Lie Algebras, Geometry, and Toda-Type Systems (Cambridge Lecture Notes in Physics)

10

.

The mapping cp* : 5(N) -* I(M) is an algebra homomorphism. Let 0 E a(M, N) and gyp/ E ,'(/N, K),, then cp o ' E a(M, K) and ((P 0 4,)* = `Y* 0 W*.

Exercises 2.1

Consider the set M - {(x1i x2) E 1182 I x1 > 0} as an open submanifold of 1182. Prove that M and 1[82 are diffeomorphic.

2.2

Show that the manifold M - JR x S' and the open submanifold N - Il82 - { (0, 0) } of Il82 are diffeomorphic.

2.3 Vector fields 2.3.1 Tangent vectors An 118 linear mapping v : 3(M) -+ 1[8 is called a tangent vector to

the manifold M at a point p E M if it satisfies the condition (2.1) v(fg) = v(f)g(p) + f(p)v(f) for any f,g E 3(M). The totality of tangent vectors to M at p is called the tangent space to the manifold M at the point p. This

space is denoted by Tp(M).

It is quite evident that for any v, u E TT(M) and a E R, the mappings v + u and av, defined as (v + n)(f) = v(f) + u(f),

(av)(f) = a(v(f)),

68

Basic notions of differential geometry

belong to TP(M). Hence, for any p E M the tangent space TP(M) has a natural structure of a vector space. Let U be an open submanifold of a manifold M and let v be a

tangent vector to M at a point p E U. It can be shown that for any function f E a(U) there is a neighbourhood V of the point p such that V C U and a function f E a(M) such that f Iv = f Iv. Define the action of the tangent vector v on a function f E '(U) by v(f) = v(f)-

It appears that this definition does not depend on the choice of the function f and we have

v(fIu) =

v(f).

(2.2)

The action of v on the elements of 3(U) that we have just described

defines a tangent vector to the manifold U at the point p. On the

other hand, any tangent vector v to U at p induces a tangent vector to M at p, acting on f E 3(M) by the rule v(f) = v(f 1U)Thus, we can identify the tangent spaces TP(M) and TP(U). Let M be a manifold and let (U, xl, ... , x') be a chart on M.

For any point p E U, define the tangent vectors (a/axi)P, i =

1,...,m, by

(axi) (f) = P

In particular, for the coordinate functions, we have

(x') = a'. It can be shown that the vectors (a/axi)P form a basis of the 19 (-)

P

vector space TP(M). Hence, any tangent vector v E TP(M) can be represented as

v=v 2 (axi)P a1 The action of v on the coordinate functions gives v(x2) = v2.

Thus, for any tangent vector v E TP(M) we have

v=v x2 a ( ) (axi)P

2.3 Vector fields

Let M and N be two manifolds and let cp E p E M define the linear mapping cp*p : Tp(M)

69

(M, N). For each T,(p)(N) by

(cP*P(v))(f) = v(W*f)

for any v E Tp(M) and f E J(N). The mapping cp*p is called the differential of the mapping cp at the point p. Let 0 E a(M, N) and co E (N, K). For any point p E M one has (cP 0 4')*p ='P*'(P) ° "*P-

From this relation it follows that if a smooth mapping cp : M -+ N is a diffeomorphism, then for any p E M the mapping cp*p is an isomorphism. A smooth mapping cp : M -a N is called regular at a point p E M if the mapping cp*p is an isomorphism. If a mapping cp

: M -* N is regular at a point p E M, then there exists a

neighbourhood U of p such that cp(U) is an open set and the restriction cpju of the mapping cp to U, considered as the mapping from U to cp(U), is a diffeomorphism. This statement is called the inverse mapping theorem.

A smooth mapping from an open interval of R to a manifold M is called a curve in M. Let A : I -+ M be a curve in M and let t e I. Introduce the notation

fi(t) = a*t (h), where x is the standard coordinate function on I. It is clear that A(t) is a tangent vector to M at the point A(t). This tangent vector is called the tangent vector to the curve A at the point p - A(t). A curve A : I -* M with 0 E I and A(0) = p is called a curve at p. It appears that for any v E Tp(M) there is a curve A at p such that A(0) = v. In other words, any tangent vector to a manifold can be considered as the tangent vector to a curve. EXAMPLE 2.10 According to example 2.8, any real vector space has a natural smooth differentiable structure. Let V be a vector

space and let u be an arbitrary vector of V. For each v E V, the mapping t E IR H u + tv E V is a curve at u. Denote by v the tangent vector to this curve at u. It can be shown that the mapping v H v is an isomorphism of vector spaces V and Tu (V ) which allows us to identify these vector spaces.

Basic notions of differential geometry

70

2.3.2 Vector fields and commutator An assignment X of a tangent vector X, E TP(M) to each point p of M is called a vector field on M. The action of a vector field X on M on a function f E "(M) is defined as a function X (f) given by the relation

(X(f))(p) = XP(f) for each p E M. A vector field on M is uniquely defined by its action on the elements of '(M). A vector field X on M is called smooth if for any f E a(M) the function X (f) is smooth. The totality of smooth vector fields on M is denoted by 1(M). Define the sum of two vector fields and the multiplication of a vector field by a function as

(X +Y)(f) X(f) +Y(f), (fX)(g) = f(X(g)) If X and Y are smooth vector fields, then X +Y is a smooth vector field. If X is a smooth vector field and f is a smooth function, then f X is a smooth vector field. With respect to these operations, the set 1(M) is a module over the algebra '(M).

From (2.1) it follows that for any X E 1(M) and any f,g E '(M) we have

X(fg) = X(f)g+fX(g) Thus, any element of 1(M) is a derivation of the algebra 3(M), and it can be shown that any derivation of a(M) is generated by an element of 1(M). Recall that the set of derivations of an algebra is a Lie algebra with respect to the commutator of derivations. In

particular. it is a vector space. Therefore, the set 1(M) is a Lie algebra over JR with respect to the commutator of vector fields, defined as

[X,1'] (f) = X (Y (f)) - Y (X (f) ) for any X,Y E 1(M) and f E a(M). In other words, the commutator of vector fields has the properties

[X,Y] = -[Y,X],

(2.3)

[aX + bY, Z] = a[X, Z] + b[Y, Z],

(2.4)

Ix, [Y, Z]] + [Y, [Z, X11 + [Z, [X, Y]] = 0

(2.5)

2.3 Vector fields

71

for any X, Y, Z E 1(M) and a, b E IR. Recall that equality (2.5) is called the Jacobi identity. Note also that the set 1(M) is an infinite-dimensional Lie algebra. The commutator [X, Y] of the vector fields X and Y is often called the Lie bracket of X and Y. Let M be a manifold, U an open submanifold of M, and X a vector field on M. As stated in the preceding section, the tangent vector XP E TP(M), p E U, can be considered as an element of TP(U). Thus, a vector field on M induces a vector field on U which

is denoted by X Iu. Here, if a vector field X is smooth, then the vector field XIu is also smooth. From (2.2) it follows that

XIu(flu) = X(f)Iu. Let (U, x1, ... , x') be a chart on a manifold M. The vectors (a/ax=)P, p E U, specify a set of m smooth vector fields on U denoted by a/axi. Using the fact that for any p E U the vectors (alax')P form a basis for TP(M), we conclude that for any vector field X on M one can write

XIu=Xi-

,

where Xz = Xlu(xi). If the vector field X is smooth, then the functions X', i = 1, ... , m, are also smooth. In other words, the vector fields alax' form a basis for the '(U)-module 1(U). Let M and N be two manifolds and let cp : M -+ N be a smooth mapping. Recall that for any point p E M there is defined the mapping co which connects the tangent spaces TP(M) and T,o(P) (N). In general, it is not possible, proceeding from the mappings co*p, p E M, to define a mapping connecting vector field on M and N. Indeed, if for a given vector fields X E 1(M) we try to define the vector field Y E 1(N) by( YW(P)

*(XP)'

we do not obtain, in general, a well-defined vector field on N. There are several reasons for this. Assume, for example, that cp is not an injective mapping; then there are at least two points p, p' E M such that W(p) = cp(p'). In this case, to obtain a welldefined vector field Y, we must have the equality W*(XP) = P.(XP'),

which is not valid for an arbitrary vector field X.

