non-commutative algebra A which allows its effective computation, it is quite natural to look at the existing algorithms in Commutative Algebra. It seems reasonable to try first the computation of the dimension for the algebra A itself. In the case of commutative finitely generated algebras over a field k, the problem is equivalent to the computation of the degree of the Hilbert polynomial of k [ x j , . . . , xn]/I, where / is an ideal of the commutative multivariable polynomial algebra k [ x j , . . . ,xn], and this can be done effectively by means of the computation of a Grobner basis for / (see [2, Section 9.3] and its references). These ideas can be exported from the case of cyclic k xi,... ,xn -modules to finitely generated ones (see [15, Section 4]). From this point of view, the notion of dimension for modules which extends properly to noncommutative algebras is the Gelfand-Kirillov dimension. The techniques from the commutative case can be easily adapted to compute the Gelfand-Kirillov dimension in the case of quantum affine spaces, namely, finitely generated k-algebras defined by relations of the type XjXi = qjiXiXj, for some nonzero scalars QJJ € k (see e.g., [17], where quantum affine spaces are called homogeneous solvable polynomial kalgebras). When dealing with more general non-commutative k-algebras, a fruitful idea is to consider R to be a finitely generated k-algebra, with finite-dimensional generating vector subspace V, in such a way that the graded algebra gr(R) associated to the standard filtration is a multivariable commutative polynomial algebra (see [6]) or a quantum affine space (see [12]).

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However, a very simple example like the Jordan plane defined by two generators x,y, subject to the relation yx — xy + x2 shows that the former techniques, based on standard filtrations with (semi)commutative associated algebras, had to be improved. In the Spring of 1996, the first author presented in the SAGA4 held in Antwerp-Brussels an algorithm to compute the Gelfand-Kirillov dimension for cyclic modules over solvable polynomials algebras ([18]) with respect to the degree lexicographical order (see [4, 5]), which is applicable to some quantum groups whose generators are subject to quadratic relations, like quantum matrices, quantum symplectic space or quantum euclidean space). A similar approach appeared in the later paper [17, Section 6]. However, there is a mistake in the basic result there [17, Lemma 6.1] which is carried on the whole section. An easy counterexample to [17, Lemma 6.1] is the aforementioned Jordan plane. The algorithm given in [4, 5] was extended to finitely generated modules over solvable polynomial algebras (called there PBW algebras) with respect to a weighted graded lexicographical order in [23, 7], which allows to handle algebras with nonquadratic relations (the simplest example is given by the commutation relations yx = xy + x3). Here we show that this algorithm can be used to compute the Gelfand-Kirillov dimension of any finitely generated module over any solvable polynomial algebra, without restrictions on the given term order. During our search for that algorithm we have discovered some interesting results on filtered algebras which, in particular, locate the solvable polynomial algebras (or, equivalently, the PBW algebras) in the theory of noetherian rings: they are precisely those algebras having a filtration (in most cases, non standard) whose associated graded algebras are graded quantum spaces. We also show that these algebras can always be re-filtered by finite-dimensional vector subspaces keeping a quantum space as associated graded algebra. This allows to use the well-known graded-filtered techniques to obtain that any solvable polynomial algebra R is an Auslander-regular noetherian algebra with exact and finitely partitive Gelfand-Kirillov dimension. Moreover, R is a Cohen-Macaulay algebra which satisfies the Nullstellensatz.

2

ADMISSIBLE ORDERS IN MONOIDEALS AND STABLE SUBSETS

Let N denote the additive monoid of all positive integers (including the neutral element 0). Let n be a strictly positive integer. We consider Nra as additive monoid with the sum defined componentwise. Let 61,... ,en be the canonical basis of this free abelian monoid.

DEFINITION 2.1 An admissible order < on (N™, +) is a total order such that, (a) 0 = ( 0 , . . . , 0) ^ a for every a e N".

(b) For all a, (3,7 e N™ with a ^ f3 it follows 0 + 7 ^ / 3 + 7. REMARK 2.2 By Dickson's Lemma (see, e.g., [2, Corollary 4.48]), admissible orders on Nn are good orders (i.e., any non-empty subset of N™ has a first element). By ^iex we denote the lexicographical order with € j ~
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Computing the Gelfand-Kirillov Dimension

35

define \a w = aiWi + - • • anwn, i.e., a\w is the dot product (a, w). The w-weighted degree lexicographical order ^w is defined by

and a ~( m ) = n N x {!,...,m} by a + (/3,i) = (a + /3,i). It is clear that any (a,i) e N n '( m ) may be written as (a, z) = a + (0, z) for any 1 < z < m and ct € N™. In the case m = 1,

we shall identify N"'(1) with N".

DEFINITION 2.3 A non-empty subset E of N"'( m ) is said to be stable if E = E + N™. The stable subsets of Nn are called monoideals of N n . The following lemma suggests that the study of some aspects of stable subsets of N™1*-"1-1 can be reduced to monoideals. This is the case of the notion of dimension that we will consider later.

LEMMA 2.4 Let E be a stable subset o/N™' ( m ) . For every i = 1,... ,m, the set Ei = {a e N"; (a,z) 6 E} is a monoideal o/N™. Moreover, E may be written as a disjoint union

where Ei = (0,i) + Ei. PROOF.

Straightforward.

D

The following result was pointed out in [15].

PROPOSITION 2.5 Let E be a stable subset o/N">( m ). There is a set (a^ii), . . . , (as,is) of elements of E such that s

^)+Nn)

(1)

Every admissible order X on Nn induces a total order on N n > ^ m \ which will also be denoted by ;< This order is defined by putting (2)

for any (a,i) and ((3,j) in N™' ( m ) . Dickson's Lemma extends to N re ' (m) , thus X is a well-ordering on N™1'"1-1. Moreover, it enjoys the following properties (a) (a,i) ^ 7 + (a,i) for every (a,i) € N"'W and every 7 € N";

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(b) for all (a,i), ((3,j) with (a,i) ^ (/3, j ) , and every 7 6 N n , it follows 7 + (c*,z) ^ 7+C/3J)

For any function / : N —> N, consider its dimension or growth degree denned as (3)

logra

which, of course, need not be finite. Some basic properties of this invariant can be found in [24], [27, Chapter 8] or [20]. DEFINITION 2.6 For any stable subset E C N">( m ) and any weight vector w with strictly positive integer components, we define the Hilbert function HF^ : N —» N of E relative to w by putting HF£(s) = card{(a, i) e N n >( m > \E

\a w ^ s}

for every s € N. In the case that w = (1, . . . , 1) we shall use the notation HF# for the Hilbert function.

In the proof of the next Lemma we shall use the notation |(o;,z)| w = |a w. By N™ we denote the subset of N" consisting of those vectors with all their components strictly positive.

LEMMA 2.7 Let E C N"'(m' be a stable subset and let w € N™ . Then d(HF£) = d(HFE). PROOF. Consider w = max{wj, . . . ,u>n}. If |(a,i)| < s, then |(a,z)|a, < ws and, thus, HFE(S) < HF^(ws). Moreover, it follows from

that \(a,i)\u < s implies |(a,i)| < s, whence HF%(s) < HFE(s).

D

By Lemma 2.7, the following definition makes sense.

DEFINITION 2.8 The dimension of a non-empty stable subset E of N n '( m ) is denned as dim(£) = d(HF^), for any w e N£. The study of the Hilbert function of a stable subset can be reduced to monoideals, as the following result shows.

PROPOSITION 2.9 Let E be a stable subset o/N™'( m ) and consider the decomposition

given in Lemma 2.4- Then

HF£ = HF£ + • - • + HF^m

(4)

dim(E) = max{dim(B1), . . . , dim(£;m)}

(5)

and

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Computing the Gelfand-Kirillov Dimension

37

PROOF. The formula (4) is clear, since (a,i) ^ E if and only if a <£ E{. The equality (5) follows now from [27, Lemma 8.1.7]. D It follows from Proposition 2.9 that the computation of the dimension of stable

subsets of N n '( m ' reduces to the case of monoideals. Let E be a monoideal of N™. In the case that E is not proper, the dimension can be computed easily as ,. ,„ / 0 dim(.E) = < . „ v ' \ n iff E = rt0

For proper ideals, we shall need the notion of support of a vector 0 ^ ex. 6 N n , defined as supp(a) = {z € {!,..., n}

at ^ 0}

Consider the set V(E) = {cr C {1, . . . , n} a n supp(a) ^ 0 Va e E} which can be computed from a set of generators ai, . . . ,ccs of E since, as can be checked,

V(E) = {ff C {!,..., n}

a n supp(a fc ) ^ 0 V/c = 1, . . . , s}

The following result allows the effective computation of the dimension of any monoideal, and hence, of any stable subset. A proof, inspired on the material of [2, Section 9.3], can be found in [5, Section 4].

THEOREM 2.10 Let E be a proper monoideal o/N™. (1) dim(£) = n - min{card(cr)

a 6 V(E)}

(2) If m is the maximum of the entries of all vectors in a finite set of generators for E, then there is an unique polynomial h(x] G Q[z] such that HF£(s) = h(s)

for every s ^ mn. REMARKS 2.11 Equivalent descriptions for the dimension dim(B) can be deduced from [11, Theorem 3.1]. The fact that HFB(S) coincides with a polynomial for s big enough was proved in [19, Lemma 16, p. 51].

3

PBW ALGEBRAS, QUANTUM RELATIONS AND FILTRATIONS

Let R be an algebra over a commutative field k, and let xj, . . . ,xn be elements in R. A standard monomial in xi, . . . ,xn is an expression xa = x"1 . . .x^™, where a = (ai, . . . , an) & N". Assume that an element r € R can be written in the form

r = Y, r<*xa

(ra e k)

(6)

aSN"

The expression (6) is called a standard representation of r. Of course, ra ^ 0 only

for a finite subset A/"(r) of N n . We will often refer as polynomials to the elements

of R having a standard representation.

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DEFINITION 3.1 Let ;< be an admissible order on N™, and consider an element r of R having a standard representation (6). The exponent exp(r) of (the standard representation of) r is defined as the maximum with respect to ;< of the finite set JV'(r). We note that exp(r) depends on the given order ^ and on the standard representation (6) of r. The type of relations that are satisfied by many interesting algebras suggests the following definition.

DEFINITION 3.2 Let R be a k-algebra generated by elements xi,...,xn. If there are nonzero scalars qji e k and polynomials pji (1 sC i < j ^ n) such that the relations Q - {xjXi = qjiXiXj +pji, 1 ^ i < j ^ n} hold in R, then we will say that the generators x\ , . . . , xn satisfy a set Q of quantum relations. If in addition, for every 1 ^ i < j ^ n for some admissible order ^ on N n , then the quantum relations Q are said to be ^-bounded.

DEFINITION 3.3 An algebra R over a field k is said to be a Poincare-BirkhoffWitt algebra if R is generated by finitely many elements x\, . . . , xn such that (PBW1) The standard monomials xa with a € N™ form a basis of R as a k-vector space. (PBW2) There is an admissible order X on Nn such that the generators xi, . . . , xn satisfy a set Q of ^-bounded quantum relations. Notice that R can then be thought of as the algebra generated by x\ , . . . , xn subject to the relations Q = XjXi = q^XiXj +PJI, 1 ^ i < j ^ n, such that no further latent relations appear. We will then say that R — k{xi, . . . ,xn; Q, :<} is a PBW algebra. Here, some pertinent comments about Definition 3.3 arise. The notion of PBW algebra is a restatement of the concept of solvable polynomial algebra introduced by Kandri-Rody and Weispfenning in [18]. It follows from [10, Theorem 1.2] that the condition (PBW2) is equivalent to

exp(/sO = exp(/) + exp(g)

(7)

for every /, g £ R. Finally, if R = k{xi , . . , , xn; Q, ^} is a PBW algebra and •<' is a different admissible order on N™, we cannot expect in general that R is a PBW with respect the new order -<' .

DEFINITION 3.4 ([26, Definition 3.7]). We will say that two elements x, y of the algebra R are semi- commuting if there is 0 ^ q & k such that yx = qxy. A graded k-algebra A is called semi-commutative if R is generated as k-algebra by a finite set of homogeneous semi-commuting elements.

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Computing the Gelfand-Kirillov Dimension

39

If A is a semi-commutative graded algebra with homogeneous generators j/i, . . . , yn, then there are non zero scalars q^ G k such that y^y, = qjiUiyj, 1 ^ i < j ^ n. Clearly, the standard monomials ya = y"1 .. .y^n with a € N™ span A as a kvector space. Therefore, every element of A has at least a standard representation. A special interesting case occurs when the standard representation of every element in A is unique. This leads to the following definition.

DEFINITION 3.5 A semi-commutative graded algebra is said to be a graded quantum affine space if the monomials ya are k-linearly independent (and hence, form a k-basis for A). Of course, every graded quantum affine space is a PBW algebra with respect to any admissible order. Our aim is to show that the PBW algebras are, precisely, those filtered algebras which have a quantum affine space as associated graded algebra. The first step will be Proposition 3.6. First, let us recall that an algebra R over k is (positively) filtered, if it is endowed with an ascending chain FR = {FnR \ n ^> 0} of vector subspaces, the filtration of R, satisfying for all n,m ^ 0 f. 1 e FQR,

2. FnR C Fn+\R, 3. (FnR)(FmR) C Fn+mR, 4

- R=Un>0FnR-

It obviously follows from the definition that FoR is a subalgebra of R. For any filtered algebra R with filtration FR, let us introduce the graded vector space

where F-iR = {0}. If r € FiR \ Fi-iR, then r + Fi^\R has degree i and a(r) = r + Fi-iR is the principal symbol of r. The multiplication on gr(R) is based, via the distributive laws, on the rule

f

cr(rs) 0

i rs £ i+j_i otherwise

where r G Fj/? \ Fi-iR and s € K,-J? \ Fj_iR. This well-defined multiplication makes gr(jf?) into a k-algebra, called the associated graded algebra.

PROPOSITION 3.6 Let R be a filtered k-algebra with filtration FR such that gr(.R) is semi-commutative generated by homogeneous elements y\,. .. ,yn with deg(j/i) = Ui ^ 0 for I ^ i ^ n. Put u = (MI, . . . , un) e Nn and, for 1 ^. i < j ^ n,

let 0 ^ <jjj G k such that j/jj/j = qjiyiyj(1) If Xi,... ,xn g J? are swc/i i/iat Xj = j/j + FUi-\R for 1 ^. i ^ n then

F,R = , therefore, R satisfies (PBW2) with respect to ^
with exp(pjj) -
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Bueso et al.

for some polynomials pji G R.

(2) If gf(R) is a graded quantum affine space, then R = k{xi, . . . , x n ; Q, ^u} is a PBW algebra.

PROOF.

1. We will prove that FSR = J^iai <« ^a?" by induction on s. Let Z

be the set of indices 1 <J i <J n such that Ui = 0. So, assume s = 0. If Z = 0, then Fo-R — k and there is nothing to prove. If Z ^ 0, then we can assume that FoR = gr(.R)o and, therefore, j/j = a;, for every i & Z. Since every element in gr(.R) has a standard representation in the monomials y a , we get

F0R= Now, assume that s > 0 and let r G -FS.R \ Fs_iR. Since r + Fs-iR € gr(/?)s and the elements yi, . . . , yn are homogeneous generators for gr(/?) we obtain that

|a|u =s

|a|u =s

By the induction hypothesis, f/

\7

T T*'~* £^ Pg _ -i1 -*/-?^ — I QtJL> tz X —

\ /

ITT*^ .IS.**'

And, therefore,

H U ^n

The equality follows from the fact that FR = {FsR}s^o is a filtration and Xi 6 -FUi-R for z = 1 , . . . , s. In particular, we have proved that xi,..., xn generate R as an algebra over k. For i,j such that 1 ^ i < j ^ s, there is a non-zero scalar qji such that yjj/i = qjiyiyj. Therefore,

x-Xi + Fu.+u.-iR = (x- + Fu.-iR)(xi + Fu.-iR) = which implies that ir jit

J

M j *•

'

j

"i i "'J

^

/

^

In particular, pji has a standard representation such that exp(pjj) ^ u e^ + e.,-. 2. We just need to prove that the monomials xa are linearly independent over k. Assume that the monomials xa are not k-linearly independent. We derive a contradiction as follows. There is a non trivial expression 5ZQewn caXa = 0, that can be chosen of minimal exponent (3. Let s = \/3\u, then 0= |a|u =s

|a|u <s

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Computing the Gelfand-Kirillov Dimension

41

which gives in gr(_R) s the following non trivial relation

E

v——^

caxa + Fs_lR=

\a\a=s

2_,

c

aya

\a\a=s

which contradicts the linear independence of the monomials ya.

D

REMARK 3.7 If the filtration FR in Proposition 3.6 is assumed to be finite dimensional then, necessarily, the vector u has all its components strictly positive.

The following result, although not explicitly stated there, was first proved by V. Weispfenning in [33, proof of Theorem 1]. We include a slightly different proof here.

PROPOSITION 3.8 Let R be a k-algebra finitely generated by x i , . . . , x n and assume that there are non zero qji € k, polynomials pji € R and an admissible order ^ on N™ such that the following ^-bounded quantum relations are satisfied for 1 ^ i < j ^ n with exp(pjj) -< €j + €j

Then there is w = (wi, . . . , wn) & N™ such that \exp(pji)\w < Wi+Wj. In particular, if R = k{zi, . . . , xn; Q, ^} is a PBW algebra, then R = k{xi, . . . , xn; Q, ^w} is a PBW algebra. PROOF. Let i,j be such that 1 ^ i < j
— Pji t ^/

Then

K.X

Write

where Cji — €j + €j — Aji. Then C is a subset of Z" such that (the unique extension to Z™ of) ^ is positive on C. By [29, Corollary 2.2] or [33, Proposition 2.1], there is some vector w = (wi, . . . , wn) £ N™ with w^ > 0 such that (w, /3} > 0 for every (3 € C. But this means that, for every 1
for every a € Aji

n After Proposition 3.8, it is possible to decide effectively if a given set of quantum relations Q is bounded by some admissible order on N" and, in the case of affirmative answer, to compute explicitly a bounding order of the form ^.w for some vector weight w = (wi, . . . , wn) £ N™ . To do this, write

c=

C

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Bueso et al.

where Cji = e, + €j — N(pji) for every 1 ^ i < j ^ n, consider the linear programming problem minimize /(to) = 101 + • • • + ws with the constraints

PROPOSITION 3.9 The set of quantum relations Q is bounded for some admissible order on N™ if and only if the linear programming problem (9) has a solution. In this case, <£> contains some vector w with integer components which gives a bounding ordering ^<w. Some explicit examples of computation of w by means of Proposition 3.9 can be found in [8].

PROPOSITION 3.10 Let R be a \i-~algebra generated by finitely many elements Xi,...,xn and assume that there are a vector w = (wi,...,wn) € N n , non-zero elements qji e k and polynomials pji e R such that the following ^<w-bounded quantum relations are satisfied for 1 ^ i < j sC n

Q EE XjXi = XiXj + pji

with exp(pij) -<w e.^ + €j

Define, for every positive integer s, F™R = ]Ciai <« kcca. Then

(1) The algebra R is filtered with filtration FWR = {F™R}. (2) The associated graded algebra grw(R) is generated by the principal symbols j/j = Xi + F^.^R forl^is^n. (3) If hji = Pji + F™i+w._1R then the following ^<w-bounded quantum relations are satisfied in grw(R) for 1 ^ i < j ^ n

Qw = yjVi = qjiViVj + hj{

with exp(%) -<w €i + 6j

(4) If R = k{o;i, . . . , xn; Q, ^<w} is a PBW algebra, then

is a PBW algebra. PROOF. 1. By [5, Proposition 1.7 (see also [18, Lemma 1.4]) we have that for every a, (3 6 N™ there exists qa^ € k such that xaxf3 = qa3Xa+P +

c

Obviously, 1 6 F^R. By (10), the monomials xa generate R as a vector space over k. This gives R = U s>0 F™R. Clearly, if a, j3 € N n are such that a\w < \(3\w then « ~<w ft- This implies that F™R C F™ R whenever s < t. For any pair of positive

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Computing the Gelfand-Kirillov Dimension

43

integers s,t, let / 6 F™R and g e F?R. We want to prove that fg e F™+tR. Clearly, it suffices to prove that xax@ € F^_tR for every a.,/3 & N™ such that a\w ^ s and \/3\w ^ t. By (10), it is enough to check that \*Y\W ^ s + t for every 7 ±>«, a + /3. But this follows from the definition of ^w. 2. For each s Jj 0, the s-th homogeneous component of the graded algebra gr(jR) is generated as a k-vector space by xa + F^.1R, with |o: w = n, and we know that xa + F^R = ya, where y > = X i + F^^R for i = 1, . . . , s. Therefore, yi , . . . , yn are homogeneous generators for gr(R). 3. We have the following computation

yjyi = (Xj + F^.^RKxi + F^R) = x,x{ + F^+W]_,R = (qjiX.Xj + F^.^.^R) + (Pji + F^.^.^R) = q^y-j + %

(11)

Moreover, since N(hji) C N(pji) we get that exp(hji) ^w exp(pji) -<w e* + €j. 4. We just need to prove that the monomials ya are k-linearly independent. Since grw(R) is a graded algebra, it suffices to prove that every linear combination of homogeneous monomials of the form Xliaj =scaya = 0 is necessarily trivial. But this is a consequence of the linear independence of the monomials xa , since we have REMARKS 3.11 1. When computing the relations Qw it is understood that we compute a standard representation of hji with respect to the standard monomials ya. This is very easy, in fact, if pj% = £QaQa:a then % = Y.\a\v =wi+Wi a «J/ Q 2. For every positive integer s it can easily be proven that

F?R={feR

|exp(/)U<s}

(12)

3. Let R — k{xi, . . . ,xn; Q, ^w} be a PBW algebra. It is easy to see that the filtration FWR is standard if and only if w = (1, . . . , 1), i.e., -^w is just the degree lexicographical order. ^ 4. There is an analog to Proposition 3. 10. (4) for the Rees algebra R associated to R = k{xi, . . . ,xn; Q, ^<w}, which was proved in [9, Teorema 2.6.3]. Thus, R = k[t,Xi, . . . ,xn;Q,^w] is a PBW algebra, where it is a new central variable, Xj = tWiXi for every 1 ^ i ^ n, w = (l,to), and the new ^-bounded quantum relations are

where — / T •TJ• — ^ - • T . ' r - 4— X j — yjz^i^j T^

\

/

,->, o^l ^A"' j

The foregoing propositions can be restated in a more compact form which is interesting from a theoretical point of view. We give two theorems. The first one explains, from the point of view of algebras given by generators and relations, when it is possible to find a filtration with semi-commutative associated graded algebra.