Basic notions of differential geometry

72

Nevertheless, let X and Y be two vector fields on M and N respectively. If for any point p E M, Y"(P) = cp*(XP),

(2.6)

we say that the vector fields X and Y are co-related, and write Y = cp*X. A vector field X E 3C(M) is called cp-projectible if there

exists a vector field Y E X(N) such that the vector fields X and Y are cp-related. If the mapping cp is not surjective, then a cpprojectible vector field X does not uniquely define a vector field Y satisfying (2.6). In this case the expression cp*X means some vector field Y on N satisfying (2.6). Let cp be a smooth mapping from a manifold M to a manifold N. It can be shown that vector fields X E X(M) and Y E .t(N) are cp-related if and only if W*(Y(f)) = X (co*f )

for any function f E a(N). Hence, if a vector field X E .C(M) is cp-projectible, then for any function f E '(M) and for any choice of the vector field cp*X we have (2.7) cp*(W*X(f)) = X(cc*f). If vector fields X, Y E X(M) are cp-projectible, then the vector fields X + Y and [X, Y] are also cp-projectible, and for any choice of the vector fields cp*X and W* Y, the vector fields W* (X +Y) and cp* [X, Y] can be chosen in such a way that

W* (X + Y) = cp*X + cp*Y' cp* [X, Y] _ [cp* X, W* Y].

If cp

(2.8) (2.9)

: M -4 N is a diffeomorphism, then any vector field X E

X(M) is cp-projectible. In this case the vector field cp*X is unique, and from (2.7) it follows that W*X (f) = cp-1*(X (cp*f))

for any f E J(N).

2.3.3 Integral curves and flows of vector fields A smooth curve A : I -* M, where I = (a, b) with a, b E 11 U {-oo, +oc}, is called an integral curve of a smooth vector field X

on M if a(t) = XA(t)

2.3 Vector fields

73

for any t E I. For any point p of a manifold M, and any vector field X E X(M), there is a unique maximal integral curve Ax (a(p), b(p)) -* M of X, such that 0 E (a(p), b(p)) and AX (0) = p. For an arbitrary t E R we define the subset Dx of M by {p E M I t E (a(p), b(p))}, and the mapping 4)X : DX -+ M by DX

4,X (p) = Ax (t).

It is clear that Do = M and o = idM. The mapping 4'X , considered as a mapping from DX to 1X (DX ), is a diffeomorphism, and x -i x

(fit)

_ -t

This equality means, in particular, that -(DX (DX) = Dxt. Moreover, f o r any t, s E Il8 the set I (Ds) n DX is a subset of Dt+3, and t xO s t+s

x= x

on 4); (Ds) nDX . Let cp E 3(M, N) be a diffeomorphism, then cp 0'0 0 cp-1 = ct .x for any t E R. There is also a useful relation lim 1 [Y t-+o t

- (")Y] = [X, Y].

(2.10)

A vector field X on a manifold M is called complete if DX = M for any t E lit It can be shown that any vector field on a compact manifold is complete. Also introduce the subset Dx of M x ll8 given by

Dx U Dx x {t}, and the mapping (Dx : Dx

M defined as

-Dx (p, t) = IpX (p) = A (t)

The set Dx is an open subset of M x R and the mapping 4x is smooth. The mapping -Dx is called the flow induced by the vector

field X. If X is a complete vector field, then Dx = M x It EXAMPLE 2.11 Consider the vector field a

X=

e-'

aX

Basic notions of differential geometry

74

defined on the manifold JR. A curve A is an integral curve of X if dA(t) = e-a(t) dt The solution of this differential equation is A(t) = ln(t + c), where c is the integration constant. Hence, for the maximal integral curves Ax, a E X we can write the expression AX (t)

= ln(t + ea).

a Therefore, Dt = (-oo, +oo) for any t > 0, while for t < 0 we have Dt = (In ItI, +oo). Thus, the vector field X is not complete.

Exercises Consider the set IEtf - (0, +oc) as an open submanifold of JR and define the mapping V : JR -* l[R+ by W(a) = ea. Prove that co is a diffeomorphism. For the vector field X = e-x8/8x on JR find the vector field co.X. 2.4 Find the maximal integral curves for the vector fields 2.3

X=(1+x2)ax on R, and for the vector fields a

a

a

a

X =x142 +x2ax 2 , X =x1a2-x28x 1

1

on R2. 2.5

Let 4X and 4jaX, a E IR, be the flows induced by the vector fields X and aX. Show that ,Dax = ,IX t

at

for any t E 1R.

2.4 Tensors 2.4.1 Cotangent space and covector fields The dual of the tangent space TT(M) is called the cotangent space to the manifold M at the point p; this space is denoted by TP (M). The elements of Tp (M) are called cotangent vectors or covectors.

2.4 Tensors

75

An assignment w of a covector wp E Tp(M) to each point p of M is called a covector field on M. For any covector field w on M we define the action of w on a vector field X on M as a function w(X) given by (w(X))(p) = wr(Xv).

(2.11)

A covector field w is called smooth if for any X E 3C(M) the function w(X) is smooth. The totality of smooth covector fields on M is denoted by .C*(M). From the definition of the action of a covector field on a vector field it follows that a smooth covector field

induces an a(M)-linear mapping from the ,a(M)-module X(M) to "(M). Furthermore, any a(M)-linear mapping from X(M) to a(M) is generated by a covector field. EXAMPLE 2.12 Let f E J(M), define the covector field df by

df(X) - X(f). It is evident that df is a smooth covector field on M, which is called the differential of the function f. The sum of two covector fields and the multiplication of a covector field by a function are defined as (w + rl) (X)

(X) + rt(X ),

(fw)(X) fw(X). If w and ' are smooth covector fields, then w + q is a smooth covector field. If w is a smooth covector field and f is a smooth function, then f w is a smooth covector field. It is clear that the set .C* (M) is a module over the algebra a(M). Let U be an open submanifold of M. Recall that for any p E U

the tangent space TT(M) is naturally isomorphic to the tangent space TT(U). This fact allows us to identify the cotangent spaces TP (M) and TP (U) and to define the restriction wlu of a covector field w on M to the submanifold U. Here we have w1U(X 1U) = W(x)1U-

On the other hand, let U be an open submanifold of a manifold M, and w E X*(U). For any point p E U there is a covector field w E 1(M) and an open neighbourhood V C U of the point p such that WIv = wIv.

76

Basic notions of differential geometry

Now let (U, x', ... , x"") be a chart on a manifold M. It can be shown that for any p E U the covectors (dxi)p form a basis for Tp (U). Actually, we have

(dx')p((a/ax')p) = 6j; and hence, the basis, formed by the covectors (dxi)p, is the dual basis for {(a/axi)p}. Further, for any covector field w on M, we can write w1u = widx',

where wi = wlu(a/axi). If the covector field w is smooth, then the functions wi are also smooth. In particular, for the differential of a smooth function f we have

(df)Iu = ax dxi. Moreover, the covector fields dxi form a basis of the j(U)-module 3C' (U).