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Bueso et al.

THEOREM 3.12 The following conditions are equivalent for a k~algebra R. (i) R is filtered by a finite-dimensional filtration with semi-commutative associated graded algebra. (ii) R is filtered with semi-commutative associated graded algebra. (in) R satisfies a set Q of ^.-bounded quantum relations for some admissible order X. (iv) R satisfies a set Q of ^<w-bounded quantum relations for some vector w € N™.

PROOF,

(i) =^> (ii) is obvious.

(ii) => (iii) is given by Proposition 3.6.(1). (iii) => (iv). This is given by Proposition 3.8 (iv) =>• (i). By Proposition 3.8, w can be assumed to be such that \exp(pji)\w < Wi + Wj for every 1 ^ i < j <; n. This implies that, in Proposition 3.10.(3), all hjiS are zero and, hence, giw(R) is semi-commutative. D

DEFINITION 3.13 An algebra satisfying the conditions of Theorem 3.12 is said to be a somewhat semi-commutative algebra. The following Theorem characterizes the solvable polynomial algebras introduced in [18] in terms of nitrations. This gives a neat connection between the solvable polynomial algebras and the filtered structures on noncommutative rings studied in [28].

THEOREM 3.14 The following conditions are equivalent for a k-algebra R. (i) R is filtered by a finite-dimensional filtration with gr(fl) a graded affine quan-

tum space. (ii) R is filtered with gr(R) a graded affine

quantum space.

(iii) R is a PB W algebra. (iv) R is a PB W algebra with respect to some admissible order of the form ^w with w 6 NJ PROOF.

This follows from propositions 3.6, 3.8 and 3.10 in a way similar to that

of Theorem 3.12.

4

D

CONSEQUENCES AND EXAMPLES

Everyone working in the field of non-commutative algebras would greatly appreciate some way to get properties of a given algebra just checking some properties on the defining relations. This is the case for the algebras satisfying a set of quantum relations. You need just to check if the relations are bounded for some admissible order on N™ and the concluding theorems in Section 3, in conjunction with the well developed theory of filtered algebras, will take care of everything. We will recall some of these interesting properties. Let R be a noetherian k-algebra. We say that the Gelfand-Kirillov dimension is exact if for every finitely generated left or right -R-module M, GKdim(M) is an integer and

GKdim(M) = sup{GKdim(AO,GKdim(M/AO}

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45

for every submodule N of M. R is left (resp. right) finitely partitive if, given given any finitely generated left (resp. right) .R-module M, there is an integer n > Q such that, for every chain M = M0 3 MI D • • • D Mm

with GKdim(Mi/Mi + i) = GKdim(M), one has m < n. R has the radical property if every element in the Jacobson radical is nilpotent. R has the endomorphism property over k if for each simple left R-module M, End(M) is algebraic over k. If both properties are satisfied by R, then R satisfies the Nullstellensatz over k. A finitely generated left or right .R-module M satisfies the Auslander condition if for every integer n and every submodule TV of Ext^(M, R), we have Ext^(Ar, R) = 0 for every i < n. A noetherian ring with finite injective dimension is said to be Auslander-Gorenstein if every finitely generated left or right .R-module satisfies the Auslander condition. If in addition R has finite global homological dimension, we say that R is Auslander regular. Finally, if M is a left or right .R-module, the grade number of M is defined as

j ( M ) = inf{i I Ext*fl(M,.R) ^ 0} e N U {+00} The noetherian k-algebra R is Cohen-Macaulay if for all non-zero finitely generated left or right .R-modules M,

j ( M ) + GKdim(M) = GKdim(R) THEOREM 4.1 Let R be a k-algebra satisfying a set of bounded quantum relations. Then the following statements hold. (1) The algebra R is left and right noetherian.

(2) The Gelfand-Kirillov dimension is exact on short exact sequences of finitely generated left or right R-modules.

(3) The algebra R is left and right finitely partitive. (4) The Krull dimension of every finitely generated R-module is bounded by its Gelfand-Kirillov dimension. (5) The algebra R satisfies the Nullstellensatz over k.

// R is any PBW algebra, then, in addition to the foregoing properties, R is an Auslander-regular and Cohen-Macaulay algebra.

PROOF. By Theorem 3.12, R has a finite-dimensional filtration with semicommutative associated graded algebra. The noetherianity of R follows from [25, 1.6.9]. The exactness of Gelfand-Kirillov dimension on short exact sequences of finitely generated left or right .R-modules follows from [32, 4.4 Theoreme] or [26, 3.8]. R is left and right finitely partitive by [16, Theorem 2.9]. The bound on the Krull dimension can be obtained from [25, 8.3.18] (see [31] for the original citation). The Nullstellensatz follows from [26, 3.8]. Let R now be a PBW algebra. Then R has a finite-dimensional filtration such that gr(.R) is an affine quantum space. Using [22,

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Bueso et al.

46

3.6 Theorem] (see also [13, Theorem 4.2]), it is easy to see that gr(/?) is AuslanderGorenstein, so by [3, 3.9 Theorem] R is Auslander-Gorenstein. Moreover, by [25, 7.5.3] (see also [14]) gr(/?) has global finite dimension, so R is Auslander regular by [25, 7.6.18] (see also [30]). Finally, by [3, 3.8 Theorem] and [25, 8.6.5], R is Cohen-Macaulay if and only if gr(R) is Cohen-Macaulay, but this follows from [22, 5.10]

D

The following are examples of PBW algebras and, hence, they enjoy the properties listed in Theorem 4.1. In each case, we include an explicit vector weight w with strictly positive integer components in such a way that the filtration given in Proposition 3.10 yields a graded quantum affine space as associated graded algebra. • Enveloping algebras of finite-dimensional Lie algebras. Let 0 be a finite-dimensional Lie k-algebra with basis xi,... ,xn. Then the enveloping algebra U(Q) = k{xi,..., x n ; Q, ^.w} is a PBW algebra, where

i < j ^ n} and w = ( 1 , . . . , 1). • Quantum matrices are a special case of the algebra H(p, A) defined in [1]. H(p, A) = k{ziQ | 1 ^ i, a ^ p; Q, ^w} is a PBW algebra, where the relations are

+ (A - l)pjiXif}Xja

if j > i, /3 > a if j > z, a < j3 if j = i, (3 > a

and where the weight vector can be expressed here as a weight matrix:

w=

.

op+i

2

P+2

(2 i+Q ) 1<: .

.

...

2

2p

<

i

• Quantum Weyl algebras. Let Q = (r (k) = k{yi,...,yn,xi,...,xn;Q,^,w} is a PBW algebra, where

(i < 3) (i < 3} (i > J) and io = ( l , l , . . . , l , l , 2 , . . . , n ) .

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Computing the Gelfand-Kirillov Dimension

47

• Quantum symplectic spaces. For every nonzero element q in k, the algebra Oq(s)f>k2n) = k{j/i,... ,yn,xi,... ,xn;Q,^.w} is a PBW algebra, where j X i = q-xixj, lyi

2

- fyiXi = i-l

- i) i=\ E (?

• Iterated skew differential operators algebras. The foregoing quantized algebras and the enveloping algebras in the solvable case are instances of the following more general construction. Quantum euclidean spaces are obtained in this way. Let

be an iterated Ore extension of k. Assume that i(x\). Put w^ = max{l, 1012} + 1. Suppose we have defined wi, . . . , ifj-i for j > 2. For k = 1, . . . ,j — l, set Wfcj = max{aiifi + - • •+a,-_iuij__i; a 6 J\f(6j(xk))} and choose u>j = maxjljU;^ — Wi; 1 < i , f c < j — 1} + 1. Then /? = k{xj, ... ,xn;Q, ^w} is a PBW algebra, where Q=

iXj + 8j(xi]

I ^ i < j ^ n}

• Polynomial Ore algebras. For 1 ^ i ^ s, 1 ^ j sj r, let a,ji,bji G k with a_jj ^ 0, and let Cjj(cc) e k [ x i , . . . , z s ] be commutative polynomials. The algebra O = k{xi, . . . , x r , tti, . . . ,us; Q, ^w} is a PBW algebra, where the quantum relations are

j + bjiUj + Cji(x)

for

1

i <j ^ 5

for for

1 1

i <j ^ r i ^ s, I ^ j

The vector weight is

to = (!,...,!, wi,...,wr) where the weights w\, . . . ,wr are constructed as follows. For each 1 ^ i ^ s > 1 ^ J ^ r > let wji be the total degree of the polynomial Cji(x). Then choose Wj = max{u>ji 1
The algebra Vq(sl3(C)). Let q be a complex number such that q8 ^ 1. Consider the complex algebra given by ten generators /i2, /is, /23, &i, ^2, ' i j

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48

Bueso et al.

/ 2 , 612, 613, 623 subjected to the following relations. 613612 = 9~ 2 6l2ei3

/13/12 = 9~2/12/13

623612 = 9 2 6l2623 - 9e13

/23/12 = ?2/12/23 ~ 9/13

623613 = 9~ 2 6l3623

/23/13 = 9~2/13/23

£2 _ /2

612/12 = /12612 +

612/01 = 9~ 6

k

= k

612/13 = /13612 + 9/23^

6

612/23 = /23612

6 13 fc 2 = 9~ 1 /C26l3

&2/13 =

e

23^1 = 9^1623

&1/23 =

623fc2 = q~2k2e23

/c2/23 =

1

613/12 = /12613 -
fcf fc| - l\l\ 613/13 = /13613 -

——5 2—————25—

~

13*1 = ^

,

2,

q-ll2el2

7 ,.

/2/12 = 9"1 /J2

613/23 = /236i3 + 9^|ei2

ei2,;2 =

623/12 = /12623

e ^ / j = qr/ i e i 3

/ j / ^ = q-/13^

623/13 = /13623 ~ 9~ /12^2

C 13 / 2 = g/ 2 6l3

^2/13 = 9/13^2

623/23 —

_

By solving the linear programming problem (9) we obtain that the displayed quantum relations are ^TO-bounded, where w = (3, 5,3,1,1,1,1,1,1,1,) Bergman's Diamond Lemma shows, after some large computations, that the standard monomials in /i2, /is, /23, ki, fc 2 , li, / 2 ,612,613,623 form a C-basis of Vq(sls(C)), and, thus, it is a PBW algebra. Notice that the quantized enveloping C-algebra of s[s(C) is a factor algebra of V^(sIs(C)).

5

GROBNER BASES FOR MODULES

Let R = k{Qi,..., Qn\ :<,} be a PBW algebra and consider the free left _R-module Rm with basis ei,...,em. Every finitely generated left .R-module is isomorphic to a factor module of the form Rm/K for some m and some submodule K of Rm which has to be finitely generated (R is noetherian). The computational treatment of finitely generated left /?-modules has its basic tool in the notion of Grobner basis of submodules of free modules of finite rank. For this purpose, we will introduce some notions and the corresponding notation. Our point of view is influenced by the theory developed for the commutative case in [21] and [15]. Every element / e Rm can then uniquely be expressed as Y^iLi fiei> where /; € R, for 1 < i < m. For every (a,i) g pj n .( m ) ) consider the element x^a'^ = xaei of Rm, where xa = x*1 • • -x%». Since B = {xa-} a g N"} is a k-basis of R, it follows that

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Computing the Gelfand-Kirillov Dimension

49

Bm — {x(a'^; (a,i) £ N"'( m )} is a basis of Rm as a k-vector space. Therefore, every element / £ Rm has a unique standard representation

We define the Newton diagram of / to be

and the exponent of / to be exp(/) = maxA/"(/). Here, the maximum is computed with respect to the order on N"'(m) denned in (2). If (a,i) is the exponent of /, then

We will refer to c^a^ as the leading coefficient of / and we will write lc(/) = C( QJ J). The exponent of the submodule K of Rm is denned as the stable subset of N ra '^ m ^

Exp(tf) = {exp(/)

feK}

DEFINITION 5.1 A subset 9l , . . . ,gt of K is said to be a Grobner basis of K if

t

+N")

(13)

By Proposition 2.5, every submodule K of Rm posseses a Grobner basis, which is expected to be a set of generators for K. This can be deduced from the Division Algorithm, which is also the fundamental tool to compute effectively Grobner bases. By [10, Theorem 1.2], for every a, j3 £ PP, there is an unique nonzero scalar qa,p € k and a polynomial pa,/3 G R such that

xax13 = qa,0xa+f3 + pa>0

(exp(p Qj/3 ) -< a + (3)

Given / £ Rm with exp(/) = (a,k) e N™'( m ) j we define the scalar exponent of / as sexp(/) = a € N™. We will write (a,i) < n '( m ) (/3,j) if and only if i = j and (/3, i) = 7 + (a, z) for some 7 £ N™. As the reader could check by using Dickson's Lemma, the Division Algorithm given in Algorithm 1 is mathematically sound. The element r obtained in the Division Algorithm is said to be a remainder of 77* __ the division of / by the set F = {/lt . . . , fs}. We will denote by / any remainder. Of course, this notation is somewhat ambiguous, as the remainder is not in general uniquely determined. We have the following consequence of the Division Algorithm.

PROPOSITION 5.2 A finite subset G = {
elements is a Grobner basis of K if and only if f — 0 for every f £ K . In particular, every Grobner basis of K is a set of generators.

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Bueso et al.

Algorithm 1 Division Algorithm for free modules_______ _____ INPUT: / , / ! , . . . , /s € Rm with /, ^ 0 (1 < i < s) OUTPUT: hi,... ,hs,r such that / = hlf1 + ••• + hsfs + r and r = 0 or Af(r) D UI=i( ex P(/i)) + N n ) = 0 and max(exp(/i!) +exp(/ 1 ),... ,exp(/i s ) + exp(/J,exp(r)) = exp(/). INITIALIZATION: hi := 0 , . . . , hs := 0, r := 0, 5 := f WHILE g ^ 0 DO IF there exists i such that exp(/J <™'( m ) exp(g) THEN choose i minimal such that exp(/ i ) < n '( m ) exp(g)

__

O-i = l C (ff)l c (/i)'? S exp(g)-sexp(/ i ),sexp(/ i ) /H := hi + aixsexP^)-^p(f,)

g

:=g

-

aiX^xp(g)-sexp(fi)fi

ELSE r := r + lm(g) g := g - lm(gr)

The effective computation of the Gelfand-Kirillov dimension of Rm/K we are interested in needs as an important ingredient the computation of a Grobner basis for K. We present a version of Buchberger's algorithm that fits our PBW algebras (see Algorithm 2). This algorithm is a special case of the one we proposed in [10] with the purpose of the effective computation of "Ext functors" for modules over the left PBW rings introduced there. DEFINITION 5.3 Let f,g be vectors in the free left .R-module Rm, and write exp(/) = (a,j), exp(g) = ((3,k). Let 7 = (71, ...,7 n ) be defined by 7* = *i,/3j} for 1
SP(/ ) f f ) = [ g-i a a i c ( f f ) a .-y-a / _ g -i

lc(/)^-/3g

Htkk.

Algorithm 2 Left Grobner Basis Algorithm for Modules INPUT: F = { / ! , . . . , f,} C Rm with /^ 0 (1 < i < s)

OUTPUT: G = {glt... ,gt}, a Grobner basis for Rfi + --- + Rfs. INITIALIZATION: G := F, B := { { f , g } ; f ^ g e G} WHILE B ^ 0 DO Choose any {/,
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Computing the Gelfand-Kirillov Dimension

6

51

HOMOGENEOUS GROBNER BASES

Let R be a filtered k-algebra with filtration FR. A filtration on a left /?-module M is a family FM = {FnM n ^ 0} of vector subspaces of M satisfying

1. FnM C Fn+iM for all n > 0, 2. (FnR)(FmM) C Fn+mM for all n,m > 0,

3- M = \Jn>QFnM. The graded vector space

is endowed with a structure of graded left gr(/?)-module in the following way. If ?7i G FjM \Fj-iM, the principal symbol of m is given by
fcr(rm) a(r)a(m) — < I0

itrm<£ FJ+i_iM otherwise

where r G ^.R \ F^fi and m G FjM \ Fj^M. Now, let # = k{:ri,...,:r n ;Q, i^} be a PBW algebra (by Proposition 3.8 any PBW algebra is of this form, after replacing, if needed, the old order ;< by a new weighted order ^w). By Proposition 3.10, grw(R) = k { x i , . . . , xn; Qw, ^w} is a PBW algebra. Let Rm be a finitely generated free left /^-module with basis BI, ..., em. This module has a canonical filtration FwRm given by F™Rm = (F™R)Rm for every positive integer s. It is immediate that F™Rm = (F™R)ei ® • • • ® (F™R)em. An immediate consequence is that grw(Rm) is a free graded left gr w (_R)-module with basis BI, ... em. Every B-submodule K of Rm inherits a filtration FWK given by F?K = K n F™Rm for every positive integer s. Also, the factor module M = Rm/K is filtered by Ff(M) = (F™Rm + K)/K. With these nitrations, the short exact sequence of left /^-modules

0 -+ K -* Rm -> Rm/K -* 0 is strict in the sense of [27], and, hence, we obtain a short exact sequence of graded left gr™(.R)-modules

0 -^ g?w(K) -* grw(Rm) -+ grw(Rm/K) -> 0 which implies, in particular, that grw(Rm/K) ^ giw(Rm)/grw(K). Therefore, a presentation of grw(Rm/K) can be constructed from a presentation of Rm/K. We will show that this construction can be done effectively.

LEMMA 6.1 For any non-zero vector f G Rm we have exp(/) = exp(<j(/)).

PROOF. First, we will prove that exp(/) = exp(<j(/)) for every / e R. So, let a = exp(/) and write |ct|u, = s. Therefore, / = caxa + 5^/3-< a C/3X^• Since R is a PBW-algebra with respect to ^w, it follows that

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52

Bueso et al. *(f)=CaXa

+

V fi -<w fi

I/3|» =s

so exp(
=

'

/j

0

if |

if|exP(/J)U<S

and'/ = ( s / i , . . . , s / ™ ) . Clearly,

and, thus, exp(/) = (CM) = (exp(/ i ),i) - (exp^/O),*) = exp(
PROPOSITION 6.2 For every submodule K of the free left R-module Rm , we have

PROOF. This follows in a straightforward way from Lemma 6.1.

D

THEOREM 6.3 1. If {glt.. .,gt} is a Grobner basis for K then

is a Grobner basis for grw(K). 2. If {hi, . . . , hs} is a Grobner basis for grw(K) and g1, . . . , gt 6 K are such that a (di) = hi for every 1 ^ i ^ t, then {gl, . . . , gt} is a Grobner basis for K . PROOF.

1. Indeed, if {g1, . . . ,gt} is a Grobner basis of K, then

As gr™(.fQ) = Exp(.K') by Proposition 6.2, and exp(g i ) — e\p(a(gi)) by Lemma 6.1, clearly

so {a(gl\... ,a(gt)} is a Grobner basis of grw(K), indeed. 2. Similar to part 1.

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

D

Computing the Gelfand-Kirillov Dimension

7

53

THE GELFAND-KIRILLOV DIMENSION

Let us recall the definition of the Gelfand-Kirillov dimension of finitely generated kalgebras (details can be found in [27], [20], [24]). If V is a finite-dimensional vector subspace of R and n is a natural number, then Vn denotes the vector subspace of R generated by all n-fold products v\ • • -vn, where Vi & V. It is understood that V° = k. Define y(n) _ V^ yi i=0

Now, assume that R is finitely generated as k-algebra by V. The GelfandKirillov dimension of R (GKdim(R), for short) measures the asymptotic rate of polynomial growth of the 'dimension function' f ( n ) = dim^ V^n\ In fact, GKdim(.R) is the infimum of the real numbers r such that f ( n ) < nr for n 3> 0. It is known that this value is independent of the choice of the generating subspace V. The GK dimension of a finitely generated left .R-module M is given by the growth of the function /(n) = dimk(V^U), where U is any finite-dimensional vector subspace of M that generates M as left .R-module. This real number coincides with the dimension of the function f ( n ) as defined in (3). Our next goal is to show that the Gelfand-Kirillov of Rm/K can be computed from a given set of generators { f i , - - - , f s } of the submodule K. The first step is to replace the given admissible order X by some order ^<w for some vector w = ( u > i , . . . ,wn) with strictly positive integer components. This is possible due to Proposition 3.10 and the effective computation of the weight vector with strictly positive components is give by Proposition 3.9. Thus, assume that R = k{xi,... ,xn; Q, ^w} is a PBW algebra, which will be filtered by the iD-filtration FWR. Any finitely generated left R-module can be presented as M = Rm/K, where Rm is a free left .R-module with basis e i , . . . , e n , and K is a submodule of Rm. We endow M with the to-filtration FWM. The Hilbert function relative to this filtration is defined as

HF^(s) = dimk(F?M)

(14)

LEMMA 7.1 With the previous notation we have GKdim(M) =

PROOF. Let HFgrw ( M ) denote the Hilbert function of the graded left giw(R) module gr w (M), namely,

3=0

By [32, Proposition 3.2, Lemme 2.2],

GKdim(M) = GKdim(gr u '(M)) = Finally, for each positive integer s,

FW(M]

dirnk(gr-(M)J) j=0

j=0

j=0

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'

54

Bueso et al.

which implies that HFgrw ( M ) = HF^.

n

LEMMA 7.2 For every positive integer s, the set

K

s, a,

is a frasis /or i/ie k -vector space F™(M). In particular,

PROOF. Let G — {g1, . . . ,gt} be a Grobner basis for K and consider f + K with |exp(/)|u, < s. By the Division Algorithm, we know that / = qig^ + • • • + qtQt + f where exp(r) ^<w exp(/) and A/"(r) n Exp(K ) = 0, i.e.,

It follows from (cc,z) ^w exp(r) for all (a,«) £ A/"(r), that KQ:,?)!^, < s, so

r =

Moreover, if (a,i) <£ Exp(/f), then a;^a'^ ^ X. We thus have proved that the set of all x^a'^ + K with the property that |(a,z)|u, < k for (a,z) ^ Exp(K) is a system of generators for FSU)(M) = {/ + K] |exp(/)| TO < s}. Let us now prove their linear independence. If

then r = E( Q ,i)^Ex P (K) c(a,i)X(a^ £ ^> w h e r e M ( r ) r \ E x p ( K ) = 0. By Algorithm 1 and Proposition 5.2, we obtain that r = Gr = Q, so C(a,i) =0. D

THEOREM 7.3 Let M be a finitely generated left R-module with a presentation M = F/K. Then GKdim(M) = dim(Exp(A")). PROOF. This is a consequence of Lemmas 7.1 and 7.2.