Let cp : M -* N be a smooth mapping. For any p E M define the mapping WP* : T;(p) (N) -* TT (M) by

(W*(a))(v) = a((p*p(v))-

Here for any 'O E 3(M, N), cp E a(N, K) and p E M one has cP0V)

For any covector field w on N and a smooth mapping cp : M -+ N we define the covector field cp*w on M by the relation (cP*w)p = cPp(wG(P))'

If the covector field w is smooth, then the covector field cp*w is also smooth. In the case where cp : M -+ N is a diffeomorphism. one has (cP*w)(X) = cP*(w(cP*X))

(2.12)

for any vector field X on M.

2.4.2 Tensor fields A tensor of type (1) on a vector space V over a field K can be defined as a K-multilinear mapping from the set

V*xV*x...xV*xVxVx...xV

2.4 Tensors

77

to K From this point of view, the tensor product of tensors t E Tlk (V) and s E T,-,(V) is the tensor t ® s E T+,, (V) given by t ® s(µ1,... µk,Vl,... ivm,v17 ...7VhU17...7Un) = t(µ1.... , µk, vl, ... IVI)s(vl, ... , vm, U1, ... , 21n) Recall that if {ea} is a basis of an m-dimensional vector space V,

and {ai} is the corresponding dual basis of V*, then the tensors

ei, ® ... ® ez, ® a'1 ®...®a", where 1 < i 1 , ... , ik, jl, ... j` < m, form a basis for Tk (V).

Now let M be an m-dimensional manifold. The space of tensors of type (k) on the tangent space TP(M) is denoted by TP(M). Let (U, x1, ... , xm) be a chart on M; the tensors asX1

1

C

where 1 _< il, ...

®... ®(/ aak®(dx" )P ®... ®(dx" )P, P

, ik, jl.... jl < m, forma basis for TP(M).

An assignment T of an element of TP E T,P (M) to each point p E M is called a tensor field of type (k) on M. In particular, a vector field is a tensor field of type (0'), and a covector field is a tensor field of type (01).

A tensor field T of type (k) on M generates an,3(M)-multilinear mapping from the set X*(M) x ... x X*(M) x X(M) x ... x X(M) k

1

to a(M), given by (T(W1) ... ,Wk,X1i...,XI))(P) TP(W1P,...,Wkp,XIP,...)XjP), and any such a mapping is generated by a tensor field of type (1) on M. A tensor field is called smooth if its action on the corresponding number of smooth covector and vector fields is a smooth function. The set of smooth tensor fields of type (k) on M is denoted .X! (M). The pointwise definition of the corresponding operations turns

the space Xk (M) into a module over the algebra '(M). Similarly, defining the tensor product of tensor fields by

(R®S)P-RP®SP, we endow the space 00

T (M) = ®3Ci (M), k,1=0

Basic notions of differential geometry

78

where Xg(M) =_ 3(M), with the structure of an associative algebra over 3(M), called the tensor algebra of M. Any tensor on a manifold can be naturally restricted to any of its open submanifolds. Let (U, x', ... , x"`) be a chart on a manifold

M. It is clear that the tensors

aa21 ®...®--k ®dx2' ®...®dx3', where 1 < i1,. .. , ik, jl, ... it < m, form a basis for the 3(M)module 3C' (M). For any tensor T of type (1) on M we have T j u = T;,::.

;

x, 0 ... 0 as

09xi"

®dx" ®... ®dx"

,

where

T],..., =Tlu(dxE',...,dx'k,C7/C7xJl,...,a/axi'). Let cp : M -+ N be a diffeomorphism and let T E Lk (M). The tensor field cp*T E Xl (M), defined by

(co T)(wl,...,wk,X1i...,X1) -1*

(T(co*wl,...,cp*wk,cO

cP

Xl,...,cps X1)) 1

is called the push-forward of T by W. Similarly, for any T E the pull-back cp*T E 3C' (M) of T by cp is defined as

.

k (N)

(cp*T)(wl,...,wk,X1i...,X,) cP* (T

(W-1*wl.... cp-1*Wk, cP*X1, ... , (p*Xl))

Here, for any diffeomorphisms has

M -* N and cp : N -+ K, one *1=c9*

)*=W.°`Y*, Furthermore, for any tensor fields R and S on M, (cP°

cp.(R ®S) = cp*R ®cp*S.

2.4.3 Differential forms Define an action of the symmetric group Sk on the space of the tensors of type (0) by t, (V1, ... , Vk) = t(vo(1)) ... , VQ(k))

Denote by E. the sign of the permutation or. A tensor t of type (k) is called skew-symmetric if tQ = Eot

2.4 Tensors

79

for any a E Sk. It is customary to call skew-symmetric tensors of type (0) on a vector space V exterior k-forms. The totality of exterior k-forms on V is denoted Ak(V). It is convenient to consider Ao(V) - IK and Al (V) - V*. In other words, the elements of the field K are 0-forms, and the covectors are 1-forms. The set Ak(V) is a subspace of TA (V ). Clearly, Ak(V) = {0} if k is greater that the dimension of V.

For a given tensor t of type (k), we can construct an exterior k-form, applying to t the alternation mapping Alt defined as Alt t

eatc. !

oESk

The tensor product of two forms is not a skew-symmetric tensor. Define the exterior product of a E Ak (V) and ,8 E At (V) by

a A,3 - (k li)! Alt(a®,3). The exterior product is a bilinear associative operation satisfying the relation a A,3 = (-1)kl[3 A a for any a E Ak(V) and /3EA,(V).

The direct sum A(V) of the spaces Ak(V) is an associative algebra with respect to the multiplication induced by the exterior product operation. This algebra is called the exterior algebra of V, or the Grassmann algebra of V. Let V be an m-dimensional vector space. For 0 < k _< m the dimension of the vector space Ak (V) is equal to (k) . The exterior algebra A(V) has dimension 2'. Let lei} be a basis for V and let fail be the corresponding dual basis for V*. The k-forms 1 < it < i2 .. < ik < m, a" A---A ask, form a basis for Ak(V). Let M be an m-dimensional manifold. We denote the space of exterior k-forms on Tp(M) by Akp(M). Let (U, xl, ... , xm) be a

chart on M, such that p E U. The tensors (dx" )p A ... A (dxik )p, 1 < it <

.

< ik < m,

form a basis for Akp(M). An assignment w of an exterior k-form wp E Akp(M) to each

point p of M is called a differential k-form on M, or simply a

Basic notions of differential geometry

80

k-form on M. Since differential forms on M are a special case of tensor fields on M, they possess the properties of tensor fields. In particular, we can define the notion of a smooth differential form. The totality of smooth differential k-forms on M is denoted by S2k (M). The pointwise definition of the algebraic operations turns Ilk(M) into the module over the algebra a(M). Defining the exterior product of differential forms by

(wA97)r-wpAgp, we provide the direct sum m

Q(M) = ®Qk(M), k=o

where 1o(M) - a(M), with the structure of an associative algebra over a(M), called the algebra of differential forms on M. Let (U, x1, ... , xm) be a chart on a manifold M. The k-forms

1
dxil

form a basis of the "(M)-module Ilk(M). For any k-form w on M, we have wI u =

, wil...ik dxi'

n ... A dxik .

(2.13)

Here the functions wi,...ik are unique if we suppose that they are skew-symmetric with respect to any transposition of the indices il, ... , ik. In this case we have wil...ik = COW-91,9X i7 , .. . , a/axik ).

The differential of a function can be considered as a particular case of the exterior derivative operation. In the general case, the exterior derivative is defined as an I[8 linear mapping d : 11(M) -* 1(M) satisfying the properties (ED1) dIlk(M) C 11k+l(M); (ED2) if f E 520 (M), then df (X) = X (f) for any X E X(M); (ED3) d o d = 0; (ED4) d(wA77) = dwA77+(-1)kwAdi7 for any w E 1k (M) and 77 E 1(M).

It can be shown that these properties define the mapping d uniquely.

If U is an open submanifold of a manifold M, then for any w E 11(M) one has d(wlu) = (dw)I u

2.4 Tensors

81

Let (U, x1, ... , x') be a chart on a manifold M. Using the proper-

ties of the external derivative and relation (2.13), for any k-form w on M we obtain (dw)I u =

1 C7Wil...ik

axa

_k!

dxi A dxit A

A

dx"'

It can also be shown that for any k-form w there is a useful formula: _

k+1

dw(Xl, ... , Xk+1) = Lam(-1)i+1Xi(w(X1, ... , Xi, ... , Xk+1) i=1

+

(-1)i+iw([Xi,Xi], Xl,...,Xi,...,Xj,...,Xk+1) 1
(2.14)

In particular, for the case where k = 1 we have dw(X, Y) = X (w (Y)) - Y(w(X)) - w([X, Y]). (2.15) It is important that the pull-back of a differential form is defined not only for diffeomorphisms. Let V : M -+ N be a smooth mapping. Define the pull-back W*w of the differential form w on N as a differential form on M given by (c,*w)P = P(wP)

It can be shown that if the differential form w is smooth, then the differential form cp*w is also smooth. Here d(cp*w) = cp*(dw),

and for any differential forms w, 77 E S2 (N) one has co*(w A77) = co*w A cp*q.

Exercises 2.6

Let w and i be 1-forms. Show that w A77 (X1,X2) = w(Xl)77 (X2) - w(X2)77(X1)-

In other words, demonstrate that

wnil =w0q-rl®w. 2.7 Let w be a 2-form and let rl be a 1-form. Prove that w A 17(X1, X2, X3) = w(X1, X2)77(X3)

+ w(X2, X3)71(Xl) + w(X3, X1)rl(X2)

Basic notions of differential geometry

82

2.8

Show that for the case where k = 2, equality (2.14) takes the form

dw(Xl,X2,X3) = X1(W(X2,X3)) +X2(W(X3,X1)) + X3(W(Xl, X2)) - w([Xl, X2], X3) - w([X2,X3],XI) -Wl[X3,X11,X2)

2.5 Complex manifolds 2.5.1 Definition of a complex manifold Let f be a continuous complex function defined in an open subset U of the space Cm. Denote by zi, i = 1, ... , m, the standard complex coordinate functions in Cm, and by xi and yi the cor-

responding real coordinate functions such that zi = xi +yi. Represent f as

f =g+V1h, where g and h are real functions on U. Suppose that for any i there exist the partial derivatives of g and h over xi and yi, and define the partial derivatives of f over xi and yi by

of

ag

+ah

= , axi axi ax= Also introduce the notation

of

-

1

of

of ayi)

ayi

+ah.

1

of

of = ag

ayi

of

-

ayi

of

kaxi ' azi = 2 kaxi The function f is called (anti)holomorphic in U if azi = 2

of =° (az Let cp be a continuous mapping from an open subset U of Cm

to C. Denote by zi, i = 1, ... , m, and w°, a = 1,... , n, the standard coordinate functions on Cm and C" respectively. The mapping cp is called (anti) holomorphic if the functions coa = cp*w°

are (anti)holomorphic. Let M be a topological space. A pair (U, cp), where U is an open subset of M and c' is a homeomorphism from U to an open subset V(U) of the space Cm, is called a complex chart on M. Let M be provided with the set of complex charts {(Ua, cpa)}-E.A such that the mappings cpa(Ua n UQ), a,Q E A, Wa 0 W,31 : VO(U. n Up)

2.5 Complex manifolds

83

are holomorphic and

U Ua=M. aEA

In this case M is said to be an m-dimensional complex manifold, and the set {(U., Wa)}.EA is called an atlas of M. It is quite obvious how to define the notions of the maximal atlas and of an admissible chart in this case. EXAMPLE 2.13 The standard structure of a complex manifold on

the space C' is specified by an atlas consisting of just one chart (Cm, idc-). The standard coordinate functions zi, i = 1, ... , m, in this case are defined as z'(c) = ci for any c= (cl,... , cm) E C-. EXAMPLE 2.14 Consider the standard two-dimensional sphere S'-. As a real manifold it has an atlas consisting of two charts defined in example 2.9. Introducing the mappings z=u1+V"-_1u2, w=v1v2, we obtain two complex charts on S2. In the corresponding domain

we have w = 1/z. Thus, we see that the manifold S2 can be endowed with the structure of a complex manifold. Since the space CCm can be identified with the space 1[82m, and

the real and imaginary parts of a holomorphic function are real analytic functions of the corresponding real arguments, any complex manifold M can be considered as a real manifold of class C'. This manifold is called the realification of M and is denoted by

M. It is clear that dim MR = 2dimM. A complex function on a real manifold M is called smooth if

its real and imaginary parts are smooth functions. A function f on a complex manifold M is said to be smooth if it is smooth as a function on MR. The set of complex smooth functions on a real or complex manifold M is denoted by (M). This set has the natural structure of a complex associative commutative algebra. A complex function f on a complex manifold M is called holomorphic (antiholomorphic) if the local expression of f with respect

84

Basic notions of differential geometry

to any admissible chart on M is a holomorphic (antiholomorphic) function. Note that not all complex manifolds admit the existence of holomorphic or antiholomorphic functions different from constants.

Let M and N be complex manifolds. A mapping o : M -+ N is called smooth if it is smooth as the mapping from the manifold MR to N. A mapping cp : M -+ N is said to be holomorphic (antiholomorphic) if its local expressions with respect to all admissible charts on M and N are described by holomorphic (antiholomorphic) functions.

2.5.2 Vector fields on complex manifolds Let M be a real manifold. A tangent vector v E Tp(M) defines a mapping from 'c(M) to C in the following way. Let g and h be

the real and imaginary parts of the function f E '(M); define the action of v on f by

v(f) - v(g) + Vv(h). It is useful to consider the complexification of the space Tp(M), which is usually denoted by Tpc(M). The elements of Tpc(M) are called complex tangent vectors. If it is necessary to distinguish complex tangent vectors and ordinary ones, we use the term real tangent vectors for the elements ofTp(M). Representing a complex tangent vector v E Tc(M) as v = u+V"-_1 w, where u, w C- Tp (M),

we define the action of v on a real or complex smooth function f by

v(f) =u(f)+

w(f). A notion of a tangent vector to a complex manifold is introduced in the same way as for a real manifold. The real vector space of all real tangent vectors to a complex manifold M at a point p E M will be denoted TRA(M). Since we identify the set of smooth functions on a complex manifold M with the set of smooth functions on the manifold MR, we can identify the space TRA(M) with the tangent space Tp(MR). Let (U, zl, ... , zm) be a chart on a complex manifold M. For each i = 1, ... , m, we can write a unique representation

zi = xi +

y2,

(2.16)

2.5 Complex manifolds

85

where xi and yx are real functions which can be considered as coordinate functions on MR. For any p E U, the tangent vectors (8/ax')p and (8/8yz) form a basis of TP(MR), and, therefore, they form a basis of TRA(M). Define the linear operator J : TRA(M) TRp(M) by

J7'((a/axi)P) = (091090,

Jp ((8/ayx)p) = -(alax1)P-

It is evident that JP is a complex structure on TRP(M). This complex structure does not depend on the choice of the chart containing the point p. We define the tangent space Tp(M) to the complex manifold M at the point p as Tp(M) = TRP(M). As it follows from (1.28), we have for the space

T1 (M) = (TRP(M))c the following direct sum decomposition: Tp (M) = TP1,°) (M) ED TPO'') (M).