D

Let us now describe the algorithm to compute the Gelfand-Kirillov dimension of a given finitely generated left .R-module M = Rm/K over a PBW algebra R = k{xi,... ,xn;Q,^<}. Given a set of generators {/ 1 ; ...,/ s } for K, proceed as follows.

1. Compute a vector weight w = (wi,..., wn) € N+ such that

R =. is a PBW algebra.

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Computing the Gelfand-Kirillov Dimension

55

2. Compute, by using the algorithms described in Section 5, a Grobner basis G for K.

3. Compute, from G, the stable subset Exp(K) of N"'( m ).

4. The Gelfand-Kirillov dimension of M is the dimension of 'Exp(K), which is computed by using Theorem 2.10.(1). REMARK 7.4 When w = (!,...,!), it follows that the ^-filtration of M = Rm/K is just the standard filtration of M. On the other hand, we have that the Hilbert function associated to this filtration is, precisely, WFExp(K)- By using the results of Section 2, we get that HFE X p(/f)( s ) 'ls a polynomial of known degree, for s big enough. By Theorem 2.10.(2) and Proposition 2.9, this polynomial can be explicitly computed by interpolation. Details can be found in [9]. This applies in particular to Weyl algebras and enveloping algebras of finite dimensional Lie algebras, which are PBW algebras with respect to -<degiexReferences

[1] M. Artin, W. Schelter, and J. Tate, Quantum deformations of GLn, Commun. Pur. Appl. Math. 44 (1991), 879-895.

[2] T. Becker and V. Weispfenning, Grobner bases. A computational approach to commutative algebra, Springer-Verlag, 1993. [3] J.-E. Bjork, The Auslander condition on noetherian rings, Sem. d'Algebre P. Dubreil et M.-P. Malliavin 1987-1988 (M.-P. Malliavin, ed.), Lecture Notes in Mathematics, no. 1404, Springer-Verlag, 1989, pp. 137-173. [4] J. L. Bueso, F. J. Castro, J. Gomez-Torrecillas, and F. J. Lobillo, Computing the Gelfand-Kirillov dimension., SAC Newsletter 1 (1996), 39-52, http://www.ugr.es/~torrecil/Sac.pdf/

[5] ____, An introduction to effective calculus in quantum groups, Rings, Hopf algebras and Brauer groups. (S. Caenepeel and A. Verschoren, eds.), Marcel Dekker, 1998, pp. 55-83. [6] J. L. Bueso, F. J. Castro, and P. Jara, The effective computation of the GelfandKirillov dimension, P. Edinburgh Math. Soc. (1997), 111-117. [7] J. L. Bueso, J. Gomez-Torrecillas, and F. J. Lobillo, Effective computation of the Gelfand-Kirillov-dimension for modules over Poincare-Birkhoff-Witt algebras with non quadratic relations., Proceedings of EACA-98, Universidad Complutense and Universidad de Alcala, 1998.

[8] ____, fiCudndo un Algebra finitamente generada es de tipo PoincareBirkhoff-Witt?, Proceedings of EACA-99, Universidad de La Laguna, Tenerife, 1999, pp. 195-204.

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[9] J. L. Bueso, J. Gomez-Torrecillas, and F.J. Lobillo, Dimension de GelfandKirillov y Polinomio de Hilbert-Samuel de Grupos Cudnticos: Metodos Algoritmicos, http://www.ugr.es/~torrecil/academia.pdf, Research Award of the Academia de Ciencias Matematicas, Fisico-Qufmicas y Naturales de Granada, 1999. [10] J.L. Bueso, J. Gomez-Torrecillas, and F.J. Lobillo, Homological computations in PBW modules., preprint, 1998. [11] G. Carra Ferro, Some properties of the lattice points and their applications to Differential Algebra, Comm. Algebra 15 (1987), 2625-2632. [12] F. Chyzak, Fonctions holonomes en calcul formel, Ph.D. thecnique, 1998.

thesis, Ecole Poly-

[13] E. K. Ekstrom, The Auslander condition on graded and filtered noetherian rings, Seminaire Dubreil-Malliavin 1987-1988, Lecture Notes in Mathematics, vol. 1404, Springer, 1989, pp. 220-245. [14] K. L. Fields, On the global dimension of skew polynomial rings, J. Algebra 13 (1969), 1-4. [15] A. Galligo, Algorithmes de calcul de base standards, Prepublication de 1'Universite de Nice, 1983. [16] J. Gomez-Torrecillas and T.H. Lenagan, Poincare series of multi-filtered algebras and partitivity, J. London Math. Soc., to appear. [17] Li Huishi, Hilbert Polynomial of Modules over Homogeneous Solvable Polynomial Algebras, Comm. Algebra 27 (1999), 2375-2392. [18] A. Kandri-Rody and V. Weispfenning, Non-commutative Grobner bases in algebras of solvable type, J. Symb. Comput. 9/1 (1990), 1-26. [19] E.R. Kolchin, Differential York, 1973.

Algebra and Algebraic Groups, Academic Press, New

[20] G.R. Krause and T.H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman Pub. Inc., London, 1985. [21] M. Lejeune-Jalabert, Effectivite D.E.A., Univ. Grenoble.

des calculs polynomiaux, 1984-85, Cours de

[22] T. Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. (1992), 277-300.

[23] F. J. Lobillo, Metodos algebraicos y efectivos en grupos cudnticos., Ph.D. Universidad de Granada, 1998.

thesis,

[24] M. Lorenz, Gelfand-Kirillov dimension and Poincare series, Cuadernos de Algebra, vol. 7, Universidad de Granada, 1988.

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[25] J. McConnell and J. C. Robson, Noncommutative noetherian rings, Wiley Interscience, New York, 1988.

[26] J. C. McConnell, Quantum groups, filtered rings and Gelfand-Kirillov dimension, Noncommutative Ring Theory, Lecture Notes in Math., vol. 1448, Springer, 1991, pp. 139-149. [27] J.C. McConnell and J.C. Robson, Noncommutative Noetherian rings, J. Wiley and Sons, Chichester-New York, 1987.

[28] T. Mora, Seven variations on standard bases, Preprint, 1988. [29] T. Mora and L. Robbiano, The Grobner fan of an ideal, J. Symbolic Comput. 6 (1988), no. 2-3, 183-208.

[30] A. Roy, A note on filtered rings, Arch. Math. 16 (1965), 421-427.

[31] S. P. Smith, Krull dimension of the enveloping algebra o/sl(2,C), J. Algebra 71 (1981), 89-94. [32] P. Tauvel, Sur la dimension de Gelfand-Kirillov, Comm. Algebra 10 (1982), 939-963.

[33] V. Weispfenning, Constructing Universal Grobner Bases, Proceedings of AAECC 5, Springer LNCS, 356, 1987, pp. 408-417.

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Some Problems About Nilpotent Lie Algebras J.M. CABEZAS and E. PASTOR, Dpto. de Matematica Aplicada, Universidad del Pais Vasco, Vitoria-Gasteiz. Spain. E-mail:[email protected] and [email protected] L. M. CAMACHO, J. R. GOMEZ and A. JIMENEZ-MERCHAN, Dpto. Matematica Aplicada I, Universidad de Sevilla, Sevilla. Spain. E-mail:[email protected], [email protected] and [email protected]

J. REYES and I. RODRIGUEZ, Dpto. de Matematicas, Universidad de Huelva, Huelva. Spain. E-m&il:[email protected] and [email protected]

Abstract

In this paper we survey the classification of some important families of nilpotent Lie algebras. We also consider some geometric problems on such families and several related computational topics. Most of the results have been obtained by at least one of the authors of this paper. So, we give an overview of the work developed on Lie theory over the last years by some members of the research group directed by J.R. Gomez.

1

INTRODUCTION

The importance of Lie algebras in different domains of mathematics and physics has become increasingly evident in recent years. For instance, in applied mathematics Lie theory remains a powerful tool for studying differential equations, special functions and perturbation theory. The study of the constructive aspects of Lie theory has become practical recently by the use of computers. Computers provide a valuable new research tool in Lie theory, which makes it possible to test (and even to formulate) various theoretical conjectures about Lie algebras. One of the first constructive problems to appear is

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related with the classification of such algebras. It is in this general framework, that our paper can be considered. A Lie algebra (Q,/J.) is a vector space 0 over a field IK, with a bilinear mapping M : 0 x 0 —> 0 denoted (X, Y) —> f-i(X, Y) — [X, Y] and called bracket product, verifying

1. [X,X} = 0, 2. [X, [Y, Z}\ + [Y, [Z, X}] + (Z, (X, Y]] = 0

for all the elements X, Y, Z of 0. The second condition is called Jacobi Identity. The dimension of the Lie algebra is the dimension of the vector space 0. Most of the Lie algebras which appear in this paper will be considered over the complex field C and will have finite dimension. By taking a basis (Xo,Xi,... ,Xn_i) in 0, the algebra is completely determined by its structure constants, that is, by the set of complex constants {C^}, defined by a(Xi,Xj) = X)fc=i ^tjXk- Then, we can identify the algebra 0 and its law fj,. Thus, the set Cn of laws of Lie algebras is an affine algebraic set defined by the polynomials expressions

(i)

£<

(2)

One could think of a program of classifying all Lie algebras by considering the above equations to be solved for unknown structure constants. This turns out to be a very complicated problem because of the non-linearity of (2). In fact, the general classification is an open problem. Levi's classical and famous theorem decomposes the classification of Lie algebras into the classification of semisimple and solvable Lie algebras. The classification of semisimple Lie algebra is well-know from the works of Killing and Cartan in 1914. The classification of the solvable Lie algebras can be reduced in a certain sense to obtain the classification of the nilpotent Lie algebras, which can actually be considered the unsolved problem. The descending central sequence of a Lie algebra 0 is defined by C°0 = 0, <^0 = [Q,Ct~lQ}. The first ideal CIQ = [0,0] is the derived algebra of g. IfC f c 0 = {0}, for some k, the Lie algebra is said to be nilpotent. The smallest integer k such that C fc g = {0} is called the nilindex of 0. The nilpotent Lie algebra g is filiform if dimC z g = n — i — 1 for 1 < z < n — 1. Therefore, an n-dimensional filiform Lie algebra 0 has nilindex n— 1. Hence, any filiform Lie algebra has a maximal nilindex, and filiform Lie algebras are considered to be the "less" nilpotent Lie algebras. The nilpotent Lie algebras with nilindex n — 2 will be called quasi-filiform. The first non-trivial classifications of some classes of low-dimensional nilpotent Lie algebras are due to Umlauf [57]. Some of the lists of nilpotent Lie algebras obtained by Umlauf contain errors and they are incomplete. Then, several authors have worked providing new lists (see [23], [24], [49], [52], [53], [54], [55], in the bibliography), but the complexity of the involved computations leads frequently to

errors. We remark that the first exact classification of the nilpotent complex Lie algebras in dimension n < 6 is probably due to Morosov [48]. It is also obtained in a different way by Vergne [59], who showed the important role of filiform Lie algebras in the study of variety of nilpotent Lie algebras laws.

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Recently a new invariant has been introduced by Goze to study nilpotent Lie algebras. Let g be a nilpotent Lie algebra. Let gz(X) be the ordered sequence of Jordan block's dimensions of the nilpotent operator a,d(X), where X £ g. In the set of these sequences, we consider the lexicographical order. Then the characteristic sequence

gz(g) =

max {gz(X}}

is an invariant of 0. An n-dimensional Lie algebra Q is filiform if and only if the Goze invariant gz(g) is (n — 1, 1). Using this invariant, Ancochea and Goze gave the classification of nilpotent Lie algebras in dimension 7 [3] and the classification of filiform Lie algebras in dimension 8 [2]. For a presentation of this problem see [44]. The list [2] contains some errors and it has been corrected by the authors [4] and by Selley [56]. The geometric approach to study nilpotent Lie algebras is one of the main methods which has been developed over the last years. The set A/"n of nilpotent algebra laws is an affine algebraic variety; two Lie algebras are isomorphic if and only if they belong to the same orbit of the natural action of the general linear group. In this approach the notion of filiform Lie algebra appears in a natural way; the subset J-n of filiform laws is an open set in J\fnAmong the geometric aspects of the variety J\fn we point out the description of their orbits, their irreducible components, etc. It is known that J\fn is reducible when n > 7 [5] , [46] . The closure of an irreducible component of Tn is an irreducible component of j\fn. Thus, we can obtain information about J\fn, by studying the variety Tn. For example, the variety }-n has at least 3 irreducible components if n > 12 [46], and consequently the variety J\fn as well. In fact, for n sufficiently large, the number of components of J\fn is at least of order n [5] . A survey of the problem is exposed in [44] , where we can find the description of the components of Fn, n < 10. In [44], one also finds the description of some components of the variety Afn, n > 12, which are obtained by using some Lie algebras belonging to JT^. These filiform Lie algebras can be graded in a certain way as we will see in Section 6. The importance of these graded filiform Lie algebras was shown by Khakimdjanov in the study of Mn [46]. In the cohomological study of the variety of laws of nilpotent Lie algebras established by Vergne [59] , the author also considers a class of graded filiform Lie algebras that plays a fundamental role. In fact, using such graded filiform Lie algebras, Goze and Khakimdjanov give in [45] the geometric description of the characteristically nilpotent filiform Lie algebras. Thus, it is clear that knowing a family of graded algebras of a certain class of nilpotent Lie algebras we get valuable information to solve problems on such a class. We now describe how this paper is structured, and the problems studied in each section. In Section 2 we present some geometric aspects about J-n. We have also summarized the classification of the n-dimensional filiform Lie algebras with n < 11. Finally, as an approach to a general filiform family, we consider the /c-abelian filiform Lie algebras, whose classification is given in any dimension. Section 3 and Section 4 are devoted to the study of nilpotent Lie algebras with small nilindex, that is opposite to the filiform ones. Concretely, in the third section

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we study the p-filiform Lie algebras, defined by the Goze invariant, in order to generalize the filiform case to nilpotent Lie algebras with given nilindex. We show how they can be useful to know the variety J\fn when p is close to the dimension, and there is very little information. In Section 4 we study the metabelian Lie algebras (nilindex 2) and some Lie algebras with nilindex 3 in order to look at relevant algebras with small nilindex where the classification is not possible. Section 5 and Section 6 return again to the study of Lie algebras of large nilindex. In fact, we consider quasi-filiform and 3-filiform Lie algebras. However, in these sections we deal with some graded families of those nilpotent Lie algebras in any dimension n. The obtained classifications are a valuable utility to study the variety Mn • Section 5 provides an extension of the graded algebras used by Vergne to the p-filiform Lie algebras, p = 2,3. In Section 6 we consider the gradation with a larger number of subspaces that the natural gradation provides, and we study the filiform and quasi-filiform case. The obtained algebras are better than the naturally graded ones to solve some cohomological problems. Finally, in the last section we show how symbolic calculus can be used to deal with some similar problems. In particular, how to make adequate use of some of the features that the sofware package Mathematica provides. The authors thank Professor Y. Khakimdjanov for his constructive suggestions. 2

FILIFORM LIE ALGEBRAS

The first non-trivial classifications of some classes of low-dimensional nilpotent Lie algebras are due to K. Umlauf. In his thesis [57] (Leipzig, 1891) he presented the list of nilpotent Lie algebras of dimension m < 6. He also gave the list of nilpotent Lie algebras of dimension m < 9 admitting a basis (Xo,Xi,... ,Xm-i) with [Xo, Xi] = Xi+i for i = 1,..., m - 2. Now, the nilpotent Lie algebras with this property are called filiform Lie algebras. Umlauf's list of filiform Lie algebras is exact only for dimensions m < 7; in dimensions 8 and 9 his list contains errors and it is incomplete. In the last years, it has been evident to study again nilpotent Lie algebras due to the importance of them in classification problems and their applications in different domains of mathematics and physics. Unfortunately, many of these papers are based on direct computations (by hand) and the complexity of those computations leads frequently to errors. M. Vergne obtained an exact classification of the nilpotent complex Lie algebras in dimension m < 6, and showed the important role of filiform Lie algebras, terminology introduced by herself, in the study of the variety of nilpotent Lie algebras laws [59]. Indeed, the subset Fn of filiform laws is a Zariski open set in the variety of nilpotent Lie algebras Afn- In spite of the developed study of filiform Lie algebras over the last years the classification problem is still open. In this section we

summarize the results obtained in the last years. 2.1

Obtaining laws of families of filiform Lie algebras

Obtaining a complete list of Lie algebras is a problem whose complexity increases with the dimension of the algebra. Thus, an approach to the study of the variety of nilpotent Lie algebras is the description of the algebraic sets of laws of filiform Lie

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63

algebras and their irreducible components. In [1] it is shown that the set of laws of 8dimensional nilpotent Lie algebras is the union of 8 irreducible components. These have been determined explicitly using some algorithms of symbolic computation developed ad hoc by Valeiras [58]. A modified implementation developed by himself in the language Mathematica can be found in [44]. Obtaining an initial handled set of polynomials, which determine the variety subject to study is an necessary first step in any subsequent study of such a variety. In [35] can be found how to associate a descent vector to each X G C J 0 which lets us write the brackets by using a recursive relation. Then we get an algorithm to obtain families of filiform laws by fixing an adapted basis. THEOREM 2.1 There exists a polynomially bounded algorithm which generates the set of laws of filiform Lie algebras, for each dimension n. By using the Jacobi identities we can obtain the conditions which the parameters have to verify to determine the family of filiform laws. Some of the obtained equations in the generated family could allow another more simple form and hence some of the parameters could be eliminated. In this way, and with an implementation of the algorithm stated in the above theorem, the 11-dimensional family of filiform Lie algebra laws was presented in [22]. Finally, in [35] one may find the study for dimension 12. THEOREM 2.2 The sets of the filiform Lie algebra laws over C11 and over C12 can be parametrized (up to isomorphism) respectively by the points of the affine algebraic sets J^u C C16, of Krull dimension 12, and F\-2 C C21 of Krull dimension 13. Finally, in [31], the polynomial equations which define the variety jFn let us determine and describe its two irreducible components. 2.2

Low-dimensional filiform Lie algebras

Ancochea and Goze gave the classification of nilpotent Lie algebras in dimension 7 [3] and the classification of filiform Lie algebras in dimension 8 [2],[4], by using the characteristic sequence introduced by Goze to study nilpotent Lie algebras. Using this invariant, several authors (see [26], [6]) gave the classification of filiform Lie algebras in dimension 9 and 10. The method of Ancochea and Goze, very well adapted to dimension 8, has been transformed in very complex computations in dimensions 9 and 10. The lists [6], [26] are again incomplete and many parameters can be eliminated. According to [46] filiform Lie algebras share some common properties starting in dimension 12. This suggests to study separately filiform Lie algebras for dimensions > 12 and to do a detailed study when the dimension is < 11. Using an approach of Vergne [59], the techniques developed by Khakimdjanov [46], and introducing the notion of elementary changes of bases we have finally given a complete classification up to isomorphism of all complex filiform Lie algebras of dimension m with m < 11 [33], In this way, for instance, the 130 families of the list [6] may be reduced to less than 65. Moreover, some families of Lie algebra laws obtained in [33] do not appear among those 130 families. Hence, the list [6] is not very useful due to the errors that such a list contains.

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The simplest (n + l)-dimensional filiform Lie algebra is Ln+i, defined by

where (Xo, Xi, . . . , Xn) is a basis of Ln+i (the undefined brackets being zero). We note that any (n + l)-dimensional filiform Lie algebra is isomorphic to a linear deformation of Ln+i (see [46]). Let A be the set of pairs of integers (k, r) such that I < k < n—I,2k+l < r
if 1 < i < k < j < n, and ^k,r(Xi,Xj) = 0 otherwise. Let JFm be the variety of filiform Lie algebras laws in an rn-dimensional vectorial space. Any (n + 1)dimensional filiform Lie algebra law fj, G J~n+i is isomorphic to /J,Q + fy, where JJ,Q is the law of Ln and $ is a 2-cocycle defined by

(fc,r)€A

and verifying the relation fy o $ = 0 with

* o #(x, y,z) = 3> (*(x, y), 2:) + * (#(?/, z), x) + # (*(z, x), y) . A basis (^Q; • • • , Xn) of an (n + l)-dimensional filiform Lie algebra with law fj, is called adapted, ii^X^Xj) = n0(Xi,Xj) + ^ > ( X i , X j ) , 0 < i,j < n. Denoting by [a\ the integral part of a from now on, all brackets of an (n + l)-dimensional filiform Lie algebra in an adapted basis (XQ, Xi, . . . , Xn) are determined by the brackets

+ a fcil X 2 fc+3 + • • • + ak>2k-2Xn,

l
If n is odd we have the supplementary bracket [^( n -i)/2; -X"(n+i)/2] = a (n-i)/2,n -XnThe non-uniqueness of an adapted basis for a filiform Lie algebra is clear, so we introduce adapted and elementary changes of bases. A change of basis / G GL(V) is called adapted to the law JJL G J-n+i(V) if the image of an adapted basis is an adapted basis. The most important observation here is that to study the closed subgroup of the group GL(V) composed of all adapted changes, denoted by GL a a(V), it is enough to study certain elementary changes of bases.

THEOREM 2.3 Let f be an element ofGL^V).

f(Xo)

Then f is given by

= aoXo + aiXi + • • • + anXn i)l 2
where agbi ^ 0. In fact, f is a product of the elementary changes of bases

f(X0) i/(a,6)= { f ( X l )

=

aX0 bXi

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a, 6 G C * .

Nilpotent Lie Algebras

f(X0)

65

= X0

f(Xl)

=

Xi

f(X0)

=

X0,

f(Xi)

=

Although the factorization of / obtained in the previous theorem is not unique, it may be chosen in a certain order. When a parameterized family of filiform Lie algebras is given, in order to eliminate the unnecessary parameters, the key is to find a sequence of elementary changes of bases. In [33] it is shown how the elementary changes of basis act. We can remark here that if a filiform Lie algebra law is given by its matrix of coefficients {a,k,m}, then the application of the change of basis i/(a, 6) gives a new law {bk,m}, with bk,m = bak,m/am~2k. So, we can transform the two nonzero coefficients ak,m,o,r,tt with ?7i — Ik ^ t — 2r, to 1 (if such a pair does not exist, we can transform only one of the coefficients in the law). The changes of bases r(a, k) and a(b, k) are basically used in parameters elimination. In each application, some coefficients are unaffected, and, under certain restrictions, we can eliminate one of the parameters. Therefore, parameter eliminations must be ordered in such a way that eliminated parameters remain zero in the following changes of bases. In this way the classification for the filiform Lie algebra with dimension less than or equal to 11 is obtained in [33].