(2.17)

The operator JP can be uniquely extended to the linear operator acting in Tp (M), which will also be denoted by JP m. The spaces TP1'0i(M) and TP°°1)(M) can be characterised as the eigenspaces of the operator JP MI corresponding to the eigenvalues + and -V,'---l respectively. Moreover, there are natural isomorphisms of the spaces TP1°0)(M) and TP°'1)(M) with the spaces Tp(M) and TP(M); in other words, we have Tn (M)

Tp(M) ® TT(M).

Since the space TP (M) is, by definition, the complexification of TR (M), the complex conjugation is defined for the elements of TT (M). This conjugation transforms elements of TP1'0)(M) to elTP°,1) (M) and vice versa. The linear operators PP and ements of Pm on TT (M), defined by PMx

2 (x -

JP x),

Pp x

2 (x + v '-1 JPMx),

(2.18)

are projection operators on TP1'0) (M) and TP°"1) (M) respectively.

Let (U, cp) be a complex chart on a complex manifold M with the coordinate functions z. Introduce the real coordinate functions x1, yi by (2.16), and for any p E U define the following

Basic notions of differential geometry

86

complex tangent vectors:

(a/azi)p = 2 ((alaxi)p - V -1(a/ayi)p) (a/azi)p = 2 ((alaxi)p +(a/ayi)p) . The vectors (a/azi)p and (a/a. ')p form bases for the spaces Tp1'0)(M) and Tp°'1)(M).

Let cp be a smooth mapping from a complex manifold M to a complex manifold N. For any p E M we have the linear mapping co*p : TRA(M) -> TR,p(p)(M). The mapping cP is (anti)holomorphic

if and only if co*p 0

fPM

(p)

'lWN

0'P*p

(cP*p 0 jPM = -', p)

O

(P*p)

for all p E M. Note also that each mapping cP*p : TRA(M) -* TR,p(p)(M) can be uniquely extended to the linear mapping from Tp (M) to T,,(p), which will also be denoted by cP*p. Return now to the case of real manifolds. Using the definition of the action of a tangent vector on a smooth complex function, we can define the action of a vector field on such a function. Moreover, the notion of a complex tangent vector leads to a natural definition of a complex vector field on a manifold. A complex vector field is

called smooth if its action on any smooth function is a smooth function. The set of smooth complex vector fields on a manifold M is denoted by Xc(M). To distinguish complex vector fields and

real ones we use the term real vector fields. The notion of the commutator of vector fields can easily be extended to the case of complex vector fields. The same can be done for the notion of co-related vector fields and for the notion of a cP-projectible vector field.

Proceed now to the case of complex manifolds. It is clear that we can identify vector fields on a complex manifold M with vector fields on the manifold MR, and we say that a vector field on M is smooth if the corresponding vector field on MR is smooth. Further, a complex vector field X on M is said to be of type (1, 0) if Xp E TOM (M) for any p E M, and it is said to be of type (0, 1) if Xp E T(°'1)(M) for any p E M. The set of smooth vector fields of type

(1, 0) (of type (0, 1)) on M is denoted by .E(l,o)(M) (X(o,l)(M)) For any vector field X we can write a unique decomposition X = X(1'0) + X(°"1),

(2.19)

2.5 Complex manifolds

87

where X(1,0) and X(°'' are vector fields of types (1, 0) and (0, 1) respectively. A complex vector field X of type (1, 0) (of type (0, 1)) is called

holomorphic (antiholomorphic) if X (f) is a holomorphic (antiholomorphic) function for any locally defined holomorphic (antiholomorphic) function f. Note that the complex conjugate of a holomorphic vector field is an antiholomorphic vector field. The commutator of holomorphic (antiholomorphic) vector fields is a holomorphic (antiholomorphic) vector field. Thus, the holomorphic (antiholomorphic) vector fields form a complex Lie algebra which is a subalgebra of the Lie algebra of complex vector fields. Let (U, z', ... , z"`) be a chart on a complex manifold M. The vectors (a/az2)P and (a/a22)P, p E U, specify a set of complex

vector fields on M, denoted by a/az2 and a/az2. Let X be a vector field X on M. It can easily be shown that

XIU=X2

a +XT a azi az2

where

X' = XIu(z&), XZ = Xlu(zi). Here and below we mark the indices corresponding to z2 by a bar. For a vector field of type (1, 0) (of type (0, 1)) we obtain X2 = 0

(X' = 0). A vector field X of type (1, 0) (of type (0, 1)) on U is holomorphic (antiholomorphic) if and only if the functions X2 (X') are holomorphic (antiholomorphic).

2.5.3 Almost complex structures and their automorphisms Let M be a smooth manifold of even dimension m = 2n. Suppose that each tangent space TP(M), p E M, is provided with a complex structure Jp . For any vector field X on M define the vector field JM(X) by

(JM(X))P = JP (XP).

For any smooth real vector field X let the vector field JM(X) also be smooth. In this case we call the a(M)-linear mapping JM :X(M) -+ X (M) an almost complex structure on M. A manifold equipped with an almost complex structure is called an almost complex manifold. By definition, the mapping JM satisfies

88

Basic notions of differential geometry

the condition ( jM)a = - ldx(nf) .

Now let M be a complex manifold. The linear mappings Jm , p E M, specify an almost complex structure on the real manifold P MR. Thus, for any complex manifold M there is the natural almost complex structure on MR which is called the canonical almost complex structure on MR. Let M be an almost complex manifold with an almost complex structure Jm. Define a mapping S : 1(M) x 1(M) -* 1(M) by S(X,Y) = 2{[JM(X),JM(Y)]

- [X,Y]

- JM([X, JM (Y)]) - JM ([JM (X) XI) I I

.

The almost complex structure J' is called integrable if S(X, Y) _ 0 for all X, Y E 1(M). For any complex manifold M, the canonical almost complex structure on the manifold MR is integrable. From the other hand, if the almost complex structure Jb1 on an almost complex manifold M is integrable, then there exists a unique complex manifold N, such that M = NR and J'M coincides with the canonical almost complex structure on M. In other words, if we have an almost complex manifold M and the almost complex structure JM is integrable, then we can always introduce on M the structure of a complex manifold, such that JM will coincide with the corresponding canonical almost complex structure. The tangent vector to a smooth curve is always a real vector, so we can find flows for real vector fields only. Considering the case of complex manifolds, it is interesting to specify those real vector fields whose flows are holomorphic (antiholomorphic) mappings. Suppose that for some vector field X E 1(M) and for any t E II8 the mappings 4)t are holomorphic. Then for any p E DX and any Y E 1(M) we have "Pt n(Jp (Yp)) = p(Yp))' p = 41t (p) This relation, taking account of (2.1), implies that (2.20) [X, JM(Y)] = Thus, if a real vector field X has the holomorphic flow, then for any real vector field Y relation (2.20) is valid. It can be shown that this condition is also sufficient, i.e., if relation (2.20) is valid for any real vector field Y, then the vector field X has a holomorphic JM([X,Y]).

flow.

2.5 Complex manifolds

89

A smooth vector field X on an almost complex manifold M, satisfying relation (2.20), is called an infinitesimal automorphism

of the complex structure JM. Suppose that the almost complex structure JM is integrable, then for any infinitesimal automorphism X of JM and any Y E X(M) we have

[JM(X), JM(Y)] = JM({JM(X),Y]) Hence, the vector field JM (X) is also an infinitesimal automorphism of JM. Moreover, the commutator of any two infinitesimal automorphisms of JM is an infinitesimal automorphism of JM. Thus, infinitesimal automorphisms of the complex structure form a Lie algebra. According to (2.20), the operator JM is a Lie complex structure for this Lie algebra. Therefore, for any almost complex manifold M whose complex structure JM is integrable, the Lie algebra of infinitesimal automorphisms of JM can be considered as a complex Lie algebra. For the case of a complex manifold 111, this Lie algebra is isomorphic to the Lie algebra of holomorphic vector fields on M. The corresponding isomorphism is realised in the following way. Let X be an infinitesimal automorphism of JM; the corresponding holomorphic vector field is X(10). On the other hand, if X is a holomorphic vector field, then the real vector field X + X is the corresponding infinitesimal automorphism of JM.

2.5.4 Complex covectors and covector fields Let us begin with the case of a real manifold. In this case we can naturally define the action of a cotangent vector field on complex vector fields. Moreover, we can consider the complexification T;c(M) of the cotangent space Tp (M) and define complex covectors, complex covector fields, and the differential of a complex function. A complex covector field is called smooth if its action on any smooth complex vector field is a smooth function. The set of smooth complex covector fields on a manifold M is denoted *c(M). This set is a module over the algebra c(M). In the case of a complex manifold M, we denote the set of real .

cotangent vectors to M at a point p by TTP(M). This space is the dual of TR (M), and the operator J" on TRP(M) induces the dual operator (JM)* on TTP(M). The operator (JM)* satisfies the relation (JM)p2 = -1; hence, it is a complex structure on TTP(M).

Basic notions of differential geometry

90

The cotangent space TT (M) to the complex manifold M at the point p is defined as TT (M)

T4(M).

The linear space Tpc(M), defined by Tpc(M) = (T;,(M))c, has the direct sum decomposition Tpc(M) = T(l,o)P(M) ® T(o,i)n(M),

where the spaces T(l,o)r(M) and T(o,l)p(M) can be characterised as the eigenspaces of the operator (Jm)* corresponding to the and -vf--l respectively. Let (U, z1, ... , zm) be eigenvalues + a chart on M and let p E U. The set {(dz')r, (dz')r} is a basis of T1c(M). A complex covector field w on M is said to be of type (1, 0) if wp E T(l,o)p(M) for any p E M, and it is said to be of type (0, 1) if wp E T(o,l)r(M) for any p E M. The set of smooth covector fields of type (1,0) (of type (0,1)) on M is denoted by X(l,o)(M) (X(o,l)(M)). For any covector field w we have a unique decomposition w = ,(1,O)

+w(°'1)

where w(1>0) and w(0,1) are covector fields of types (1, 0) and (0, 1), respectively. Here w(1

°)(X) = w

(X(1,0) )

,

w(o 1)(X) __ w (X(0,1) l

)

for any vector field X. J A complex covector field w of type (1, 0) (of type (0, 1)) is called holomorphic (antiholomorphic) if w(X) is a holomorphic (antiholomorphic) function for any holomorphic (antiholomorphic) vector field X. Let (U, z1, ... , zm) be a complex chart on a complex manifold M. The differentials dz' (dz') are holomorphic (antiholomorphic) covector fields on U. For any complex covector field w on M we have

wIU = wi dz' + wldz'.

Moreover, the covector fields dz' and dz', i = 1, ... , m, form the basis of 'c(U)-module X*c(U).

2.5 Complex manifolds

91

Using the relations dzi (a/az') = 6 dzi (a/az3) = 0,

dzi (a/az') = 0; dzi (a/azj) = 6!,

wi =

w! = wlu(a/azi).

,

we see that wIu(a/azi)

The covector fields dzi form a basis for 3C(1,0) (M), while the covec-

tor fields dzi form a basis for L(0,1)(M). A complex covector field

w on M is holomorphic (antiholomorphic) if and only if wi = 0 (wi = 0) and wi (wl) are holomorphic (antiholomorphic) functions for any complex chart on M.

2.5.5 Complex differential forms Starting with the complexified tangent spaces TP (M), we obtain complex tensor fields on M. Actually, a complex tensor field of type (i) can be considered as an ac(M)-multilinear mapping from

X*c(M) x ... x .C*c(M) x XC(M) x ... x Xc(M) k

l

to ''(M). The set of smooth complex tensor fields of type (k) on

M is denoted by .ic(M). We are especially interested in complex differential forms. De-

note by Q '(M) the subspace of 3C°c(M) formed by all smooth totally skew-symmetric complex tensors on the complex manifold M. Let w be a complex 2-form. Using decomposition (2.19), we obtain the relation w = w(2'0) + w(1'1) + w(0'2)

where (2,0)(X,Y) = w

(X(1,0), Y(1,0))

(X, Y) = w (X(1,0), Y(0'1)) w(°'2) (X, Y)

+ W X(0,1), Y(1'0))

(X(0'1), Y(0'1))

(2.21)

,

.

,

(2.22)

(2.23)

It can easily be shown that w(2'0), w(1'1) and w(°'2) are 2-forms.

Basic notions of differential geometry

92

For an arbitrary complex k-form w we have (2.24)

w

(p'q)w =

p+q=k

where the k-forms w(p'q) are defined by the relations similar to (2.21)-(2.23). Here p and q are the numbers of the arguments of w having types (1, 0) and (0, 1) respectively. The k-form w is said to be of type (p, q) if in decomposition (2.24) only the component w(p'q) is different from zero. The set of all k-forms of type (p, q) forms a linear subspace of 1 (M) which is denoted SZ(p,q) (M) and there is the following direct sum decomposition

Qk(M) = ® Q(p,q)(M)p+q=k

Applying the operator of the external derivative d to a form of type (p, q), we obtain a sum of two differential forms, one form of type (p + 1, q) and the other one of type (p, q + 1). Define the operators d(1'0) : SZ(p,q) (M) _+ Q(p+i,q) (M) and d(0'1) : St(p,q) (M) _+ Q(p,q+1) (M) by

d(1'0)w = (dw)(P',q),

d(°'1)w = (dw)(p,q+1).

From this definition it follows immediately that d = d(1'0) + d(°'1).

Furthermore, it can be shown that d(1,°) o d(1,°)

=

0,

d(°,1) o

d(°,1)

= 0,

d(1'0) o d(°'1) + d(°'1) o d(1'0) = 0.

Let (U, z1, ... , zm) be a chart on M. For any k-form w we can

write WIU =

E p+q=k p'

wil p .,.. q) _l

1 q.1

zPj ... s

dzi'A...Adzi'Adz"n.. Adz'Q, (2.25)

and this expansion is unique if we suppose that the functions are skew-symmetric with respect to any transposition either of the indices i1i ... , ip, or of the indices j1i ... , jq. In such a case we have wip,giP31...jy

= wI U(a/aztl, ... ,

a/aztP, a/aP l,

... , a/a2-9).