THEOREM 2.4 Every n- dimensional complex filiform Lie algebra law, with n < 11, is isomorphic to a law p?n of the List of Laws given in [33]. This approach develops the method of Ancochea and Goze [2] and allows to simplify the computations, but when dimension increases we cannot obtain reliable results without the help of computers. The symbolic programming language Mathematica is useful as an assistant to get the classification. This can be seen in [32]. Finally, as an important application, the classification obtained in [33] has been used to study symplectic structures on the filiform Lie algebras [34]. 2.3

/c-abelian filiform Lie algebras

In addition to know some families of filiform Lie algebra laws in concrete dimensions, we also know the classification of filiform Lie algebras g with the condition CIQ abelian, obtained by Bratzlavsky [7]. The notion of k-abelian nilpotent Lie algebra is introduced in [27] (if CkQ is abelian) to consider the most general situation. This is a synthesis of two properties in the theory of nilpotent Lie algebras: the filiformity (the nilpotent Lie algebras with this property are "less" nilpotent) and the commutativity (the nilpotent Lie algebras with this property are "most" nilpotent). The classification of 2-abelian filiform Lie algebras is obtained in [27] by using the properties of the elementary change of bases in the 2-abelian case.

THEOREM 2.5 Every (n + 1)- dimensional (n > 6) 2-abelian filiform Lie algebra law is isomorphic to one of the laws n i i S ^ ( f l ) , H2,a,t(^}> M3,s,t(^)> /-M, 3,3+2(^)1

Mis.t, Mi4,s(^), Mi5,s(fi), fj,ie,s, Mi?, which are defined in [27].

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A Lie algebra is called characteristically nilpotent if all of its derivations are nilpotent. The definition and first example of such a Lie algebra has been given by Dixmier and Lister [25]. Khakimdjanov has shown that the family of characteristically nilpotent Lie algebras, which had been considered as narrow, is in fact very large [46]. As an application of the obtained classification, and by using the criterion of characteristically nilpotence given by Goze and Khakimdjanov [47],[45], we can conclude that all Lie algebra laws described in Theorem 2.5, except the 2-abelian algebras of laws fig,Sla+2(^o), Mis.t, Mie.s, Mi?, are characteristically nilpotent.

3

THE FAMILY OF p-FILIFORM LIE ALGEBRAS

The study of the variety of nilpotent Lie algebras through the filiform ones in the previous section is centered in those algebras of maximal nilindex among the nilpotent Lie algebras having the same dimension. When we want to study that variety by considering nilpotent Lie algebras with small nilindex the difficulties increase more and more as we can see in Section 4. To study some problems about nilpotent Lie algebras with small nilindex we have chosen in each concrete nilindex a family which allows us to use it in a similar way like the filiform family is used to obtain results on the complete variety. Namely, an n-dimensional nilpotent Lie algebra g is said to be p-filiform if its Goze invariant is gz(g) = (n — p, 1, /?., 1). Thus, the family of p-filiform Lie algebras is a large family of Lie algebras, including the filiform ones as a particular class. Indeed, the filiform and quasi-filiform Lie algebras are the p-filiform ones with p — 1,2 respectively. In this section we are interested in p-filiform families with p close to the dimension of the algebras of such families, i.e., when the family of algebras considered has small nilindex and where the complete variety of nilpotent Lie algebras is not very well-known. 3.1

p-filiform Lie algebras with p > n — 3

When we consider the family of p-filiform Lie algebras with dimension equal to n, it is obvious that 1 < p < n — 1. If p = n — 1 the situation is trivial because the family is just reduced to the abelian algebra in the appropriate dimension. If 0 is a p-filiform Lie algebra, we can always choose an adapted basis of the algebra g. That is, a basis (Xo, Xi,..., Xn-p, Y j , . . . , Yp-i) such that XQ e g— [g, g] and verifying [X0,Xi] = Xi+1 l
{X0,Xn_p}

[X^Yj]

= 0

= 0

l<j
We now describe how the family of p-filiform Lie algebras can be determined for the n-dimensional nilpotent Lie algebras when p = n — 2 and p = n — 3. The results obtained in [13] show how it becomes more difficult to obtain these classications as p decreases. We have adapted techniques already applied to specific dimensions; that is, we have considered appropriate arguments about nilpotentcy (generally, adjointnilpotency), characteristic sequences and those conditions which are derived from assumption that the algebras be p-filiform. Then, with an adapted selection of

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changes of bases, we have obtained the explicit classification for the p-filiform families with p > n — 3. Indeed, firstly we have considered some Jacobi identities in which XQ appears to get a preliminary expression of the brackets which define the family of algebras considered on an adapted basis. Those brackets can be dramatically simplified by considering some conditions which the structure constants have to verify in order to guarantee that Yj £ Im ad(Xo), I < j < p — I , for the vectors in the adapted basis considered, and to preserve the appropriate nilindex of the algebras. Secondly, as the Goze invariant of the algebras considered is (n — p, 1, 1, . . . , 1), there cannot exist a nonzero minor of order n — p in ad(Z), for all Z <£ [g,g]. Thus, some appropriate choices for Z (for example: XQ + AYj , Xi + AYj , AXo + Xi, . . . , A € K) with a process of finite induction, lead to some more restrictions. Finally, some adjoint-nilpotency considerations on any vector X of g (for instance, the characteristic polynomial P(A) of the adjoint matrix corresponding to such a vector has to be A"), some relations obtained from the remaining Jacobi identities, and some appropriate changes of basis (from simple change of scale to other, very complicated, ones) we obtain the goal of the classification by eliminating unnecessary parameters. In [13] we can see that the (n — 2)-filiform Lie algebras are direct sums of abelian algebras and Heisenberg algebras. THEOREM 3.1 In dimension n > 3, there are exactly [(n— 1)/2J nilpotent, real or

complex, pairwise non-isomorphic (n — 2) -filiform Lie algebras, denoted as gL 1 and whose laws in the basis (XQ, X\, . . . , Xn-p, Y\, . . . , ^p-i) are given by 4

[X0,*i]

=

^

X2

One also finds in [13] the classification of the family of (n — 3)-filiform Lie algebras, which are obtained in a similar way. We can see how the different dimensions of the center and of the derived algebras show that they are pairwise non-isomorphic algebras.

THEOREM 3.2 In dimension n > 5, there are exactly n — 1 nilpotent, real or complex, pairwise non-isomorphic (n — 3) -filiform Lie algebras, denoted as g/ 3 1 ^, and whose laws in the basis (Xo, Xi, . . . , Xn_p, Yj, . . . , Yp_i) are given by

(X0,Xi}

= Xl+1

[*1>^~4]

=

Y?k-i,Y2k — —

l
X3

= X3

,withl<

l
V ln-4-

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S

<

n

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As an application we have studied some cohomological properties of the p-filiform Lie algebras obtained above in which the determination of the algebra of derivations of each p-filiform Lie algebra is essential. They can be found in [20], [12] where it is also described how symbolic calculus can be used to solve similar problems. 3.2

(n — 4)-filiform Lie algebras

We now show the increasing difficulties involved in the problem when we consider p = n — 4. Obtaining the classification of (n — 4)-filiform algebras by using the method considered above was not successful. Thus, it has been necessary to develop new techniques. Actually, we have used central extensions of certain algebras with dimension smaller than the dimension of those algebras that we want to study. Let g be a p-filiform Lie algebra of dimension n. If we obtain its quotient by a one-dimensional central ideal, the resulting quotient algebra has dimension n — 1 and its Goze invariant is either (n — p — 1,1,..., 1) or (n — p, 1 , . . . , 1). In this second case we can obtain again the quotient algebra by a one-dimensional central ideal, obtaining a quotient algebra of dimension n — 2 and Goze invariant either (n — p — 1 , 1 , . . . , 1) or (n — p, 1 , . . . , 1). If we follow the process (that finishes because dim(Z(g)) is finite), it is obvious that we always get a quotient algebra of Goze invariant (n — p — 1,1,...,!) of dimension n — k. Conversely, we invert the process and from the algebras of characteristic sequence (n — p — 1,1,...,!) and dimension n — 1 we construct the central extensions that have Goze invariant equal to (n — p, 1,..., I ) . These p-filiform Lie algebras will be called algebras of first generation. If we apply the same process to the same algebras, but in dimension n — 1, we obtain p-filiform algebras that will be called of second generation. The process will continue until an algebra appears in some generation. In [11] we can find that all nilpotent Lie algebras of dimension n and Goze invariant (4,1, (?.~~.4,1) are obtained as central extensions of a first generation of only three algebras of dimension n — l and Goze invariant (3,1, ^r.4,1). Specifically, the "first" of each one of the two existing families with dimension of the derived algebra 2 and the only one with dimension of the derived algebra 3. Any (n — 4)filiform nilpotent Lie algebra of dimension n, that is a central extension of second generation, is a central extension of the first generation as well.

THEOREM 3.3 In dimension n > 8, there are exactly 6n — 29 pairwise nonisomorphic complex (n — 4) -filiform nilpotent Lie algebras. They are distributed in twelve families, denoted by 0/ 4 ^

j \ , 1 < i < 12, whose laws can be found in [11].

We can see in [11] how these algebras are pairwise non-isomorphic by considering some invariants as the dimension of several appropriate subspaces: ZQ, C1^, C 2 0, f)i = Cen(ClQ), etc. Finally, it is shown in [11] that the p-filiform Lie algebras can be obtained as extensions by derivations of filiform Lie algebras of dimension n — p + I for p > n — 4. The case p = n — 3 appears in [10]. 3.3

p-filiform Lie algebras with n — 6 < p < n — 5

In a way similar to the cases studied above, we can obtain a general simplified

expression for the families of p-filiform Lie algebras when n — 6 < p < n — 5, but the classification of such families is by now unsolved. However, we have studied some

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interesting subfamilies in each case in order to get an approach to the complete family. The generic family of laws of the (n — 5)-filiform Lie algebras is obtained in [8], and the generic family of laws of the (n — 6)-filiform Lie algebras can be found in [18]. The laws of the family of (n — 5)-filiform Lie algebras allows us to get the classification when we consider some restrictions on the derived subalgebras of the family. Indeed, we can see in [8] or [16] the algebras g obtained when dim[g,g] = 6, that is when such derived subalgebras have dimension as large as possible.

THEOREM 3.4 If Q is an (n — 5)-filiform Lie algebra of dimension n > 9 with dim(C 1 g) = 6, then it is isomorphic to one of the algebras 0/ 5 ^ ^, I < i < 5, which are defined in [8]. We have also considered the families of (n — 5)-filiform and (n — 6)-filiform Lie algebras when n = 8. The classification for these families completes the p-filiform family in that dimension, which shows more information about the variety A/S. All of the 3-filiform Lie algebras can be reduced to only three [17]. The classification of 2-filiform Lie algebras (quasi-filiform Lie algebras) is obtained in [19], where it is shown how symbolic calculus was used to get the goal. Concretely, by using algorithms implemented in the Mathematica programming language [60].

4

LIE ALGEBRAS WITH SMALL NILINDEX

We consider here some type of nilpotent Lie algebras with small nilindex. Only the abelian algebra is obtained in each dimension having nilindex equal to 1. However, when the nilindex is greater than 1 the difficulties to determine all pairwise non-isomorphic algebras increase more and more. Indeed, the classification of the nilpotent Lie algebras with nilindex 2 is a very hard problem. The classification of those (2p + l)-dimensional algebras which have Goze invariant ( 2 , . . . , 2,1) is equivalent to the classification of the bilinear forms on Cp with values in Cp, which include all Lie algebras of dimension p. Thus, it is impossible to obtain a complete classification of such nilpotent Lie algebras [44], Therefore, to know examples of families of such Lie algebras with small nilindex is a valuable utility to study the variety of nilpotent Lie algebras J\fm.

A nilpotent Lie algebra of nilindex 2 is called a metabelian Lie algebra. Now, we are interested in certain types of metabelian Lie algebras and certain types of algebras with nilindex 3 determined by their Goze invariants. Goze's invariant allows to separate a family of nilpotent Lie algebras with a fixed nilindex into some families which we can study more easily. In this way, we can separate the family of the n-dimensional metabelian Lie algebras by considering those having Goze invariant equal to (2,2, . P A,2,1 n7?P\l), where p goes from 1 to [(n — 1)/2J. The simplest case to be considered is the family determined by ( 2 , 1 , . . . , 1). These are the (n — 2)-filiform Lie algebras and they have been studied in [13]. The classification can be found in Section 3, where we have surveyed the p-filiform Lie algebras. In the same way, the nilpotent Lie algebras with nilindex 3 which have Goze invariant equal to ( 3 , 1 , . . . , 1) are the (n — 3)-filiform Lie algebras, and these are also classified for arbitrary dimension in [13].

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We present in this section some general results about the metabelian Lie algebras, including the classification for a general family of such algebras under certain assumptions. We also study some concrete families of nilindex 2 and 3 in order to demonstrate how and where the difficulty increases in these classification problems. 4.1

Metabelian Lie algebras

To study the metabelian Lie algebras we separate them into families with Goze invariant (2,2,. P A,2, 1 n7??\ I). Then, if Q is an n-dimensional Lie algebra with nilindex 2 there is p 6 TL such that 1 < p < [(n - l)/2j and gz(g) = (2,2,. p) . ,2,l n r. 2 p \l). When p = 1, that is gz(g) = (2, 1, . . . , 1), the algebra Q is one of the (n — 2)filiform Lie algebras determined in [13]. In the general case, the difficulty to obtain the classification lies in the fact that there are too many parameters with very few restrictions. Let Q be a metabelian Lie algebra. A basis

is called adapted if

[Xo,X2i-i] = Xx, l
by n-2p-l

k=l

k=l

Let s denote the rank of the matrix (of), where af is the appropriate coefficient a 2t-i 2j-i fr°m the vector Yk in the bracket [ X z i - i ^ X ^ j - i ] above. Then, the derived algebra C J g has no more than s different elements X^i, 1 < i < p from the adapted basis considered. It follows that s < ( p ), and then we can bound the dimension of the algebra C1Q. THEOREM 4.1 Let Q be an n-dimensional metabelian Lie algebra and let p € N verify n > 2p + 1. Then p
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difficult as dim(C 1 0) decreases. In fact, when we consider dim(C 1 0) to have the largest possible value, the classification of such a family of Lie algebras can be obtained in any dimension. Indeed, let g^ p denote the algebras defined in the basis

(Xo,Xi, X2, • • • , X-2P, YI, . . . , Yn_2p-i)) by

where n > 2p + 1 > 5. For every p, the algebra g^ p

has Goze invariant equal to

n

gz(0) = (2, 2, . )., 2,1 7??\ 1) and its derived algebra verifies dim(C10jll)p) = p+ (%). Then, we can see in [43] how any n-dimensional metabelian Lie algebra g with Goze invariant gz(g) = (2, 2, p)., 2,1 n7??\ 1) and dim (C x g), having the largest possible value, is a trivial extension of an appropriate Lie algebra g^ . THEOREM 4.2 Let g be an n-dimensional metabelian Lie algebra with Goze invariant gz(g) = (2, P)., 2,1 n72?\ l},n>2p+l + (P) and dim(C J 0) =P+(%). Then the algebra is g = g^p ® Cn~m, where m = ( p f 2 ). It remains an unsolved problem, to obtain the classification when dim(C 1 g) decreases. Indeed, if the derived algebra has the smallest possible value we always find a family of algebras in which every algebra is a linear deformation of one of the model algebras For instance, a metabelian Lie algebra g with Goze invariant gz(g) = (2, 2 , 1 , . . . , 1) and dim(C 1 g) = 2 belongs to the family

[X0,X2i-i]=X2i, [ X i , Yi] =6jj^2 + &ii^4>

1 < i < 2, 1 < z < n — 5,

[Yi, Yj]

1 < i < j < n - 5.

=0^X2 + 4X4,

But we cannot separate non-isomorphic algebras from the family so far. However, a cohomology study about the algebras g° p was introduced in [41], and it can be found in [51], in order to illustrate the dimension of the orbits of such algebras. Moreover, we can also see in [51] how the number of metabelian Lie algebras obtained increases, when we consider dim(C 1 g) ^ p. THEOREM 4.3 Let g be an n-dimensional metabelian Lie algebra, n > 10, with Goze invariant g = ( 2 , 2 , 2 , 1 , . . . , ! ) and dim(C 1 g) > 3. Then g is isomorphic to a trivial extension of one of the following algebras defined in [51]: 0^ 3, g^ 3, g^, 3; 0n,3> 3n,3> 8n,3> 8n',3> ^ —T — l~T~ \ > Sra',3> ^ —r — {.^~2~ \ ' 3rt'3' ^ —r — l^2~ \ '

0^,3, 0 < r < l1^^; Qn^, 0 < r < L11^12]; g^f 3 , 0 < r < L2^11] • We remark that when we consider a metabelian Lie algebra g with Goze invariant ( 2 , 2 , 1 . . . , 1) and dim(C 1 g) > 2, then g has to be a trivial extension of gjj 2 -

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4.2

Lie algebras with nilindex 3

The study of the Lie algebras with nilindex 3 has been started in [51], in order to check if the results about metabelian Lie algebra could be generalized to other algebras with small nilindex. In this case, Goze's invariant could be one of the following (3, .p)., 3, 2, .q)., 2, 1, n -^T 2
defined in [51], we can separate the study of these algebras from the possible values (p,q). For example, the pair (1,0) corresponds to the (n — 3)-filiform Lie algebras which have been studied in [13]. Although for these algebras we have no relation to bound dim(C 1 g) as in Theorem 4.1 for the metabelian ones, we can see how the dimension of the derived algebra plays an important role in those algebras with small nilindex. Thus, we can express the family determined by (p, q) = (1, 1) in an adapted basis [51], and hence we can obtain the classification of those families of algebras with dim(C 1 0) > 3. For the non-split algebras mentioned in Theorem 4.4 below, we can also find some cohomological results in [51].

THEOREM 4.4 LetQ be an n- dimensional Lie algebra, n>8, with Goze invariant (3, 2, 1, . . . , 1) and dim(C 1 g) > 3. Then g is isomorphic to a trivial extension of one of the algebras Qln iti, 1 < i < 4n — 24, defined in [51]. To classify the Lie algebras having Goze invariant (3, . P A,3, 1, n7??\ 1) seems a much harder problem than its "similar" (2, . P A,2, 1,n7??\ 1) in metabelian Lie algebras. Indeed, in the simplest case (p,q) = (2,0) we have to consider for a Lie algebra 0 its derived algebra verifying 4 < dim(C 1 g) < 8. However, when we consider the largest value dim(C 1 0) = 8 the algebra Q just can be a trivial extension of one determined Lie algebra [42].

THEOREM 4.5 Let Q be an n- dimensional Lie algebra, n > 11, with Goze invariant (3,3, 1 n.~.6}, 1) and dim(C 1 0) = 8. Then Q is isomorphic to a trivial extension of the algebra Q^ 2 0 , defined in the basis (Xo, Xi,Xz, • • • , XQ, YI, . . . , Yn-?) by [Xo,Xi\=Xi+i, i €{1,2,4,5}, [X1)x2]=y1,

[X1,X4}=Y2, [XltX5=Y3,

To know if the algebras with Goze invariant (3, .PA, 3, l,n7?P\ 1) and having derived algebra as largest as possible can be determined as extensions of an appropriate family g^ p 0 is a very interesting question which should be a generalization of Theorem 4.2.' '

5

NATURALLY GRADED NILPOTENT LIE ALGEBRAS

In the cohomological study of the variety of laws of nilpotent Lie algebras established by Vergne [59] the classification of a class of graded filiform Lie algebras plays a

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fundamental role. The achieved classification allows for an easy expression of a filiform Lie algebra. The gradation considered by Vergne is provided in a natural way from the descending central sequence of any nilpotent Lie algebra. Now, the algebras obtained in this way are called naturally graded Lie algebras. Vergne proves that, up to isomorphisms, there is only one naturally graded filiform Lie algebra in odd dimensions and two of them when the dimension is even. This fact also allows other authors to deal with different aspects of the theory. For example, using such graded filiform Lie algebras, Goze and Khakimdjanov give in [45] the geometric description of the characteristically nilpotent filiform Lie algebras. Thus, it is clear that knowing the graded algebras of a certain class of nilpotent algebras we get valuable information about the structure of such a class. This may facilitate later the study of several problems that can appear within the whole of the class. In this section we study the naturally graded Lie algebras for the nilpotent Lie algebras with nilindex near their dimension. Thus, we concretely are interested in p-filiform Lie algebras for whose values of p the methods used in the sections above fail. 5.1

Naturally Graded filiform and Quasi-filiform Lie Algebras

If g is a nilpotent Lie algebra of dimension n and nilindex k, it is naturally filtered by the descending central sequence of g, (Cl&)0 k. Associated to g there exists a graded Lie algebra grg = ©iezgi, taking g; = 5j/5i+1. Thus, we have

When grg and g are isomorphic, denoted by grg = g, we will say that the algebra is naturally graded. So, if g is a naturally graded Lie algebra, such a natural gradation is finite, that is grg = gi © g 2 © • • • © g^, with [g^, QJ] C Qi+j, for i + 3 £ k, and the number of subspaces jjj is as large as the nilindex of g. The filiform case, characterized by its maximum nilindex fc = n — 1, in each dimension was studied by Vergne [59], who introduced the notation, proving that there are only two algebras of this type, Ln and Qn, when the dimension of the algebra is even, and there is only one, Ln, if n is odd. The algebras Ln and Qn are those defined on the basis (Xo, Xi, . . . , Xn-i), by

[X0,Xi]=Xi+1, (The undefined brackets, except for those expressing anti-symmetry, are supposed to vanish). As we have seen in Section 2, any n-dimensional filiform Lie algebra is a linear deformation of the algebra Ln [44]. Now, we want to study the naturally graded quasi-filiform Lie algebras, that is we will consider the n-dimensional Lie algebras with nilindex equal to n — 2. In

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order to make easy the structure of such a family we have to find a basis in which the brackets of the algebras of the family are suitable. We can see in [29] how to obtain an homogeneous adapted basis of the algebra 0, i.e., a basis whose properties are stated in the theorem below. THEOREM 5.1 Let Q be an n- dimensional naturally graded quasi-filiform Lie algebra. Then, there exists a basis (Xo, Xi, . . . , Xn_2, Y) of the algebra Q such that XQ, X\ G gi, Xi
z
Thus, we can use the laws of the family to reduce the unnecessary parameters. When g is an n-dimensional naturally graded filiform Lie algebra such that dim(C 1 0) = n — 3, we can consider a natural gradation where gi =< Xo,Xi,Y >. Then, the algebras obtained are reduced to a trivial extension of the naturally graded filiform Lie algebras. In fact, g = Ln-\ © C, if n is even, and either g = Ln_i © C or g = Qn-i © C, if n is odd. These graded quasi-filiform Lie algebras will be called split, and their only interest is to emphasize the natural gradation underlying the filiform subalgebra of the extension. We will denote by Q(n,r} an n-dimensional quasi-filiform Lie algebra in which we can choose an adapted basis so that the natural gradation verifies Y & Qr (note that dim(g r ) = 2, r ^ 1). Thus, we can view the pair of integers (n,r), I < r < n — 2, as an invariant for any naturally graded quasi-filiform Lie algebra. Then, the result in the previous paragraph says that if g = g( n ,i), then the algebra g is a split algebra. Moreover, if g = Q(n,r) with r even we can see in [30] that dim(C 1 g) = n — 3, so g is also a split algebra. Then, we can describe the family of the n-dimensional non-split naturally graded quasi-filiform Lie algebras. LEMMA 5.2 Let g = Q(n,r) be a non-split naturally graded quasi-filiform Lie algebra.

Then r is odd and g belongs to the parameterized family defined in an adapted

basis (X0,Xi,...,Xn-2,Y)

by

[Xi,Y]=aXi+r

where a £ C if 3 < r < n — 3, and a = 0 if r = n — 2. We can see how the quasi-filiform Lie algebras L( n>r ) and Q( n ,r) defined in the basis If n > 5, and 3 < r < 2 {(n - 1)/2J - 1, r odd,

If n > 7, n odd, and 3 < r < n — 4, r odd,

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[X0,Xi} = (n,r)

I

belongs to the family in the lemma above. So, these algebras are naturally graded algebras. They play an almost similar role as the algebras Ln and Qn in the filiform family. In each dimension n, for each acceptable (n,r), 3 < r < n — 5 we obtain that the naturally graded quasi-filiform Lie algebras are just L( n>r ) when n is even, and I/( n]r ) or <5( n>r ) when n is odd. Those cases in which r is close to the dimension n are special. When r = n — 2 (n odd), then g = £( n ) n _2)- When r = n — 3 (n even) or when r — n — 4 (n odd), we obtain also one more algebra noted T( n)7 .), which is called terminal and is defined in [30], where the algebras £(7,3), £}g 5 \, and £?g 5^ are defined as well.

THEOREM 5.3 Every n-dimensional naturally graded quasi-filiform Lie algebra is isomorphic to one of the following algebras: If n is even to Ln-\ ® C, T(n,n-3) or ^(n,r)i with r odd and 3 < r < n — 3. If n is odd to Ln—\ ® C, Qn-i ® C, L(n,n-2)> r (n,n-4)i ^(n,r) or Q(n,r)> with r odd and 3 < r < n — 4. When n = 7 or n = 9 we add the algebras £(jtz)> £}g $\i £?g 5)As an application, we refer to [40], [21] for a characterization of the space of derivations of the naturally graded quasi-filiform Lie algebras and a description of how symbolic calculus can be used to solve similar problems [28]. 5.2

Naturally Graded 3-filiform Lie Algebras

The complete n-dimensional family of nilpotent Lie algebras with nilindex n — 3 can be separated in the subfamilies characterized by the Goze invariants (n — 3,1,1,1), and (n — 3,2,1). The first one is the family of the 3-filiform Lie algebras. We are now interested in the naturally graded Lie algebras for such a family in order to look for a family of graded algebras which extends the results in the section above. In fact, the situation for naturally graded 3-filiform Lie algebras is a generalization of the filiform and quasi-filiform Lie algebras. There is one locally finite family of terminal algebras (depending on one parameter) for n odd or n even; furthermore, there are one or two locally finite families (depending on two parameters) for n odd or n even, respectively. Precisely, for n > 11, there are O(n 2 ) non-split Lie algebras.

THEOREM 5.4 Let Q be a naturally graded 3-filiform Lie algebra of dimension n, grg = ®igs0i and let (Xo, Xi,... ,Xn^3, YI, Yj) be an adapted basis of Q . Then

gi Qi

D D

< X0,Xi > <Xi>

Bi

=

{0}

2 n - 2.

More precisely, the decomposition obtained is g = gi © 32 © • • • © fln-s with [0i,0j] C Qi+j, for?+j < n-3, verifying that 2 < dim(gi) < 4 and 1 < dim(flj) < 3, 2 < z < n-3.

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The vectors YI, Y% are special. The triple (n, 7-1,7-2) will represent the case of naturally graded 3-filiform Lie algebras of dimension n, where the integers r± and r-2 indicate the positions of the coordinates of (dim(gi),dim(02), • • • , dim(g n _3)) different from those of (2, !,...,!). And 0( ra ,n,r 2 ) represents an algebra of the appropriate family determined by (71,7-1,7-2), 1 < r\ < r% < n — 3. Some general

facts about (71,7-1,7-2) can be found in [14], [15]. • The cases (71,7-1,7-2), where either r\ or r-2 is even are not admissible. Moreover, when r\ = 7-2, then r\ is odd. • The admissible cases (71,1,7-2) produce split Lie algebras: either a direct sum or a direct sum g n _2 ©C 2 , where Qn-i and Qn-2 correspond to a naturally 0 ra _i graded 2- filiform and 1-filiform Lie algebra of dimension n—l and n — 2, respectively. • The case (4, 1, 1) corresponds to the abelian Lie algebra. Moreover, in the case (5, 1, 1) two algebras are obtained: one split and the Heisenberg algebra. Of course, the general structure of the family of naturally graded 3-filiform Lie algebras is more complicated than that of the quasi-filiform ones.

LEMMA 5.5 Any naturally graded 3-filiform Lie algebra of dimension n is isomorphic to one whose law can be expressed, with (Xo, X\, . . . ,Xn^s, Yj, Yjj) an adapted basis, by 1 < i < n-4,

+ (-

[Xi,Y2] = fXi+r^ [Yi , Y2] = hXn_3

1
l
where 3 < ri < r 2 < TI — 3, ri,r2 od
by If n > 8, and 3 < r\ < r<2 < n — 3, n, r-2 odd, r

;l = X,A

If n > 10, n even, and 3 < r\ < r^ < n — 5, r\,r-2 odd, 1 < i
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The naturally graded 3-filiform Lie algebras of dimension n and type (n,ri,r2) when r-2 is "terminal", i.e., when r% is close to n — 3 (so Y? € Qi, Xn-s G QJ in a gradation of the algebra, where jjj is near QJ) has been studied carefully in [14]. The terminal families r(n,r, n — 5) (n even, n > 12, r odd, 3 < r < n — 7) and r(n,r,n — 4) (n odd, n > 9, r odd, 3 < r < n — 6) can also be found in [14]. As in the quasi-filiform case, we have to consider some exceptional algebras in low dimensions (n < 10): one for each dimensions 8 and 9 and two infinite families for n = 10 [9]. In the general case (n > 11), we will always suppose to be working with Lie algebras g with n > 11 and 3 < r\ < r%, ri,r2 both odd. We can see in [14] that, for odd dimension, there is one bi-parameter locally finite family of algebras, i( n ,r 1 ,r 2 ) (parameters n, r ^ ) , and one terminal one-parameter locally finite family, T( n ; r i ] n _ 4 ). For even dimension another bi-parameter family, <3( ra ,r-i,r 2 ) and °ne terminal oneparameter family, T(n;r.1]TJ_5) appear, but T(,i ]T . lin _4) disappears (r-2 = n — 4 is even if n is even and this is impossible). In order to obtain the desired classification, it is essential that any algebra g should belong to the family is an extension for ideals of g' = 0/C l ™~ 4 (g), which is in a natural way a graded 3-filiform Lie algebra of dimension n — 1 and type (n — I,ri,r2). Thus, the general classification is obtained using induction on the dimension of the algebras of such a family. The following theorem shows that there are no terminal algebras when r^ < n — 7.

THEOREM 5.6 Any naturally graded 3-filiform Lie algebra Q of type (n,TI,^) with n > 11, 7*1, TI both odd, 3 < TI < TI < n — 7, is isomorphic to an algebra of the family L(n, TI,^) if n is odd or to an algebra of the families J^( n , r i i r 2 ) or Q( ra ,rj,r 2 )

if n is even.

Thus, the results obtained for the naturally graded 3-filiform family are a generalization of the filiform and quasi-filiform families. This allows to conjecture about the structure of the naturally graded p-filiform Lie algebras when p is greater than 3.

6

LENGTH OF NILPOTENT LIE ALGEBRAS

The difficulties to obtain the classification of a class of nilpotent Lie algebras lead to study the algebras which can give useful information about such a class. In this way, the graded Lie algebras play an important role. We have considered in Section 5 the naturally graded algebras for some families of p-filiform Lie algebras. In that natural gradation on nilpotent Lie algebras, the subspaces in the gradation and the existence of an appropriate homogeneous basis (necessary to obtain the classification) are a natural consequence of the central descending sequence of the Lie algebras considered. However, it introduces a restriction by fixing the number of subspaces in the gradation by means of the nilindex. Several authors, for instance Goze, Khakimdjanov, and some of the authors of this paper, have considered graded Lie algebras which possess not just one natural gradation, but a gradation with a large number of subspaces, because this condition facilitates the study of some cohomological properties for such algebras (see [13], [44]). For such a "length" of the gradation, the main interest is in algebras whose length is as large as possible.

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In this section we are interested in the study of the filiform and quasi-filiform Lie algebras with length larger than their nilindex. Thus we have obtained other graded Lie algebras different from those in the section above. These graded Lie algebras with a greater number of subspaces than the naturally graded ones can be useful examples of filiform and quasi-filiform algebras to approach some problems about the variety of nilpotent Lie algebras. 6.1

Connected gradations

We suppose in this section that all Lie algebras are defined over the field of complex numbers. We also consider Z-graded Lie algebras; that is, admitting a decomposition Q = ®j 6 g Si> where the subspaces Qi satisfy [0i,gj] C Qi+j for all i,j 6 Z. We will say that a nilpotent Lie algebra Q admits a connected gradation 0 = Qni © • • • ® Qn2, when each Qi, with n\ < i < 77-2, is nonzero. The number of subspaces l(($Q) = n^ — n\ + 1 will be called the length of the gradation. DEFINITION 6.1 The length /(g) of a Lie algebra g is defined as I(Q) = max {/(®g) = n2 - n\ + 1 : g = g ni ® • • • © g na is a connected gradation} . This means that /(g) is the greatest number of subspaces from the connected gradation which can be obtained in g. Thus, every Lie algebra g has at least length equal to 1, because we can consider the connected trivial gradation g = go- On the other hand, an algebra cannot have length greater than its dimension. Thus, for every nilpotent Lie algebra g we have 1 < /(g) < dimg. For example, the filiform Lie algebra Ln admits the gradation Ln = Si ® • • -®Qn, where the subspace Qi = < X j _ i >, with 1 < i < n, is generated by the element Xi-\ of the basis (Xo, Xi,..., Xn-i) over which the algebra Ln is defined. Although such a connected gradation on Ln is not natural (QI ^ Q/Cls), it could be more useful to be considered because each subspace Qi has dimension equal to 1. Moreover, this gradation has the greatest possible number of nonzero subspaces, and this shows that the length of such an algebra is l(Ln) = n, If we consider a family of nilpotent Lie algebras and we look for the graded algebras with a given length, then we cannot suppose an adequate basis to express the algebras in a easy way (as Ln in an adapted basis). This is a first problem to solve in order to obtain the classification of such a graded family of Lie algebras. Thus, firstly we will obtain an homogeneous adapted basis that allows us to determine the structure of the family of Lie algebras considered. Secondly, we will separate that family in subfamilies determined by the gradations which we must consider. Finally, we will obtain the non-isomorphic algebras. 6.2

Filiform Lie Algebra of maximum Length

In addition to Ln, there are other filiform Lie algebras that admit this gradation. The algebras Rn and Wn are defined in a homogeneous basis (Xo,..., Xn-i), by n

~

f {X0,Xi} =Xi+1

1
=X2+J

2<j
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Nilpotent Lie Algebras

79

[X0,Xi]=Xi+l

l
r v- y i _ 6 (i-i)! (j-i)! (j-i)

- ———

——— i < j < n — 2 — i.

As for L n , we can check that l(Rn) = n and l(Wn) = n. Therefore, Rn and Wn are also maximum length filiform Lie algebras. The importance of these algebras was shown in the study of reducibility of the variety of laws of the nilpotent Lie algebras A/"ra. Khakimdjanov [46] describes in the variety j\fn, with n > 12, two irreducible components which contain an Rn and Wn respectively. Thus, the determination, up to isomorphism, of the filiform Lie algebras with length equal to their dimension can provide us, then, with new information about the variety J\fn. To obtain the naturally graded algebras, Vergne [59] uses the fact that if g is a filiform Lie algebra of dimension n, there exists an adaptedbasis (Xo, Xi,..., Xn_i), such that

[A-0,AVi] = 0 is verified. It can be observed that the basis considered in the definitions of Ln, Rn and Wn are adapted bases. When g is an n-dimensional filiform Lie algebra with 1(0) = n, we can see in [37] how to get an adapted basis by choosing homogeneous vectors from the subspaces of any decomposition 0 = ffig such that l(ffig) = n. This allows us to obtain the structure of the graded filiform Lie algebras considered.

THEOREM 6.2 If Q is an n-dimensional filiform Lie algebra of maximum length I(Q) = n that admits the decomposition g = g ni ffi • • • © Sni+n-i, then there exists

an adapted and homogeneous basis of g. Of course, we cannot obtain different Lie algebras from the gradations considered by just reordering the indices. In fact, we can see in [37] that g admits the decomposition g = g _ n + 2 f f i - • -ffigi if and only if g = g_i ffi- • -ffig n _2, and g admits

the decomposition g = gi ffi • • • ffi g n if and only if g = g_ n ffl • • • ffi g_i. In the first case the only algebra of the family is g = Ln, but the algebras Kn and Q'n verify 0 = 81 ® • • • ffi 0m where Kn (n > 8) and Q'n (n > 7, n odd) are defined in the basis (X0,Xi,...,Xn-.i) as follows: V l V [Ao,VA,] = Ai+i,

1 < z < n - O2,

F

fj, X

r

• X

j n-2 | _ . l

1 ^ " ^--' ,«

= (-1)*"1 ( I ^^ I - Z) X

I n-2 I

•] = ( — ! Y ('~ 1 )("~ 3 ~') a v

,

I

<J

<\ "~ 4 I ,

o < 7 < "~3

with a = 0, if n is even and a = 1, if n odd.

I [Xo,Xi] = Xi+i,

l
Moreover, the algebras Ln, Rn, Wn, Kn and Q'n, which admit the decomposition 0 = 0i © • • • ® Qni determine the filiform Lie algebras of maximum length.

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80

Cabezas et al.

THEOREM 6.3 Let g be an n- dimensional filiform Lie algebra of maximum length /(g) = n and n > 12. Then, either Q = Ln, g = Rn, Q - Wn, g = Kn or g = Q'n (n odd). In addition to these filiform Lie algebras, we must consider the algebras with dimension less than 12 and maximum length, obtained in [36]. 6.3

Quasi-filiform Lie algebras of length greater than their nilindex

The number of subspaces for the natural gradation in the quasi- filiform case is n— 2, where n is the dimension of the algebra [29]. So, in addition to the Lie algebras with length n, the algebras with length equal to n — 1 can also provide useful information to know better the complete quasi-filiform class. As in the filiform case, to obtain the n-dimensional naturally graded quasifiliform Lie algebras we can see in [29] how to get an homogeneous adapted basis (X0,Xi, . . . ,Xn-2,Y), such that

[X0,Xi}=Xi+1,

i
In [38], we have shown that the adapted basis can also be chosen homogeneous for any possible gradation with the maximum length on quasi-filiform Lie algebras.

THEOREM 6.4 Let g be an n-dimensional quasi-filiform Lie algebra of maximum length l(g) = n that admits a decomposition g = g ni © • • • © Qm+n-i- Then there exists an adapted and homogeneous basis (Xo, Xi, . . . , Xn_2i Y) °f 3 such that: (a) If the decomposition o/g is g = g _ n + 2 ® - • - ® g o ® 0 i , then g =< Y > © < X\ > © < X2 > © • • • ffi < ^o > or Q =< Xi>®<X2>®---®®<X0>. (b) If the decomposition of Q is g = Q-n+3 © • • • © 01 © 32, then g =< Xi > © < X-2 > © • • • © < ^0 > © < ^ > •

(c) If the decomposition of Q is g = g0 ©gi © • • • ©0 n -i, then g =< Y > © < X0 >

© < Xi > © • • • © < Xn_2 > •

(d) If the decomposition o/g is g = gi©g2©- • -®Qn, then g =< X0>®®< Xi > © • • • © < Xn-2 > orQ=<X0>®<X1>®---®< Xn^2 > ® . Obviously, every trivial algebraic extension g = g'©C of a filiform Lie algebra g' of maximum length is also a quasi-filiform Lie algebra of maximum length, because we can join the subspace C at the end of the gradation of g' to get the appropriate gradation for g. Thus, the algebras Ln_i © C, Rn~i © C, W n _i © C, Kn_i © C and Q'n_i © C are quasi-filiform Lie algebras of maximum length equal to n [37]. All these algebras are examples of split Lie algebras g © C, whose decompositions lie actually in g'. The non-split quasi-filiform Lie algebras of maximum length have been determined in [38], where we can also find the study of some geometric problems about the graded quasi-filiform Lie algebras obtained. THEOREM 6.5 (Quasi-filiform Lie algebras of maximum length) Let Q be an n-dimensional non-split quasi-filiform Lie algebra of maximum length l(g) = n, with n > 13 . Then the algebra g is either g = gL 2_n-. (n odd), g = g? ^ or Q = s f n i ) ; where the algebras 0 L 2 _ r a ) > 0 L i ) ; 0L, i)

(Xo, Xi, . . . Xn-2, Y) as follows:

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

are

defined in the basis

Nilpotent Lie Algebras

81

// n > 5 and n is odd,

Ifn>5, 2 0(n 1}

'

' j\ V i , yv Vi II — _ yv V j_|_3)

9 _
Now we consider an n-dimensional quasi-filiform Lie algebra g of length l(g) = n — 1. We remark that the natural gradation for this algebra has n — 2 subspaces. Moreover, we do not know if such an algebra is naturally graded. If g does, then the algebra is one of the algebras obtained in Section 5 with another appropriate gradation different from the natural one. Thus, the most interesting point is to study families with few naturally graded Lie algebras. Then we could obtain other graded algebras to study the chosen family. When the derived algebra of g has the minimal dimension dim(C 1 g) = n — 3, then g = Ln-\ © C or g = <5 n _i © C [29], so we first consider the family of quasi-filiform Lie algebras verifying such a condition about their derived algebras. In [39] we have studied the possible gradations to be considered, and we can see that if g is an n-dimensional quasi-filiform Lie algebra of length n — 1 and dim(C 1 g) = n — 3, then it is always possible to choose an homogeneous adapted basis of the algebra (Xo,Xi,... ,Xn-2,Y) for any gradation g = ©g such that /(0g) = n — 1. This allows us to get the structure of the family, and finally to determine the algebras of the family considered. The theorem below shows how hard it is to obtain a classification.

THEOREM 6.6 Let g be an n-dimensional quasi-filiform Lie algebra with length = n— 1, dim(C ll g) = n — 3 and n > 13. Then either g = Qn-\ ffi C, g = gL -,, Q

Q

/

lT~l QllfTL

_ J _ J _ J

CLTLQi T) Odd

A 1

OT T) ~~ TL — 4 /

4

0 —- Q/

/

\

r

'

^

^

^

x

O 1

ITL — t3 "^ T) "^ TL — o /

'ra •. (n — 5 < p < n — 3) for 3 < p < n — 3, which are defined in [39]. The low-dimensional cases (n < 13) and an approach to the study of the ndimensional quasi-filiform Lie family of algebras g with length l(g) = n — 1, verifying dim(C 1 g) ^ n — 3, can be found in [50].

7

SYMBOLIC CALCULUS ON LIE ALGEBRAS

Some of the problems studied in this survey require a lot of computations in order to reach the goal we are aiming at. A family of Lie algebras is usually determined by a set of polynomial equations involving the structure constants of the family, and it is obtained from some restrictions which have to be considered about such algebras arising from Jacobi's relations, nilindex, Goze's invariant, etc. Then, to

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82

Cabezas et al.

determine a family of Lie algebras in a reliable way is a very complicated problem. Moreover, the basic object to study is an n-dimensional Lie algebra, whose law is usually given in an adequate basis. Often, we have to change the considered basis and we have to determine how the family is modified. Obviously, a general change of basis in arbitrary dimension cannot be done. So we have to "guess" the change of basis which simplifies the law of the family. When a family is determined in an easy way, for instance when we know the different algebras up to isomorphism, we are often interested in several problems to approach the complete variety of nilpotent Lie algebras by the family that is considered. Thus, we usually want to know the dimension of the orbits of those representative algebras in the family, their spaces of derivations, etc. In some cases the theorems come from the algebras whose laws are very simple, but in many cases we must make a lot of computations to get examples which allow us to find out the right way to the solution of the considered problem. Thus, we have developed many algorithms to assist us where a computation "by hand" could be either dangerous or almost impossible. Most of them have been implemented in the symbolic programming language that the package Mathematica provides [60]. The software package Mathematica has been very useful. It has been used for its programming capacities and as a powerful symbolic calculator. It has been considered always as an assistant to obtaining examples from concrete algebras in which we are interested for high dimensions. Moreover, we have used the developed Mathematica packages to check some conjectures that could be made in order to modify the packages when a general result was stated. Thus, we have used Mathematica in an interactive way, and we often had to solve some problems different from the initial ones. Mathematica has become a well-know symbolic software tool to perform computations in a reliable way. We have shown in several sections of this survey how some of the results obtained have been presented in Computational Algebras Conferences or Mathematical Symposiums. The programs we developed can be found in the appropriate references or directly through the authors of this paper.

REFERENCES

[1] J. M. Ancochea, J.R. Gomez, M. Goze, G. Valeiras, Sur les composantes irreductibles de la variete des lois nilpotentes, J. of Pure and Appl. Algebra, 106(1996), 11-22. [2] J. M. Ancochea-Bermudez, M. Goze, Classification des algebres de Lie filiformes de dimension 8, Arch. Math., 50(1988), 511-525. [3] J. M. Ancochea-Bermudez, M. Goze, Classification des algebres de Lie nilpotentes de dimension 7, Arch. Math., 52:2(1989), 157-185. [4] J. M. Ancochea-Bermudez, M. Goze, On the varietes of nilpotent He algebras of dimension 7 and 8, J. of Pure and Appl. Algebra, 77(1992), 131-140.

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83

[5] J. M. Ancochea-Bermudez, M. Goze, You. B. Hakimjanov (Khakimdjanov), Sur la reductibility de la variete des algebres de Lie nilpotentes, C. R. A. S. Paris, 313:1 (1991), 59-62.