2.5 Complex manifolds

93

For the action of the operators d('°0) and d(°"') on w we obtain the following expressions: (1 p)

(d

1

_ w)jU

aw(p,q) i,...3g

p+q=k p! q! (d(°')w)I

1

U

p+q=k p. q.

az1

aw(p,q)

dz

i

Adz" n

a (2.26) A dz3,

A dzi, A

A dzj9. (2.27)

dz,

az

A complex differential form w of type (p, 0) ((0, q)) is called holomorphic (antiholomorphic) if d(°'')w = 0 (d(''0)w = 0). Let w be a complex differential form of type (p, 0). In this case relation (2.25) takes the form wI U =

A ... A dztP.

P!

Now using (2.27), we conclude that the form w is holomorphic if and only if the functions wi,...iP are holomorphic for any choice of the chart (U, z', ... , z'). For a differential form w of type (0, q) we obtain WJU =

A... Ad2'9,

q

and relation (2.26) implies that the form w is antiholomorphic if and only if the functions w;7...,q are antiholomorphic for any choice

of the chart (U, z', ... , zm).

Exercises 2.9 Prove that the operators a/azi and a/azi are C-linear and satisfy the Leibnitz rule, i.e.,

a(flf2) azi

- afl f2 + f af2 azi azi

8(f1f2) = afi azi azi f2

afl + f azi

for any complex functions f, and f2. 2.10 Prove that azi i azi azj = b3' azj azi azi

azi azi

94

Basic notions of differential geometry

2.11 Show that a complex function f = u +v is holomorphic if and only if

au

av =

axi - ayi

an + av aye

axi

2.12 For any complex function f = g + Vr--1h define the function

f by f - g -h. Show that of azi

of

azi

2.6 Submanifolds 2.6.1 Definition of a submanifold A mapping cp E 5(M, N) is called immersive at a point p E M if the mapping W.p : T,(M) -+ T,(p)(M) is injective. A smooth mapping cp : M -4 N is called an immersion if it is immersive at any point of M. Let M and N be two manifolds such that M C N. The manifold M is called a submanifold of the manifold N if the inclusion mapping t : M - N is an immersion. It is clear that an open submanifold of a manifold is a submanifold. Since for any point p of a submanifold M of a manifold N, the mapping t,1, is injective, we will identify the tangent space Tp(M) with the subspace t.p(Tp(M)) of the tangent space Tqp)(N) = Tp(N).

Now let M and N be complex manifolds. The manifold M is said to be a (complex) submanifold of the manifold N if the inclusion mapping t : M -+ N is a holomorphic immersion. From this definition we see, in particular, that for any point p of a submanifold M of a complex manifold N, JN(Tp(M)) = TT(M). There is an inverse statement. Let N be a complex manifold and let k be a submanifold of NR such that for any p E M the relation JN(TT(M)) = TT(M)

is valid. In this case there exists a complex submanifold M of N coinciding with k as a set. In other words, the set of all complex submanifolds of a complex manifold N coincides with the set of all submanifolds M of NN such that for any p E M the tangent space Tp(M) is stable under the action of the operator JP .

2.6 Snbmanifolds

95

Let M be a manifold and let cp be an injective immersion from M to a manifold N. Denote by 5 the mapping cp considered as a mapping from M to co(M). The mapping cp is bijective. Introduce a

topology on cp(M) = (p(M) supposing that the subset U C p(M) is open if the set P-1(U) is open. Further, endow cp(M) with a differentiable structure with respect to which the mapping cp is a diffeomorphism. It can be shown that such a differentiable structure exists and it is unique. Denote the resulting manifold by M. It is clear that cp = t o (p, where t : V(M) -+ N is the inclusion mapping. Since cp is a diffeomorphism, the mapping t is an immersion, and M is a submanifold of N. Thus, any injective immersion

from a manifold M to a manifold N defines a submanifold of N. EXAMPLE 2.15 Define a mapping cp from the manifold IR to the standard two-dimensional torus T2 by W(a) = ((cos aa, sin aa), (cos,la, sin,6a) ),

where a and 0 are some nonzero real numbers. The mapping cp definitely is an immersion. Suppose that for some a1, a2 E JR such that a1 a2 we have W(a1) = cp(a2). It is clear that it is possible

only if a(a2 - al) = 2kir and ,3(a2 - al) = 21ir for some integers k and 1. In this case we have the equality a/,6 = k/l which implies that a/,3 is a rational number. Therefore, the mapping cp is injective if and only if a/,3 is an irrational number. In such a case cp(R) is a submanifold of T2 called an irrational winding of the two-dimensional torus.

Consider a submanifold M of a manifold N. The inclusion map-

ping t : M -4 N is a smooth mapping; hence, it is continuous. Therefore, for any subset U open in N, the subset m f1 U = t-1(U) is open in M, but, in general, not any open subset V C M can be represented as M n U for some U open in N. Thus, the topology of M is, in general, stronger than the induced topology. For example, an irrational winding of the two-dimensional torus has topology

stronger than the induced topology. An abstract subset M of a manifold N may have different differentiable structures allowing one to consider it as a submanifold of N. Actually, a topology on M fixes its differentiable structure. Namely, if two submanifolds coincide as topological spaces, then they coincide as manifolds. In

particular, if the topology of a submanifold M of a manifold N

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Basic notions of differential geometry

coincides with the induced topology, then there is no other differentiable structures on the set M, endowed with the induced topology, which turn it into a submanifold of N. A submanifold is called an embedded submanifold if its topology coincides with the induced topology. An injective immersion cp : M -+ N is called an embedding if it is a homeomorphism from M to cp(M), where W(M) is considered as a topological space with respect to the induced topology. If cp : M -+ N is an embedding,

then the corresponding submanifold M of the manifold N is an embedded submanifold. A mapping cp E '(M, N) is called submersive at a point p E

M if the mapping V,,p : Tp(M) -+ TT(P)(N) is surjective. If a mapping cp E a(M, N) is submersive at any point of M, it is called a submersion. If a smooth mapping cp : M -4 N is submersive at

a point p E M, then the point p is called a regular point of the mapping cp. A point q E M is called a regular value of a smooth mapping cp

: M -* N if any point p E M such that f (p) = q

is a regular point of W. If a point q E N is a regular value of a smooth mapping cp : M -+ N, then W -'(p) can be endowed with the structure of an embedded submanifold of M. EXAMPLE 2.16 Let M and N be two manifolds. Consider the manifold M x N and define the projections prM : M x N -+ M

andprN:MxN -* Nby

prM (p, q) = p,

PrN (p, q) = q.

Any point p E M is a regular value of the mapping prM; hence, the

set prM (p) = {p} x N has the natural structure of an embedded submanifold of M x N, which is certainly diffeomorphic to the manifold M. Similarly, for any q E N the set prN1(q) = M x {q} is an embedded submanifold of M x N difleomorphic to M. Let us also show that for any point (p, q) E M x N there is the following isomorphism T(P,q) (M x N) _ TP(M) ® Tq(N).

(2.28)

To this end, define the inclusions c9 : M M x N and tP : N -4

MxNas

LN

t9 (p) = (p, q), (q) = (p,q). These mapping are immersions; so one has LP*q(Tq(N)) ^ Tq(N). tq P(TP(M)) TP(M),

(2.29)

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97

Further, the sum tq r(TP(M))+t q(Tp(N)) is direct. Indeed, suppose that for some nonzero u E T,(M) and v E Tq (n) we have (2.30)

11 qp(u) = tp q(v).