[6] L. Boza, F. J. Echarte, J. Nunez, Classification of complex Filiform Lie Algebras of dimension 10, Algebra, Groups and Geometries, 11:3(1994), 253-276.

[7] F. Bratzlavsky, Classification des algebres de Lie nilpotentes de dimension n, de classe n — I , dont I'ideal derive est commutatif, Acad. Roy. Belg. Bull. Cl. Sci., 560 (1974), 858-865. [8] J.M. Cabezas, L.M. Camacho, J.R. Gomez, R.M. Navarro, A class of nilpotent Lie algebras, to appear in Communications in Algebra, 2000. [9] J.M. Cabezas, L.M. Camacho, J.R. Gomez, R.M. Navarro, E. Pastor, Lowdimensional naturally graded 3-filiform Lie algebras, Preprint MA1-01-OOVI, Universidad de Sevilla, 2000.

[10] J.M. Cabezas, J.R. Gomez, Las algebras de Lie (n — 3)-filiformes como extensiones por derivaciones, Extracta Mathematicae, 13(3) (1998) 383-391. [11] J.M. Cabezas, J.R. Gomez, (n — 4)-filiform Algebra, 27(10) (1999) 4803-4819.

Lie algebras, Communications in

[12] J.M. Cabezas, J.R. Gomez, Cohomological properties ofp-filiform Submitted to Journal of Lie Theory.

Lie algebras,

[13] J.M. Cabezas, J.R. Gomez, A. Jimenez-Merchan, Family of p-filiform Lie algebras, In Algebra and Operator Theory, 1997. Ed. Y.Khakimdjanov, M. Goze, Sh. Ayupov, 93-102, Kluwer Academic Publishers, 1998.

[14] J.M. Cabezas, J.R. Gomez, E. Pastor, Structure theorem for naturally graded 3-filiform Lie algebras, I Colloquium on Lie Theory and Applications, Vigo (Spain), 2000. [15] J.M. Cabezas, J.R. Gomez, E. Pastor, Naturally graded 3-filiform Lie algebras, Submitted to Proceedings of Edinburgh Mathematical Society, 2000. [16] L.M. Camacho, Algebras de Lie p-filiformes PhD thesis, Universidad de Sevilla, 2000.

[17] L.M. Camacho, J.R. Gomez, R.M. Navarro, 3-filiform Lie algebras of dimension 8, Ann. Math. Blaise Pascal, 6(2) (1999) 1-13. [18] L.M. Camacho, J.R. Gomez, R.M. Navarro, Family of laws of (n — 6)-filiform Lie algebras, Preprint MA1-02-OOVI, Universidad de Sevilla, 2000. [19] L.M. Camacho, J.R. Gomez, R.M. Navarro, The use of Mathematica for the classification of some nilpotent Lie algebras, IMACS-ACA'99, Meeting El Escorial (Spain), 1999.

[20] L.M. Camacho, J.R. Gomez, R.M. Navarro, Algebra of derivations of (n — 3)filiform Lie algebras, ILAS'99 Meeting Barcelona, 1999.

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84

Cabezas et al.

[21] L.M. Camacho, J.R. Gomez, R.M. Navarro, Cohomology of some nilpotent Lie algebras, Extracta Mathematicae, 15 (2000). [22] F.J. Castro, J.R. Gomez, A. Jimenez-Merchan, J. Nufiez, How to obtain families of laws Lie algebras, Proc. EACA'95, 29-37. Serv. Publ. U. Cantabria, 1995. [23] A. Cerezo, Les algebres de Lie nilpotentes reelles et complexes de dimension 6, preprint N. 27, Nice, 1983.

[24] J. Dixmier, Sur les representations unitaires des groupes de Lie nilpotents III. Canad. J. Math., 10(1958), 321-348. [25] J. Dixmier, W.G. Lister, Derivations of nilpotent Lie algebras, Proc. AMS, 8(1957), 155-158.

[26] J. R. Gomez, F. J. Echarte, Classification of complex filiform nilpotent Lie algebras of dimension 9, Rend. Sem. Fac. Sc. Univ. Cagliari, 61(1) (1991), 21-29. [27] J.R. Gomez, M. Goze, Y. Khakimdjanov, On the k-abelian Filiform Lie Algebras, Communications in Algebra, 25(2) (1997) 431-450. [28] J. R. Gomez, A. Jimenez-Merchan, The graded algebras of a class of Lie algebras in Mathematics with Vision (ed. V. Keranen, P. Mitic), 151-158, Computational Mechanics Publications, 1995.

[29] J. R. Gomez, A. Jimenez-Merchan. Algebras de Lie graduadas y cdlculo simbolico. Actas del Primer Encuentro de Algebra Computacional y Aplicaciones, EACA'95, 57-65, 1995.

[30] J.R. Gomez, A. Jimenez-Merchan, Naturally Graded Quasi-Filiform Lie Algebras, Submitted to Journal of Algebra. [31] J. R. Gomez, A. Jimenez-Merchan, You. B. Hakimjanov (Khakimdjanov), On the Variety of Nilpotent Lie Algebra Laws of Dimension 11, Rendiconti Cagliari

66 (2) (1996) 137-142. [32] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimjanov, Classifying Filiform Lie Algebras with Mathematica in Innovation in Mathematics, (ed. A. Hietamaki, V. Keranen, P. Mitic), 169-176, Computational Mechanics Publications, 1997.

[33] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimdjanov, Low-Dimensional Filiform Lie Algebras. Journal of Pure and Applied Algebra 130 (1998), 133-158. [34] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimdjanov, Symplectic Structures on the Filiform Lie Algebras. To appear in Journal of Pure and Applied Algebra. [35] J. R. Gomez, A. Jimenez-Merchan, J. Nufiez-Valdez, An algorithm to obtain laws of families of filiform Lie algebras, Linear Algebra and its Aplications 279 (1998), 1-12.

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[36] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Tratamiento simbolico de algebras de Lie filiformes graduadas conexas. Actas del Segundo Encuentro de Algebra

Computational y Aplicaciones, EACA'97, 136-142, 1996.

[37] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Filiform Lie Algebras of Maximum Length. Submitted to Acta Mathematica Hungarica. [38] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Quasi-Filiform Lie Algebras of Maximum Length. Submitted to Linear Algebra and Applications. [39] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Quasi-Filiform Lie Algebras of Length dim[g, g] + 2. ILAS'99 Meeting Barcelona, 1999. [40] J.R. Gomez, R.M. Navarro, Espacios de derivaciones de algebras de Lie con Mathematica, Proc. of IV Encuentro de Algebra Computational y Alicaciones, (EACA'98). Sigiienza, 1998. [41] J.R. Gomez, I. Rodriguez. Geometrical properties of model metabelian Lie algebras, Submitted to Algebras, Groups, and Geometries. [42] J.R. Gomez, I. Rodriguez. A special case of family of Lie algebras with nilindex 3. ILAS'99 Meeting Barcelona, 1999. [43] J.R. Gomez, I. Rodriguez. Metabelian Lie algebras with maximal derived. ILAS'99 Meeting Barcelona, 1999. [44] M. Goze, You. B. Hakimjanov (Khakimdjanov), Nilpotent Lie algebras, Kluwer Academics Publishers, 1996.

[45] M. Goze, You. B. Hakimjanov (Khakimdjanov), Sur les algebres de Lie nilpotentes admettant un tore de derivations, Manuscripta Math. 84 (1994), 115224. [46] You. B. Hakimjanov (Khakimdjanov), Variete des lois d'algebres de Lie nilpotentes, Geometriae Dedicata, 40(1991), 229-295. [47] Yu. Hakimjanov (Khakimdjanov), Characteristically nilpotent Lie algebras, Mat. Sbornik, 181:5 (1990), 642-655 (russian). English transl. in Math. USSR Sbornik, 70:1(1991). [48] V. Morozov, Classification of nilpotent Lie algebras of sixth order. Izv. Vysch. U. Zaved. Mat., 4:5(1958), 161-171. [49] O. A. Nielsen, Unitary representations and coadjoint orbits of low-dimensional nilpotent Lie groups, Queen's Papers in Pure and Appl. Math., 63, 1983. [50] J. Reyes. Algebras de Lie casifiliformes

graduadas de longitud maximal. PhD

Thesis, Universidad de Sevilla, 1998.

[51] I. Rodriguez. Algebras de Lie con invariante de Goze dado. PhD Thesis, Universidad de Sevilla, 2000. [52] M. Romdhani, Classification of real and complex nilpotent Lie algebras of di-

mension 7, Linear and Multilinear Algebra, 24, 167-189, 1989.

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[53]

Saffiullina, On the classification of nilpotent Lie algebras of dimension 7, Kazan, 1976 (in Russian).

[54]

C. Seeley, Seven-dimensional nilpotent Lie algebras over the complex numbers, PhD. Thesis, Chicago, 1988.

[55]

C. Seeley, Degenerations of 6-dimensional nilpotent Lie algebras on C, Commun. Algebra, 18:10(1990), 3493-3505.

[56]

C. Seeley, Some nilpotent Lie algebras of even dimension, Bull. Austral. Math. Soc. 45(1992), 71-77

[57]

K. A. Umlauf, Uber die Zusammensetzung der endlichen continuierlichen Transformationsgruppen insbesondere der Gruppen vom Range null, Thesis, Leipzig, 1891.

[58]

G. Valeiras, Sobre las componentes irreducibles de la variedad de leyes de algebras de Lie nilpotentes complejas de dimension 8, PhD. Thesis, Univ. Sevilla, 1992.

[59]

M. Vergne, Cohomologie des algebres de Lie nilpotentes. Application a I'etude de la variete des algebres de Lie nilpotentes, Bull. Soc. Math. France 98(1970), 81-116.

[60]

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, 1991.

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On L*-Triples and Jordan #*-Pairs A. J. CALDERON-MARTIN, Departamento de Matematicas. Cadiz. 11510-Puerto Real. Cadiz, Spain. E-mail: [email protected]

Universidad de

C. MARTIN-GONZALEZ, Departamento de Algebra, Geometria y Topologfa. Universidad de Malaga. Apartado 59, 29080-Malaga, Spain.

Abstract In [12], Lister introduced the concept of Lie triple system. He classified the finite-dimensional simple Lie triple system over an algebraically closed field of characteristic zero. Neher also studies Lie triple systems and their relations with Jordan triple systems in [13]. In order to study infinite-dimensional Lie triple systems, we introduced the notion of //-triple and obtained a classification of //-triples admitting a two-graded //-algebra envelope in [3]. However, the problem on the existence of L*-algebra envelopes is still open. We prove in this paper, using Jordan //"-pairs techniques, that every infinite-dimensional topologically simple //-triple, verifying a purely algebraic additional property, has a two-graded L*-algebra envelope and then we classify them.

1

PREVIOUS RESULTS ON //-TRIPLES

Let A be a C-algebra and * : A —» A a conjugate-linear map, for which (x*)* = x and (xy)* = y*x* hold for any x, y e A. Then * is called an involution of the algebra A. We recall that an H*- algebra A over C is a nonassociative C-algebra, which is also a Hilbert space over C with inner product (• •), endowed with an involution * such that (xy\z) = (x\zy*) = (y\x*z) for all x,y, z € A. A two-graded H*-algebra, is an //"-algebra which is a two-graded algebra whose even and odd part are selfadjoint closed orthogonal subspaces. We call the two-graded //'"'-algebra A topologically simple if A2 ^ 0 and it has no nontrivial closed two-graded ideals. In the sequel an //-algebra will mean a Lie //*-algebra. The classification of topologically simple //-algebras is given in the separable case by Schue (see [14], [15]) and later in the

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88

Calderon-Martin and Martin-Gonzalez

general case in [8], [1] and [13]. If HI and H-2 are Hilbert spaces with scalar products ( • | - ) i , z = l,2 and / : H i —> H-2 a linear map such that

for any x, y € H I , then we will say that / is a k-isogenic map. We can define an H*-triple as a complex triple system (T, < -, •, • >) provided with: 1. A conjugate-linear map * : T —> T such that (x*)* = x and <x,y,z >* = < z * , y * , z * >

for any x, y, z G T. 2. A complex Hilbert space structure whose inner product is denoted by (•[•) and satisfies

(<x,y,z>\t) = (x ) = (y| ) = (z ) for any x, y, z,t €. T.

From [5] it follows that the triple product < - , - , - > of T is continuous.

We define the annihilator of an //"-triple (T, < •, •, • >) as

Ann(T) = {x e T :< x,T,T >=< T,x,T >=< T,T,x >= 0}. This set Ann(T) turns out to be a closed ideal of T. The definition of topologically simple H "-triple is similar to that used in the case of //"-algebras. The structure theorems for //"-triples given in [5] reduce the interest on H *-triples to the topologically simple case. Essentially, these theorems claim that under suitable conditions, one can split any //"-triple as the orthogonal direct sum of its annihilator plus the closure of an orthogonal direct sum of topologically simple //"-triples. We define an L* -triple as a Lie //""-triple. If L is an L*-algebra, then it can be viewed as an L*-triple by defining the triple product [x,j/,zj = [[x,y],z] for any x,y,z <E L, this L*-triple will be denoted by LT . Furthermore, for any two-graded L*-algebra, L = Lo-LLi, its odd part LI is an L*-subtriple of LT . If T is an //"-triple isometrically *-isomorphic to the L*-triple LI for some two-graded L*-algebra L, we shall say that L is a two-graded L*-algebra envelope of T iff LQ := [Li, L I ] . It is

also easy to check that if T is an L*-triple with two-graded L*-algebra envelope L, then T is topologically simple if and only if L is topologically simple in the graded sense. We refer to [3, Theorem 1] for the following classification of topologically simple L*-triples that admit a two-graded L*-algebra envelope.

THEOREM 1.1 Let T be a topologically simple L* -triple admitting a two-graded L* -algebra envelope. Then T is one of the following: 1. The L* -triple associated to an L*- algebra L by defining the triple product [a, 6, c] := [[a, 6],c] and the same involution and inner product of L.

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2. Skw(A,r) with A a topologically simple associative H* -algebra, T an involutive *- automorphism of A, the involution and inner products induced by the ones in A, and the triple product as in the previous case. 3. Sym(A,a)

with A as in the previous case but now a is an involutive *-

antiautomorphism, and the triple product, involution and inner product induced by the ones in A. 4- Skw(A,T)r\Skw(A,a) with A as before, T an involutive *- antiautomorphism, a an involutive ^-automorphism such that ar = ra , and the involution and inner product induced by the ones in A.

The problem on the existence of L*-algebra envelopes is still open. However, if T is an L*-subtriple of A~ for a topologically simple ternary //"-algebra A, that is, for any x, y, z & T one has

[x, y, z] =< x, y, z > - < y, x, z > - < z, x, y > + < z, y, x > where < •, •, • > is the triple product of A, then it can be proved that it has a twograded L*-algebra envelope. Indeed, from the classification of topologically simple ternary //"-algebras (in the complex case) of [7, Main Theorem, p. 226], one may see that there is an associative topologically simple two-graded //"-algebra B = Bo-LBi (see [6] for classification theorems) such that A is the ternary //"-algebra associated to BI (with triple product < x,y,z >= xyz for all x,y,z 6 BI). Let // = Lo-L^i be the two-graded L*-subalgebra of B" generated by T. It is easy to prove that LI = T and LQ = [T, T], hence the topological simpleness of T implies that of L in the graded sense.

2

PREVIOUS RESULTS ON JORDAN //"-PAIRS

Let A = (A+ , A~) be a pair of modules over a commutative unitary ring K, and < - , - , - >: Aa x A~a x A" —» A",

two trilinear maps such that

(x,y,z) H->< x,y,z> for a g {+,—}• Then A is called an associative pair if the following identities are satisfied: « x,y,z >,u,v >=< x, < u,z,y >,v >—< x,y,< z,u,v » for x,z,v € A" and y, u 6 A~a .

Let A = (A+,A ) be a pair of /iT-modules and Q° : A° —> homK(A~CT,A'7) two quadratic operators for a € {+,—}• We define the trilinear operators

and the bilinear operators

Da : Aa x A~a —> End(Aa)

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as {x,y,z}° = D"(x,y)z := Q°' (x + z)y - Qa' (x)y - Q°' (z)y , for z, z £ Aa , y € A~a and cr 6 {+,—}. We shall say that A = (A+,A~) is a Jordan pair if the next identities and its linearizations are true: Q°(Q°(x)y) = Q°(x)Q for x, z e A° , y <E A~a and cr e {+, -}. One can find a classification of prime, nondegenerate Jordan pairs with nonzero socle in [10, Teorema 6] or [9, Theorem 7]. If A is an associative pair, then AJ will denote the symmetrized Jordan pair of A, that is, the Jordan pair whose underlying /('-module agrees with that of A, and whose quadratic operators are given by Qa (x)y =< x,y,x >a .

Let A = (A+,A~) be a complex pair and * = (* + ,*~) a couple of conjugatelinear mappings *°" : Aa —> A~a for which *CT o *~CT = Id and

for xa , za 6 Aa and y a G A C, endowed with an involution * = (* + ,*~) such that

t-cr for xa,za,ter e ^ and y~a e A" We also recall that an H*-psdr A is said to be tope/logically simple when

< Aa,A~a,A° > ^ 0 and its only closed ideals are {0} and A. If A is an associative H*-pair, then AJ can be canonically equipped with a Jordan H*-pair structure. Moreover, if £ : A —» Aop is an involutive *-antiautomorphism, then both Sym(A, £) and Skw(A, ^) can also be structured as Jordan //"-pairs in obvious ways. Given an infinite dimensional and topologically simple Jordan //*-pair, we can always describe it from a Jordan H*algebra or from an associative //*-pair. Indeed, every topologically simple Jordan //"-pair J is prime, non-degenerate and with non-zero socle (see [2, Proposition 1]), then, the classification of Jordan pairs given in [10, Teorema 6] implies that forgetting the H*-structure, the underlying Jordan pair J is one of the following: Type (i). J = (V, V) with (V, {-, •}) a quadratic Jordan algebra, being the triple products {x, y, z}a = {x, { y , z}} - {y, {x, z}} + {z, {x, y}} for any x, y, z e V. Type (ii). J is a subpair of (L(X,Y),L(Y,X))J containing (F(X, Y),F(Y,X))J a with the triple products {x,y,z} := xyz + zyx , where (X,X') and (Y,Y') are dual pairs over an associative C-division algebra A. Type (iii). J is a subpair of

(Sym(L(X, Y), |), Sym(L(Y, X), H))

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containing (Sym(F(X,Y),$),Sym(F(Y,X),W where (X,YJ), (X,YJ°r) are a dual pair and its opposite over an associative C -division algebra (A,r) with involution T, where the triple products are {x, y, z}" = xyz + zyx and with ft the adjoint operator. Type (iv). J is a subpair of

(Skw(L(X, 7), fl), Skw(L(Y, X ) , «)) containing (Skw(F(X,Y),$),Skw(F(Y,X),$)) where ( X , Y , f ) , ( X , Y , f ° P ) are a dual pair and its opposite over C, where the triple products are {x, y, z}a = xyz + zyx and with ft the adjoint operator. If J is a topologically Jordan //*-pair of Type (i), then one can define an involution and inner product on V so as to obtain that V is a Jordan //"-algebra. We prove, in [2], that any topologically Jordan //"-pair of Type (ii) is of the form J = AJ , where A is a topologically simple associative //*-pair. Similar arguments apply to a topologically simple Jordan //*-pair J of types (iii) or (iv), give us that J is of the form J = Sym(A, £) or J = Skw(A, £) respectively, with A as above, and £ an involutive *-antiautomorphism from A to Aop, (see [4] for more details). The concept of polarized triple system of a pair will also be used with the same meaning as in [13].

3

MAIN RESULTS

Neher finds in [13] the relation between simple polarized Lie triple systems, simple polarized Jordan triple systems and simple Jordan pairs. We can obtain similar results in an H "-context so as to give the next

THEOREM 3.1 (a) Let L = L+ J_ L~ be a topologically simple polarized L* -triple. Then J(L) : — L with the product

is a topologically simple polarized Jordan H* -triple. (b) Let J = J+ _L J~ be a topologically simple polarized Jordan H* -triple. Then L ( J ) :— J with the product x,y,z := {x,y,z} — {y,x,z} is a topologically simple polarized L* -triple. (c) The operations L —> J ( L ) and J —> L(J) described in (a) and (b) are inverses of each other.

PROOF.- (a) and (b). According to [13] we have that J(L) := L (resp. L(J) := J) is a Jordan triple system (resp. Lie). It is easy to check that the involution and inner product of L (resp. J) make J(L) := L (resp. L ( J ) := J) an //"-triple. Since any ideal of J(L) is an ideal of L, we conclude that J(L) is a topologically simple Jordan //"-triple. Our next claim is that L(J) is topologically simple. We first observe that AnnL(J) = 0. Indeed, from [13], AnnL(J) is a IT-invariant ideal of L(J), being II the automorphism of L ( J ) defined as H((x+,x~)) : — (x+, -x~), hence [13] shows that AnnL(J) is an ideal of J and therefore AnnL(J) = 0. Finally, let J be a closed ideal of L(J), as [ I , L ( J ) , L ( J ) } is an ideal of J ([13]), either [7, L( J), L(J)\ = 0 and then / = 0, or [I, L ( J ) , L ( J ) ] = J which implies / = L( J ) .

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The item (c) is immediate to check.

It follows, easily, that a Jordan H*-pair is topologically simple if and only if its associated polarized Jordan _ff*-triple is also topologically simple. The next result is clear by [13].

PROPOSITION 3.2 Let J; = (J+, Jr) (i e {1,2}) be topologically simple Jordan H*-pairs, and let L(Ji) be their associated polarized L*-triples. Then, if J\ and J% are *-isometrically isomorphic, L(J\) and L(J^) are also *-isometrically isomorphic. If L(Ji) and £(1/2) are *-isometrically isomorphic then J\ is *-isometrically isomorphic with J% or J%p.

THEOREM 3.3 Let (T, [ - , - , • ] ) be an L*-triple. Then there exists an associative algebra U such that [x, y, z] = xyz — yxz — zxy + zyx for any x,y,z € T. PROOF.- The proof consists in the construction of algebra envelope of T and then, denning U as the L. Define LQ to be the span of ad(Li,Li), where product operator. We have that L = LQ ® LI with

L = LQ ® LI, a two graded Lie universal enveloping algebra of a d ( x , y ) ( z ) = [ x , y , z ] is the left the product defined by

[(ad(x,y),z),(ad(u,v),w)}

=

= ( a d ( [ u , v , y ] , x ) - ad([u, v , x ] , y ) + ad(z,w), [z,y, w] - [ u , v , z ] ) is a two graded Lie algebra envelope of T, that is, {x, y, z] — [[x, y]z] for any x, y, z € T. (In fact, one can construct an involution *, preserving the grading, and a nondegenerate hermitian form /, in such a way that LI is isometrically *-isomorphic to T considered as an L*-triple with the restriction of * and of /, however, it does not seem easy to complete this construction so as to produce on L an i*-algebra structure). The universal enveloping algebra of L, see [11, Chapter V] is the algebra U we are looking for.

THEOREM 3.4 (MAIN THEOREM) Let (T, [•, -, •]) be an infinite dimensional topologically simple L*-triple system. Let U be an associative algebra such that [x, y, z] = xyz —yxz — zxy + zyx for any x,y,z e T. If xyx G T for every x,y e T, then T has a two-graded L*-algebra envelope and therefore is one of the L*-triples described in Theorem 1.1. PROOF.- By section 1, we only have to prove that T is an L*-subtriple of A~, being A a topologically simple ternary H ""-algebra. It is straightforward to prove that T ® T with the product [(a, 6), (c, d), (u, v)} := (adu + uda — cbu — ubc, bcv + vcb — dav — vad),

involution (a, 6)* := (6*, a*) and inner product < ( a , b ) \ ( c , d ) >=< a\c > + < b\d > is a polarized L*-triple and that

T

—>

T®T

x

H->

(x,x)

is a fc-isogenic *-monomorphism of L*-triples. Now, the structure theorems for polarized L*-triples (analogous to the ones in [5]), allow us to assert that T is

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*-isomorphically /c-isogenic to an L*-subtriple of a topologically simple polarized L*-tripleT+ LT~. Let (J = J+ _L J~, {-,-,•}) be the polarized Jordan H*-triple associated to + T _L T~ (see Theorem 3.1). Following the description of topologically simple Jordan #*-pairs given in Section 2 we have to study the following four possibilities: (a) If J = (V, V), where V is a quadratic Jordan //*-algebra, then we have

4> : T+ ±T~

—»

L(V ± I/)

an *-isometric isomorphism of //-triples. One easily shows the existence of

+ : and 4>~ : T —> also check that

V

T

—»

V

such that 4>(x,y] = (4>+ (x) , <j>~ (y)) for x,y € T. We can

for any x + , y ~ , 2 + £ T. Therefore ~ (4>+}~l , and a suitable involution. However, (V, [•, •, •]') turns out to be a nontopologically simple L*-triple (moreover, Ann(V) ^ 0 since [I, a, b]' = 0 for a, 6 £ V), which is impossible. Thus, J has to be one of the other types of topologically simple polarized Jordan //""-triples, that is: (b) J = (A+ _L A~)J with (A+ A. A~ , < • , - , - > ) a topologically simple ternary ff*-algebra. Then, for any (x+,x~), (y + , y~), (z+,z~) £ T, we have that

<

that is, T is a subtriple of (^4+ ± A~)~, with A+ X A~ a topologically simple ternary ff* -algebra, as we wanted to prove. (c) and (d). If J = Sym(A+ _L A~,£) or J = Skw(A+ ± A~,£), with (>1+ 1 ^4"", < - , - , • > ) a topologically simple ternary _ff*-algebra, we can prove as in case (b) that T is also a subtriple of a topologically simple ternary H* -algebra. This completes the proof. Note that, following the proof of the last theorem, we have that every nonquadratic topologically simple L*-triple system that embeds in a polarized i*-triple system (particularly topologically simple polarized L*-triple system) has a twograded L*-algebra envelope.

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REFERENCES [1] M. Cabrera, A. El Marrakchi, J. Martmez and A. Rodrfguez Palacios, An Allison-Kantor-Koecher-Tits construction for Lie #*-algebras. Journal of Algebra. 164 (1994), no.2, 361-408.

[2] A. J. Calderon and C. Martin, Dual pairs techniques in H* -theories, Journal of Pure and Applied Algebra 133 (1998) 59-63.

[3] A.J. Calderon Martin, C. Martin, Two graded L*-algebras, preprint, Universidad de Cadiz (Spain), 1999. [4] A.J. Calderon Martin, C. Martin, On associative ff*-structures induced by Jordan JY*-pairs, preprint, Universidad de Cadiz (Spain), 1999. [5] A. Castellon, J. A. Cuenca, Associative #*-triple Systems. In "Workshop on Nonassociative Algebraic Models'. Nova Science Publishers (eds. Gonzalez S. and Myung H. C.). New York, 1992. pp. 45-67. [6] A. Castellon, J. A. Cuenca, and C. Martin, Applications of ternary #*-algebras to associative /f*-superalgebras, Algebras, Groups and Geometries, 10, (1993), 181-190.

[7] A. Castellon, J. A. Cuenca, and C. Martin, Ternary H*-algebras. Bolletino U.M.I. (7) 6-B (1992), 217-228. [8] J. A. Cuenca, A. Garci'a and C. Martin, Structure theory for L*-algebras. Math-Proc. Camb. Phil. Soc. (1990), 361-365. [9] A. Fernandez Lopez, E. Garcia Rus and E. Sanchez Campos. Prime Nondegenerate Jordan Triple Systems with Minimal Inner Ideals. Nova Science Publishers Inc. (1992) 143-166. [10] A. Garcia, "Nuevas aportaciones en Estructuras Alternativas", Tesis Doctorales/Microficha num. 157. Secretariado de Publicaciones de la Universidad de Malaga. 1995. [11] N. Jacobson, Lie algebras. Interscience 1962.

[12] W.G. Lister, A structure theory of Lie triple systems. Trans. Amer. Math. Soc. 72 (1952), 217-242. [13] E. Neher, On the classification of Lie and Jordan triple systems. Comm. in Algebra. Vol. no. 13 (12), 2615-2667.

[14] J. R. Schue, Hilbert Space methods in the theory of Lie algebras. Trans. Amer. Math. Soc. vol 95 (1960), 69-80.

[15] J. R. Schue, Cartan decompositions for I/*-algebras. Trans. Amer. Math. Soc. vol 98 (1961), 334-349.

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Toric Mathematics from Semigroup Viewpoint A. CAMPILLO, Departamento de Algebra, Geometrfa y Topologfa, Universidad de Valladolid. 47005-Valladolid. Spain. E-mail: campillo @agt. uva. es P. PISON, Departamento de Algebra, Computacion, Geometrfa y Topologfa, Universidad de Sevilla. Aptdo. 1160. 41080-Sevilla. Spain. E-mail:ppison@cica. es

I

INTRODUCTION

Toric geometry is a subject of increasing activity. Toric varieties are objects on which one usually can check explicitly properties and compute invariants from algebraic geometry. This happens for the so-called normal toric varieties, i.e. algebraic varieties which are constructed from rational fans in a euclidean space. In the last 10 years the theory of non normal toric varieties has also been developed providing a very different and new scope as well as interesting and beautiful new applications. Normal toric geometry mainly uses techniques from convex geometry, as it is technically founded on the concepts of fan and cone. Fans are sets of polyhedral cones in such a way that each cone provides an affine chart of the toric variety. Namely, those charts have, as coordinate algebra, the algebra of the semigroup of lattice points lying inside the dual cone of the corresponding cone of the fan. To study non normal toric geometry one needs to be more precise than to consider only cones. In fact, what one needs is to consider affine charts where coordinate algebras are semigroup ones for more general classes of semigroups. Thus, convex geometry should be used only as a tool by taking into account that nice semigroups generate concrete polyhedral cones. The purpose of this paper is to show how mathematics in toric geometry can be understood as the theory of appropriate classes of commutative semigroups with given generators. This viewpoint involves the description of various kinds of derived objects as abelian groups and lattices, algebras and binomial ideals, cones and fans,

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affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics. Our approach consists in showing the mathematical relations among the above objects and clarifying their possibilities for future developments in the area. For that purpose, we will survey some recent results and concrete applications.

2

SEMIGROUP AND GENERATORS OF TORIC GEOMETRY

The central object we will consider along the paper should be finitely generated cancellative commutative semigroups with a specified system of generators. By a commutative semigroup we understand here a set S endowed with an internal commutative operation denoted by + having a zero element denoted 0. Semigroup homomorphisms are maps preserving the operation + and the element 0. Thus, one has the category of semigroups. Cancellative for 5 means that S is isomorphic to a subsemigroup of an abelian group, or in other words that the semigroup homomorphism S —> G(S), where G(S) is the abelian group generated by S, is injective. Here G(S) is the abelian group of classes of pairs (m,n) G 5 x 5 for the relationship (m,n) ~ (m',n') iff m + n! — m' + n. Thus, our central object should be the data of a semigroup S as above plus a surjective semigroup homomorphism

TTO : Nh -> 5, where N is the semigroup of nonnegative integers. Notice that TTQ is just the same data than the choice of a generator system of the semigroup S, namely the generator system ni,... ,nh, where rij = TTQ(CJ), ej being the /i-uple with j-coordinate 1 and other coordinates 0. Since toric geometry is a subject providing explicit computations and results, one can think that toric mathematics essentially consists in the detailed study of maps TTo of the above type. Along the paper it will be shown how the above statement stands when dealing with affine or projective toric objects. For studying such a map TTQ one needs to understand the structure and behavior of its fibers 7r^"1(?7i) for m & S. This is an elementary and difficult problem which,

for many purposes, becomes the key problem of toric geometry. A first remark is that one should consider some kind of finiteness hypothesis, namely requiring that the fibers TT^^TO) be finite for every m. The following result gives some distinct characterizations of that hypothesis.

PROPOSITION 2.1 (see [4]) Let TTQ : N'1 —> 5 be a surjective semigroup homomorphism where S is a cancellative commutative semigroup. Then, the following conditions are equivalent: 1. KQ (TO) is finite for every m.

2. There is no infinite sequence m € S, TOI, . . . , m j , . . . € S — {Q}, such that m — TOI — • • • — mi 6 S for every i. 3. Sn(-S) = {0}.

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4- There exists a semigroup hornomorphism A : 5 m = 0.

N such that A (TO) = 0 iff

Semigroups satisfying conditions in Proposition 2.1 are called, in the literature, of different forms according to the property one wants to emphasize. Thus, they are said to be combinatorially finite in view of (1) (see [5]), Nakayama in view of (2) (see [18] and [24]), strongly convex in view of (3) (see [11]), or positive in view of (4). The terminology which will be used along this paper is that of "positive". The description of the fibers is related to the study of relations among the chosen generators of S. Since the "kernel" of TTQ does not exist in the category of semigroups, to describe the relations one needs a different object, the congruence F of TTo, to define those relations. The congruence F is the binary relation on N^ consisting of those pairs (u, v) g N'1 x N'1 such that u, v belong to the same fiber TTQl(rn) for some m e S. Congruences are binary equivalence relations on semigroups allowing to give a semigroup structure on the quotient, i.e. with the property that (u, v) 6 F and w is in the semigroup (i.e. Nh in our case) then (u + w,v + w) e F. Since 5 is a finitely generated semigroup, by [13, 1.6], one has that the congruence F is finitely generated, i.e. that F is the least congruence containing one finite set of elements in it. In other words, one can say that 5 is a finitely presented semigroup. In the rest of the paper we will show how to treat and exploit the information in a semigroup with their generators and relations. This will involve several fields of mathematics on each of which one will derive concrete perspectives and consequences. The figure below shows the scheme of the spirit of our discussions.

\

T Semigroups (Generators and relations)

I Arithmetics!

3

ABELIAN GROUPS AND LATTICES

Consider a map TTQ : N'1 —> 5 as in section 1. Since the assignment to a semigroup of the group generated by the semigroup is functorial, one has an induced exact sequence of abelian groups given by

0 -> L -» G(Nh) = Zh -> G(S) -» 0, where L is a subgroup of Zh, so L is finitely generated and torsion free. We will refer to L as the lattice associated to the data TTQ. Notice that L is nothing but the kernel of surjective induced group homomorphism -K : Zh —> G(S), so that L is the object keeping the information of the group theoretical relations among the semigroup generators n\,..., n^.

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The relation between the congruence F and the lattice L can be described easily. In fact, if (u, v) e F then there is a unique element w € N'1 such that (u — w, v — w) 6 F and the supports of u — w and v — w are disjoint. Here by support of an element of Zh we mean the set of indices whose coordinates are nonzero for such an element. Notice that if < denotes the componentwise product ordering on Zh, then w is nothing but the infimum of u, v for that ordering. Thus, one has a well defined map b : F -> N'1 x L,

given by 6(u, v) = (w, u — v). If (w,l) € N'1 x L, set 1 = 1+ - I", where 1+ = sup(l,Q), l~ = sup(-l,0) and sup(,) stands for the supremum relative to the ordering <. Then the assignment to the element (w, 1) of the couple (1+ + w, 1~ + w) is a map N h x L —> F which is, by construction, inverse to 6. So one has the following result. PROPOSITION 3.1 The map b is a bijection. It follows from Proposition 3.1 that the information in F is just the same as in L and how one can get one from the other. Moreover, from free abelian groups and their sublattices one can study the semigroups we are interested in. In fact, if a lattice L C Zh is given, then from the obvious exact sequence

0-> L-+Zh -^Zh/L->0,

one can consider the subsemigroup S of the group Zh/L given by the image of Nh and generators given by the images of the elements ei,... ,e^. Also notice that the condition on S to be positive is equivalent to the condition L D Nh = (0). Notice that, in general, the abelian group G(S) = Zh/L can have torsion, so the semigroup S can also have torsion in the sense that it can contain elements ?7i, n € S, m ^ n and integers a G N such that am = an. If T is the torsion subgroup of G(5), the image of S in G(S)/T is a new semigroup S of the same kind than 5. Notice that S is positive iff S is so. This follows from the fact that LnN' 1 = (0) iff LnN' 1 = (0), L being the lattice for the induced map 7f0 : Nh -> 5. Finally, we remark that S is not only the image of N'1 by TT, but S is also the image of other subsets of Zh, in particular of the set N^ + L. This new set is also a semigroup which has an obvious structure of N^-module. As semigroup it is not positive (except in the trivial case L = 0 for which 5 = (0)), however if S is positive then N^ + L has the property analogous to (2) in Proposition 2.1, i.e. there is no infinite sequence of elements m = TOO > TOJ > . . . > TOJ > . . . in N'1 + L. In other words, if 5 is positive then N ft + L is generated by its minimal elements for the ordering <. Note that such minimal elements are nothing but the primitive elements of the set N'1 + L, i.e. those elements which are not a sum of a nonzero element of Nh with another element of N'1 + L.

4 SEMIGROUP IDEALS AND ALGEBRAS In this section we will fix a commutative field k. Then, one has a functor from the category of semigroups to that of /c-algebras taking each semigroup to its semigroup /c-algebra. Notice that, for any semigroup S, the semigroup /c-algebra k[S] consists

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of the vector space generated (freely) by the symbols x m > one f°r each m e 5, endowed with a multiplication given on symbols by the rule xm ' Xn = X m + n > f°r m,n € S. Now, consider a map TTQ : N'1 —> 5 as in section 1, and apply the above functor to it. One gets an exact sequence

0 -» 7 -> A = k[Nh ?$R = k[S] -» 0, where / is the kernel of the fc-algebra homomorphism • • •) x" h i so that / can be understood as the ideal of polynomial relations of such symbols. The ideal 7 is binomial as it is generated by the binomials X" — X" for (u, v) ranging over the congruence F. Using Proposition 3.1 one sees that it is also generated by X' — X' where 1 ranges over the lattice L. Anyway, notice that to generate / it is enough to take a finite number of binomials X" — X w , where the couples (u, v) generate the congruence F. Now, assume that S is positive. Then nice properties occur. First, by 3) in Proposition 2.1, one has that the irrelevant MR = ®TOJO kxm and MA = ©m-*;o A™ are ideals of R and A respectively. Second, by 1) one has that each Am is a finitely dimensional vector space. Third, by 2), Nakayama's lemma holds for Sgraded modules; in particular one can speak about minimal systems of homogeneous generators for / which are nothing but those inducing a basis of the vector space I/MA!- It is clear that one can consider minimal sets of binomial generators for /. In fact, one can consider S'-graded free homogeneous resolutions of R as an A-module. If 5 is positive, Nakayama's lemma shows that one can consider the minimal free resolution (which is unique up to isomorphism) which is one of the type n —> ^ pfp^ —> p ••• ^ lp p = AA —> fS. riH —> ^. U, r\ U —> p 1*2^—> f\ ^—>l -TQ where each Fi is a free S-graded finite A-module, the ^ are graded of degree 0 and p is the projective dimension of R as A-module, i.e. the least integer p such that Fp ^ 0. The Auslander-Buchsbaum theorem shows the relation p + r — h, where r is the depth of R. The integer r ranges over the values 0 < r < d, where d is the Krull dimension of R. Notice that the Krull dimension of R coincides with the rank of the abelian group G(S). The last statement follows from the computation of dimensions in terms of transcendence degrees. It implies, in particular that the dimension of the /c-algebra k[S] does not depend on the field k. This is not the case

for the integer r which could depend on k.

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Commutative algebra provides interesting particular cases. First, when r = d, the ring k[S] is said to be Cohen-Macaulay. This property depends on S and k but not on the map TTQ. If k[S] is Cohen-Macaulay and, moreover, Fp has rank 1 as an A-module, then k[S] is said to be Gorenstein. This is a case in which the minimal resolution is self-dual, i.e., by applying the functor Hom( — ,A) and considering the natural grading, the induced exact sequence

0 -> Eom(FQ, A) -> Hom(Fi, A) -> • • • • • • -* Eom(Fp, A) -» Coker(<4) -> 0, is S-graded isomorphic to the minimal resolution of R. Again the Gorenstein property depends on S and k, not on TTQ. Finally, k[S\ is said to be a complete intersection if / can be generated by h — d homogeneous elements (in fact binomials). Equivalently, complete intersection means that the congruence F can be generated by h — d pairs. The complete intersection property only depends on 5 and implies the Gorenstein property.

5

CONES AND FANS

With assumptions as above, the next object one can associate to a semigroup 5 is the cone C(S) generated by S, i.e. the cone generated by the image of S in the Q-vector space VQ := G(S) (8>z Q. Since the base ring extension from Z to Q kills the torsion, the cone C(S) obviously coincides with that of its image 5 in G(S)/T. If 5 is not positive, then C(5) is equal to the whole VQ, so it contains trivial information. Thus, the interesting case turns out to be the case in which S is positive. Note that S is positive if and only if C(S) is a strongly convex cone (i.e., if one has C(S) n —C(S) = 0). This fact justifies the terminology in (3) section 1. Now, if one takes into account the generators of 5, then one has that the cone C(S) is the rational polyhedral one generated by (i.e. it is the convex hull of) the images in VQ of the generators n\,..., n^. Thus, convex geometry occurs as a useful technique of toric mathematics. There is a very important case, in which the cone C(S) determines the semigroup 5. In fact, a semigroup is said to be normal if it is torsion free and if, moreover, one has 5 = C(S) n G(S). It is well known that 5 is a normal semigroup if and only if the semigroup fc-algebra A;[5] is an integrally closed domain, i.e. a normal ring. Hochster's theorem [14] shows that if S is normal then, in fact, k[S] is CohenMacaulay. A trivial example of normal semigroups are the free semigroups, i.e., those which are isomorphic to N* for some integer t. In fact, free semigroups are the only ones such that the fc-algebra k[S] is a regular ring. The terminology "regular" is coherently used also in convex geometry, being applied to a cone on Q* which is generated by a basis of the lattice Z*. Notice that a semigroup is free if and only if it is normal and if the cone it generates is regular. Toric geometry appears initially as the study of normal toric varieties. Thus, the development of normal toric geometry is based on convex geometry and, therefore, one can say that normal toric mathematics is convex geometry mathematics. Coming back to the general case, the convex cone C(S) provides a new interesting invariant for a semigroup 5, namely the number of edges e of C(S). Comparing

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with the dimension, one has e > d, and the equality holds whenever the cone C(S) is simplicial. Thus, semigroups for which e = d will be called simplicial along the paper. Free semigroups are a very special case of simplicial semigroups. Toric varieties also include non affine ones. Affine toric varieties are nothing but the affine varieties X with coordinate fc-algebra equal to a semigroup fc-algebra with the assumptions of section 1. General toric varietes are algebraic varieties which can be covered by affine toric varieties with overlappings which are also affine toric. Normal toric varieties are usually given in terms of convex geometry. The data consists in a fan $ of rational polyhedral cones in Q™, i.e, a set {crj^g^, where <& is a finite set, each a a strongly convex polyhedral cone in Q n , the faces of each a in $ are also in $, and the intersection of every couple of two cones in $ is a common face of both of them. The variety is constructed in the following way. For each a in <&, consider the semigroup 5CT of integer coordinates points which lie inside the dual cone of a, and let Xa be the affine toric variety given by Sa. Then the toric variety X is the join of the affine varieties Xff, the intersection of any two Xa, XT

of those affine charts being the toric variety Xanr. Thus, for a normal toric variety, the fan <& not only determines the variety but it represents and exhibits its geometry. In fact, cones in the fan correspond to affine charts in such a way that intersection of cones correspond to the overlapings of the corresponding charts. For non normal toric varieties one can proceed in a similar way, but taking a further precision on the semigroups. Thus, one needs a fan 4> as above plus, for each cone a, a subsemigroup S'a of Sa generating the same cone as Sa and in such a way that the intersection of two charts with respective coordinate algebras k[S'a] and k[S'T] is the affine chart with coordinate algebra /c[S^.nr]. Thus, one sees that, also in the global case, toric mathematics is not only convex geometry but again involves finitely generated cancellative semigroups. The support of a fan <£> is defined to be the union of the supports of the cones in the fan. The fan is said to be complete if its support is Q n . Toric varieties built from complete fans are complete algebraic varieties. The next section is devoted to the particular case of projective varieties, a subclass of complete toric varieties.

6

AFFINE AND PROJECTIVE TORIC VARIETIES

Toric varieties are algebraic ones, so algebraic geometry is naturally related to toric mathematics. Particularly interesting algebraic varieties are the affine and projective ones. When some data TTQ : N^ —-> S, is given, the semigroup 5 gives rise to the (abstract) affine toric variety X = Spec(k[S]), whereas the choice of generators provided by TTQ gives rise to an embedding of X into the affine space A.h. The dimension of X is just the rank d of the abelian group G(S). Below, we discuss and emphasize how abstract and embedded projective toric varieties can also be described in nice terms. Let S be a finitely generated cancellative commutative semigroup. Assume that 5 is endowed with a semigroup map A : 5 —> N such that the semigroup is generated by the elements in the set 5j = A - 1 (l). Then, for any choice of the field k, the couple (S, A) gives rise to an (abstract) (d — l)-dimensional projective algebraic scheme, namely Z = Proj(k[S}), where k[S] is now viewed as an Ngraded algebra by relaxing its natural 5-grading via the map A (in other words,

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degree i 6 N homogeneous elements are the sum of homogeneous elements of Sdegrees in A~ 1 (z)). Along the paper, couples (S, A) as above will be referred to as polarized semigroups. For a polarized semigroup (5, A), one has the property that m G 5 is a sum of i > 0 elements of Si if and only if one has A(m) = i. This property has two immediate consequences. First, the set Si (and hence any fiber X ~ 1 ( i ) ) is finite, as 5*1 is nothing but the set of irreducible elements in S. Second, the semigroup S is, a fortiori, positive. For the last statement, notice that to prove positiveness, when a map A as above already exists, one only needs to check A~ x (0) = 0 which follows from the afore mentioned property. Now, assume that 5 is torsion free. Then, since k[S] is a domain, the projective algebraic scheme Z is, in fact, a projective algebraic variety.

PROPOSITION 6.1 Let (S, A) be a polarized semigroup such that S is torsion free. Then Z = Proj(k[S}) is a projective toric variety. In fact, since S is torsion free, it can be viewed as a subset of VQ. On the other hand, the map A extends to a group homomorphism AZ : G(S) —> Z and to an R-linear map AR : VR —» R, where VR = G(S) ®z R» Now, let QI be the convex hull of the set Si in VR, and let S° C Si be the vertex set of S x . Notice, that S°,Si and QI lie in the affine hyperplane in VR given by A^^l). Fix m° € S°. Then, the semigroup S(m°) generated by the set of elements of type TO — TO° with m G Si is a new positive finitely generated semigroup whose associated group is A^^O). In particular, it follows that the dimension of the affine toric variety X(m°) given by S(m°) is d— 1 where d = rankG(S), i.e. the dimension of the projective variety Z. Moreover, X — Spec(k[S}) being the projecting cone of Z, the construction shows that the affine toric varieties X(m°), when TO° ranges over Sf, form a covering of Z as affine charts, making Z into a projective toric variety. This shows the proposition. For projective normal varieties it is possible to describe which Cartier divisors are ample and very ample ones. By a polarization of a projective variety one means picking a very ample Cartier divisor class. It provides an embedding of the variety in a projective space. When the variety is toric, one sees that the polarization produces a polarized semigroup (S, A) in such a way that the variety is isomorphic to the one given by the couple (S, A). See [11] for details. Thus, it is equivalent to give an embedded projective toric variety and to give a polarized semigroup. Notice that, for a given polarized semigroup (S, A), the set Si is the set of irreducible elements of S, so it is the only generator set contained in Si which gives the embedding of the affine toric variety X = Spec(k[S\) which is the projecting cone of Z. Thus, a polarized semigroup provides a canonical embedding of the projective toric variety into ph~l where h is the cardinality of Si. We remark that the fan giving rise to the projective variety Z lies in the dual space of the hyperplane Aq^O). Namely, the cones of the fan are exactly the duals of the cones generated by the semigroups S(m°). By construction, it is easy to see that such a fan is a complete one which corresponds to the algebraic geometric fact that any projective variety is complete. Finally, as it occurs for affine toric varieties, the main algebraic geometric characteristics of projective toric varieties are recognized in terms of the polarized semigroup (S, A). Thus, Z = Proj(k[S}) is said to be arithmetically Cohen-Macaulay

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(resp, Gorenstein) if and only if the algebra k[S] is Cohen-Macaulay (resp. Gorenstein). In the same way, Z is projectively normal if and only if k[S] is normal, i.e., if the semigroup 5 is normal. Finally the variety Z is normal (resp. regular) if and only if each semigroup S(m°) is normal (resp. free). Notice that to be projectively normal means that Si = 5,, where Si = A~ 1 (z) and Si = C(S) n X ~ 1 ( i ) , i.e. if every element in 5, is a sum of i elements of Si for all i's. Normalness can be characterized in rather similar terms, using the Ehrhart and Hilbert functions. The Ehrhart (resp. Hilbert) function is the map E (resp. H): N —> N given by E(i) = card(Si) (resp. H(i) = card(Si)), which coincides with a polynomial map of degree d—1 with coefficients in Q for i big enough. Then, under the most general conditions, the leading terms of the polynomials for E and H are equal, and the variety Z is normal exactly when both polynomials are equal. Obviously, in those terms, projective normalness is characterized by the property E = H.

7

POLYTOPES, SIMPLICIAL AND CELLULAR COMPLEXES

Once one has an embedded affine or projective toric variety, one looks at describing and computing, when possible, equations and syzygies for the embedding. Most results in this direction are recent and they use combinatorial objects such as simplicial and cellular complexes or polytopes. Note, from section 5, that the projective case is reduced to the affine one, as for a given polarized semigroup, the equations (and syzygies) of the embedded projective variety it defines are the same as the equations (and syzygies) for its projecting cone affine variety. Such an affine variety is nothing but the (affine) toric variety given by the semigroup S of the polarization with Si as chosen system of generators. Along this section, we will assume that a map TTQ : Nh —> S, as in section 1, is fixed, and that 5 is a positive semigroup. Denote by A the generator system of S given by TTQ, by II the set of primitive elements of the N^-module M = N'1 + L, and, for every m £ S, by Tm the set of monomials of 5-degree equal to m. Notice that the set Tm can be identified with the fiber iiQl(m). Recall that the fact that S is positive implies that M is generated by II and that each TTO is finite. Then, there are several combinatorial objects with vertex set one of A, II or Tm which are naturally associated to TTQ as described below. Associated to any fixed element TO in S one has the simplicial complexes A TO , Qm and the polytope Qm defined, respectively as follows. First, Am is the simplicial subcomplex of parts F of A such that m — np € 5", where np = X^eF n- Second, @ m is the simplicial subcomplex of parts G of Tm such that all the monomials of G have a non unit greatest common divisor (i.e. those monomials share at least one variable). Third, fim is the polytope in VR = Zh ®z R- given by the convex hull of the set Tm = 7r^ 1 (rn). Notice that on the set S one can consider an ordering ^ defined by TO' ^ m if and only if TO — m' € 51, and that, if TO' ;< TO then one has A m / C A m , 0m/ times a monomial of degree TO — m! is a subcomplex of Qm and, finally, the translation of fl m / by any vector in the fiber i^Ql(m — m!) is a subset of f2 m . Associated to the whole of 5, one has two useful regular cellular subcomplexes of parts of II. Namely, on one side one has the so-called Taylor complex H which

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is nothing but the (simplicial) complex of all parts of II, and, on the other hand, the hull complex S which is the subcomplex whose faces are the subsets of II which correspond with some unbounded face of the convex hull of the set of points of VR. of type ta = (tai,...,tah) for a = (ai,...,a^) € M where t is any big enough real number. The mentioned correspondence is the obvious one taking into account that any vertex of the above convex hull is necessarily one of type tb with b e II. See [2] for details on the construction and properties of the hull cellular complex. Sometimes, a subcomplex of E, the so-called Scarf complex is considered. It is, in fact, a simplicial complex which is defined to be the set of parts H of II satisfying the property a# ^ a/// for every H' ^ H where, a# stands for the supremum of the elements in H for the ordering < of section 2. The hull and the Scarf complexes coincide when the data TTQ is generic, i.e. when the congruence F can be generated by couples (u, v) such that the unions of the supports of u and v is the set {1, 2 , . . . , h}. In the sequel, we will often use reduced homology with values in the field k for simplicial and cellular complexes. The corresponding z'-th reduced homology vector spaces will be denoted by Hi. The description of equations has to do, in practice, with the determination of sets of binomial generators of the semigroup ideal / (section 3) which are either a minimal set of generators or a Grobner basis. For each monomial ordering (i.e. a total order on the set of monomials for which the monomial 1 is the minimum and which is closed under multiplication by constant monomials) one has a well denned reduced Grobner basis with respect to such an ordering, which also happens to be generated by binomials (see [22] for details). Thus, each such reduced Grobner basis can be understood either as a subset of the congruence F or of the lattice L (sections 1 and 2). The union of reduced Grobner bases for all the possible monomial orderings is called the universal Grobner basis, and it has the property of being, simultaneously, a Grobner basis for all monomial orderings. Again the universal Grobner basis can be seen as a subset of F or L. A reduced Grobner basis with respect to a concrete ordering can be computed from any other generator system by means of the well known Buchberger algorithm. The description of the universal Grobner basis becomes more difficult and it will be stated precisely just below. To find the universal Grobner basis, consider the subset U of S consisting of those elements m G S such that the polytope fi m has an edge which is not parallel to some edge of some fi m / for some TO' -< m. Then, for each m 6 U consider the binomials of type Xu — Xv, where the coordinates of u — v are relatively prime and the segment [u,v] is an edge of £2m. A result by Sturmfels, Weismantel and Ziegler [21] shows that the set of all binomials one obtains in this way when m ranges over U is exactly the universal Grobner basis of /. Such universal basis is finite as one can see that it is contained into the so-called Graver basis which is itself finite. The Graver basis consists of the binomials corresponding to the primitive elements of the lattice L, i.e. those elements 1 = 1+ — 1~ in L for which there are no other 1' = 1'+ - I'" in L such that 1 ^ 1' and 1'+ < 1+ and l'~ < 1~. To find minimal sets of homogeneous generators of / one can proceed as follows. Consider the set C of elements m £ S such that Ho(Qm) ^ 0, i.e. those elements for which the complex Qm is not connected. The set C is finite. For each TO 6 C pick a monomial Xu in each connected component of Qm and distinguish the monomial Xv picked for one concrete component. Then the binomials Xu — Xv, where Xu ranges over the picked monomials for the other components, are the degree m terms

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of a minimal system of homogeneous generators of /. Thus, when TO ranges over the set C, the whole set of obtained binomials Xu — Xv is a minimal set of homogeneous generators for the ideal. A different way to find homogeneous generators for /, which involves the complexes A m , is also available for higher order syzygies, and it will be discussed next in this higher order context. We do not know if such discussion could also be reasonably done in terms of the complexes 0m as well as of the complexes Am or whether Qm could be used to describe the universal Grobner basis. The description of syzygies consists in obtaining either the minimal 5-graded resolution (section 3) or concrete resolutions with other special properties, for example, the property of preserving the symmetries relative to the action of the lattice L. With notations as in section 3, the z-th order syzygy module is the 5-graded module A^ = ker((fi). Notice that one has A^o = /. For each degree TO G S, the number of generators of degree m in any minimal set of generators for Ni is, by Nakayama's lemma, the dimension of the /c-vector space Vi(m) = (Ni}m/(MANi)m. A first and key connection between syzygies and toric geometry is a result due initially to Hochster, [15], and considered again by several authors in [7], [1], [5], which asserts that one has an explicit and natural vector space identification of type

V-(m) = ffi(A m ), where Hi stands for the reduced simplicial homology with coefficients in the field k. Moreover, computations of direct and inverse images by the isomorphisms giving rise to the above identification are available. This result illustrates how combinatorics play a natural role also for describing syzygies, and, therefore, how one has many reasons to include combinatorics among toric mathematics. The first direct applications of the above result are given by Briales, Campillo, Marijuan and Pison in [4] to give an effective algorithm to compute minimal systems of binomial generators of the ideal /. To apply for i > I the above natural isomorphisms, the main difficulties which arise are first to compute those values of m such that ffj(A m ) is nonzero, and, second, to determine the homology. If one is able to avoid these two difficulties in concrete cases, then from the fact that the isomorphisms are explicit, one can derive successive methodic constructions of minimal sets of generators for the syzygy modules in the minimal resolution of R (see [5] for details). To approach the first difficulty, we will mention that, recently, Briales, Pison and Vigneron [6] ( [19] for the case i = 1) determine appropriate finite subsets Cj of S with the property that m $. Ci implies //j(A m ) ^ 0. As a consequence, they obtain an algorithm for computing the minimal resolution, (see [6] for details), because the second difficulty is quite well understood from a computational viewpoint, as concrete homologies can be calculated by means of linear algebra and integer linear programming as pointed out in [18], [19] and [6]. However, integer programming being also a technique to whose development toric geometry also is contributing (as we will show at the end of the paper), it is convenient to try to better understand the explicit structure of the homologies H_(Am). This is treated by Campillo and Gimenez in [8]. For it, one considers a partition A = £ U C, where £ is a subset of generators whose image in Vq generates minimally the cone C(S), in the sense that, for each edge of C(S), £ contains exactly

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one element whose image generates such an edge. Notice, that one has e = card(£), e being nothing but the invariant of S in section 4. Algebraically, one deduces that k[S] becomes a finite extension of k[£]. Thus, the minimal graded resolution of k[S] as an ^4-module can be compared with its minimal resolution as a B-module, where, now, B = fc[N e ] corresponds, as in section 1, to the semigroup generated by the set £. This situation puts in evidence two kinds of objects. First, one has the Apery set relative to £, which is nothing but the set Q of elements q 6 S such that q — n $_ S for every n E £. In other words, the Apery set is nothing but the set of exponents whose corresponding symbols generate minimally k[S] as a fc[£]-module, and, therefore, it is a finite set. Second, for each m € S one has the analog of Am for this relative situation, namely the simplicial subcomplex Tm of parts J of £ such that m — nj e S. Thus, one can see that the dimension of Hi(Tm) is exactly the number of degree m elements in a minimal set of £ -homogeneous generators of the z-th-order syzygy module in the above minimal resolution of the S-module k[S]. Now, for a fixed m € S, one has a key long exact sequence of type

. . . -> Hi+l(Qm) -^Ki^ ffi(A m ) -> Hi(Qm) -+#<_! ^ ... where H.(Qm) and K, are appropriated vector spaces of the following nature. First, H.(Qm) is the homology of a complex associated to the vertex m of a graph QQ with coloured edges constructed from the knowledge of Q, which has C as colour set and which is called the Apery graph. The vertex set of QQ consists of the elements m of type q + nj where q & Q and I C C. Edges of colour n £ C join a vertex

m1 to another m whenever m — m' = n. The complex associated to m has as z'-th chain space the one freely generated by the subsets / C C of cardinality i + I such that TO — n/ E. Q, the boundary map being the projection of the usual simplicial boundary. Second, the spaces K. are much more difficult to describe and we avoid the details. However, they can be computed in successive steps in two different and complementary ways. One, in terms of new graphs of exactly the same type than QQ but with other concrete sets instead of Q. Another in terms of homologies of type H.(Tmi) where the elements m' are of type TO — n/ with / C C. See [8] for the details and some applications. An extra consequence of the construction of the complexes Tm is that one has a way to characterize the depth r of the ring k[S}. Recall that the three integers r, d and e associated to a positive semigroup are such that r < d < e. The integers d and e are easily obtained from 5. To obtain r, in [8] it is proved that if TO is an integer with 1 < r$ < d, then the inequality r > TO is equivalent to the fact He-ro(Tm) = 0 for every m & S. In particular, for ro = d one gets a characterization of the Cohen-Macaulay property by the property

He-d(Tm) = 0 for every m € S, which, for the simplicial case e = d means that all the complexes Tm are connected. From this, it is easy to recover the well known characterization due to Goto [12] with asserts that, for simplicial semigroups, the Cohen-Macaulay property is equivalent to the property

TO € G(S),n, n' e £,n ^ ri ,m + n e S,m + n' 6 S1 => TO e S.

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Other characterizations of the Cohen-Macaulay property for nonsimplicial cases are given in [23] and [20]. A standard application which illustrates the use of the technique of the above long exact sequences is to the case of simplicial Cohen-Macaulay semigroups (i.e. those for which r = d = e). Since, in that case, the complexes Tm are connected, one can deduce that Ki = 0 for every i, so that one gets

for every m and i. Thus, the minimal resolution for simplicial Cohen-Macaulay semigroups can be derived from a unique combinatorial object, the Apery graph. For the general case, there are other ways to derive free resolutions for R from a unique combinatorial object. Namely, as shown in [2] this can be done either from the Taylor or from the hull complexes, H and E respectively. Let us explain how this works. For it, consider for each of the above cellular complexes an associated complex of ^-modules given as follows. The z-th order chains are the elements of the free A-module generated by the i-dimensional faces of the considered cellular complex, and the boundary map is given on any such face

# by

y^e(H,H') —— H' a V "' where the sum ranges over all the faces H' of the considered cellular complex, e(H,H') € {0,1,—!} denotes the incidence index for the cellular complex, and &H •, a/H are the elements defined above. Recall that, from the definition of regular cellular complexes, the incidence index satisfies the properties e(H, H') — 0 unless H' is a facet of H, so the above sum is extended only to facets of H in the cellular complex. Because of the properties of the Taylor and hull complexes, one has that what one actually gets are free A-module resolutions of k[M] = k[Nh + L}. Moreover, the resolution, which is Z^-graded by construction, is in fact invariant by the action of the lattice L induced from its action on II. This means, that each one of the chain of A-modules is in fact also a free .AfLj-module, where A[L] is the algebra of the group L on the coefficient ring A. Notice that one has A[L] = k[Nh x L], so the first projection N ft x L —> N^, seen as an N^-module homomorphism, gives rise to a surjective A-linear map A[L] —> A. Now, by extending scalars via the map A [L] -—> A and taking into account that

k[M] ®A[L] A = k[S] = R, one gets a complex which is in fact S-graded and exact. Thus, according to the considered cellular complex, one gets two S'-graded resolutions of R, which are respectively called the Taylor and the hull resolution.

Both resolutions are, in general, far from being minimal; however, if the data TTQ is generic, then the hull resolution is (isomorphic) to the minimal one. However, they are interesting and useful since they keep the action of the lattice L. Notice that, as commented before, the hull complex is equal to the Scarf complex for the generic case, so, in that situation the Scarf complex can be directly used instead of the hull one for constructing the above resolution, which is minimal besides. Nevertheless, we remark that the generic case is combinatorially characterized by the fact that

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the simplicial complexes Am which are not connected have connected components which are full simplices. This is a strong assumption from the combinatorial point of view, so, in general, if one wants to know about the minimal resolution the only available description (by the moment) is that discussed before, based on the study of the simplicial complexes A m . In the general (non generic) situation the hull resolution has, nevertheless, another nice property, as in fact it is a finite one, i.e. the free ^-modules involved are or finite rank and the number of them is finite. This is a non obvious statement which follows from the fact that the Graver basis is finite. See [2] for details.

8

MULTINUMERICAL SEMIGROUPS

In practice, the toric data TTQ (of a semigroup with a given generator set) is often given in arithmetical terms. In fact, the group G(S) being finitely generated, it is nothing but, up to isomorphism, one of type

Zd x Z/giZ x ... x Z/giZ for convenient integers d,l,gi,.. • ,giThus, if such an isomorphism is, a priori, considered, then TTQ becomes equivalent to the specification of the coordinate (d + /)-tuples (in the above product group) of the generators HI ,..., n^ of S. A semigroup given by such a specification is called a multinumerical semigroup. For the simplest case d = 1 and / = 0, they are usually referred to as numerical semigroups in the literature. What one would need, therefore, is to study toric varieties within arithmetics from multinumerical semigroups. This means to deduce the behaviour and geometrical properties of those varieties from arithmetic properties of the (d + Z)-tuples of integers or modular integers given by the semigroup generators. Such an arithmetical study becomes, nevertheless, difficult and it is an open problem except in rather few cases. The difficulties arising can be explained if one looks at the discussion in the above section on how combinatorics are involved in the development of toric geometry. In fact, using objects such as polytopes or simplicial or cellular complexes avoids having to deal with delicate relations among numbers. However, mathematically speaking, once that combinatorial methods have grown up and produced nice results, one can hope and try to interpret them in the framework of arithmetics. This strategy is used in [8] for affine and projective toric curves and in [5] for affine and simplicial projective toric surfaces. For the general case, good computational results dealing with equations are also derived by Vigneron in [18]. To show the possibilities of the above strategy, we will discuss, here, such results for curves. An affine toric curve is given by the numerical semigroup S given by a set A of h nonnegative integers. One has r = d = 1 and, since the cone C(S) has only one edge, also e = 1. Thus, this case is a simplicial Cohen-Macaulay one. Then, pick a partition of A in a set £ consisting of any single element s € A and as complementary set C the set of the h — 1 remaining elements. Now, consider the Apery set Q consisting of those integers q & S such that q — s £ 5, and from it construct the coloured graph QQ. It is not difficult to translate the graph structure into arithmetical relations, so that the homologies //j(A m ) = Hi(Qm) for

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Toric Mathematics from Semigroup Viewpoint

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the vertices m of QQ can be derived from such relations. One concludes that the minimal resolution for affine toric varieties can be obtained in complete arithmetical terms from the set of generators of the given numerical semigroup. See [8] for details. A projective toric curve of degree s is given by a subsemigroup of N2 generated by a set A = £ U C, where £ is the set consisting of the two elements (s, 0) and (0, s) and C consists of elements (GI, s — c\),..., (c^_2, s — 0^-2) for different values Cj with 0 < Cj < s. The semigroup S can be polarized by the function A given by \(c, c') — (c + c')/s, so that 5 defines an embedding of the projective toric curve in p/i-i_ Notice that one has d = e = 2 and that either r = 2 or r = 1, depending on the projective curve to be or not to be arithmetically Cohen-Macaulay. Let Si be the numerical semigroup generated by GI, . . . ,Q l _2,s, and for each c 6 Si denote by /z(c) the least number of the above generators of Si needed to achieve the sum c. Notice that the function fj, satisfies the property /j,(c) < /z(c—s) + l for every c e S whenever c — s S S. By translating into arithmetics the methods in [8], in [9] it is shown that the projective toric curve is arithmetically CohenMacaulay if and only if one has /i(c) = n(c — s) + 1 for every s e Si such that c - s e S. In general, from the knowledge of the function fj, it is easy to find the Apery set Q relative to the above partition A = £ U C as well as the set D consisting of those elements m in S such that m — (s, 0) € S, m — (0, s) e S, m — (s, s) ^ S. One can consider a coloured graph Q-p in an identical way as QQ but replacing Q by D. In [8] it is shown that the vector space Ki in the long exact sequence of the previous section can be identified with the homology Hi(Dm], where this last homology has also an identical construction than that for the case of the set Q. Thus, one deduces the long exact sequence

. . . -> Hi+1(Qm) -> Hi(Dm) -> tf z (A m ) -> Hi(Qm} - > . . . . The involved homologies as well as the image maps in this exact sequence can be given in aritmetical terms from the given data s, GI, ..., c^-i- From here, this is so for the reduced homologies H. (A m ), therefore, the minimal resolution of the projective toric curve is obtained from arithmetics.

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APPLICATIONS

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