For any f E a(M) the function prM f belongs to '(M x N), and (2.30) implies

u(tq *(prM f)) = v(tP *(pr* f

On the other hand, prM of

= idM and prM otN is a constant

mapping. Therefore,

u(tq *(prvr f)) = u(f), v(t *(pry f)) = 0, that contradicts to our assumption. Since the dimension of M x N coincides with the sum of the dimensions of M and N, we obtain

T(r,9)(M x N) = q r(T'r(M)) ®t q(Tq(N)) Now, taking (2.29) into account, we obtain (2.28).

Let N be an n-dimensional manifold and let if 1, ... , f k}, k < n, be a set of smooth functions on N. Consider a subset M of N defined as

M - {q E N I f'(q) = 0,...,fk(q) = 0},

(2.31)

and suppose that the differentials dff..... dff are linearly independent at any point p E M. Define the mapping cp : N R' by

cp(q) = (f 1(q), ... , f k(q)),

then M = cp-1(0,... , 0). According to our assumption, for any p E M the mapping V* is injective, therefore the mapping cp*P is surjective. Hence (0, ... , 0) is a regular value of cc, and M can be

provided with the structure of an embedded submanifold of N. In such a case we say that M is an embedded submanifold of N, defined by the equations

f1=0, ...

,

0-

The dimension of the manifold M is equal to n - k. EXAMPLE 2.17 The standard sphere Sm can be considered as a submanifold of R1+1, defined by the equation m+1

E(xi)t i.=1

- 1 = 0,

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98

where xi, i = 1, ... , m + 1, are the standard coordinate functions

on R-+'. Now let N be an n-dimensional complex manifold and let

if ', ... , f k}, k < n, be a set of holomorphic functions on N. Define the set M by relation (2.31) and suppose that the complex differentials dfP , ... , df are linearly independent at any point p E M.

In this case M can be endowed with the structure of a complex embedded submanifold of N of dimension n - k.

2.6. 2 Distributions and the Frobenius theorem Let M be an m-dimensional manifold and let n be an integer such that 1 < n < m. An assignment D of an n-dimensional subspace

Dp C Tp(M) to each point p E M is called an n-dimensional distribution on M. An n-dimensional distribution D on M is said to be smooth if for any point p E M there is an open subset U containing p, and the set of smooth vector fields X 1 ,.. . , Xn on U such that for any q E U the set {Xiq}i=1,,,.,,, is a basis for Dq. We say that a vector field X belongs to a distribution D if Xr E Dp for any p E M. A smooth mapping co from a manifold M to a manifold N is said to be tangent to a distribution D on N if co,,p(v) E V ,(r) for any pE M and v ETP(M). A smooth distribution D is called involutive if for any smooth vector fields X and Y belonging to D the commutator [X, Y] belongs to D. A submanifold N of a manifold M is called an integral manifold of a distribution D on M if TT(N) = Dp

for any p E N. Suppose that D is a smooth distribution on a manifold M such that for any point p E M there is an integral manifold of D containing the point p. In this case D is an involutive distribution. An integral manifold of a distribution D is said to be maximal if it is not a proper subset of any other connected integral manifold of D. The Frobenius theorem states that for any n-dimensional smooth involutive distribution D on a manifold M and for any point p E M there exists a unique maximal n-dimensional integral manifold of D containing the point p.

2.7 Lie groups

99

In the case of a complex manifold M, a smooth distribution D on M is said to be complex if JM(Dp) = Dp for any p E M. Let a real vector space V be endowed with a complex structure

J and let W be a linear subspace of V such that J(W) = W. In this case the restriction of J to W is a complex structure on W, and the complex vector space W can naturally be considered as a linear subspace of the complex vector space V. Recall that in the case under consideration the vector space Ve has a direct sum deV(°,1) and composition Vc = ® V(1'0), where the subspaces V(1,0) are naturally isomorphic to the vector spaces V and V respectively. The corresponding isomorphism allows one to identify the subspace W of V with a subspace of V(1,0) The above discussion shows that if D is a complex distribution on a complex manifold M, then for any p E M we can identify the subspace Dr with a subspace of Tp',O) (M) which will also be V(°,1)

7Z7

denoted by V. A complex distribution D on a complex manifold M is called holomorphic if for any point p E M there is an open neighbourhood U of p and a set of smooth holomorphic vector fields X1,. .. , X,,, on U such that for any q E U the set {Xiq}i=1,...,n is a basis for Dq. The maximal integral manifolds of an involutive holomorphic distribution are complex submanifolds.

2.7 Lie groups 2.7.1 Definition of a Lie group A group G is called a Lie group if it is a manifold and the mapping

(a, b) E G x G Hab-1 is smooth. From this definition it follows that for any Lie group G the mappings

(a,b) EGxGHabEG, aEGHa-1 EG are smooth. EXAMPLE 2.18

(i) The set W< - R - {0} is an open submanifold of R; it is a Lie group with respect to the multiplication of real numbers.

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(ii) The vector space Mat(m, Ilk) of all real mxm matrices can naturally be considered as a smooth manifold diffeomorphic to 7 . Here the standard coordinate functions g2j, i, j = 1,... , n, are defined by Rr"2

92i (a) = ata ,

(2.32)

where a = (at3). Denote by GL(m, IR) the set of all real nondegenerate mxm matrices. Considering GL(m, IR) as an open submanifold of the manifold Mat(m, IR), and as a group with respect to

the matrix multiplication, we see that it is a Lie group. This Lie group is called the real general linear group. (iii) Let V be an m-dimensional real vector space. Consider the set GL(V) of all nondegenerate linear operators on V. It is a group with respect to the product of linear operators. Fixing a basis for V, we associate with any nonsingular linear operator on V a real nondegenerate mxm matrix. Hence, one can introduce on GL(V) a smooth differentiable structure which obviously does not depend on the choice of a basis for V. With respect to this differentiable structure, the group GL(V) is a Lie group. Let G and H be Lie groups. The direct product G x H considered as the direct product of the groups G and H and the direct product of the manifolds G and H is a Lie group which is called the direct product of the Lie groups G and H. A group G is called a complex Lie group if it is a complex manifold, and the mapping

(a,b)EGxGHab-'EG is holomorphic. Since any complex manifold can be considered as a real smooth manifold, it is clear that any complex Lie group G is simultaneously a real Lie group which will be denoted by GR. EXAMPLE 2.19

(i) The set Cx - C - {0} is an open submanifold of C; it is a Lie group with respect to the multiplication operation. (ii) The set Mat(m, C) of all complex m x m matrices can be considered as a complex manifold diffeomorphic to C"`2 with the standard coordinate functions 9 'J (c) = cj,

2.7 Lie groups

101

where c = (ci;). The complex general linear group GL(m, (C) is, by

definition, the set of all complex nondegenerate mxm matrices considered as an open submanifold of the manifold of all complex mxm matrices, and as a group with respect to the matrix multiplication. It is clear that GL(m, C) is a complex Lie group. (iii) The set of all nondegenerate linear operators on a complex vector space V is a complex Lie group isomorphic to the complex general linear group GL(m, Q, where m = dim V.

2.7.2 Lie algebra of a Lie group With any element a of a Lie group G we can associate the following differentiable mappings of G onto itself:

left translation La : b E G H ab e G; right translation Ra : b E G H ba E G. For any a, b E G we have L. o Lb = Las, Ra o Rb = Rba, La o Rb = Rb o La.

(2.33) (2.34)

From these equalities if follows, in particular, that (La)-1 = La-1, (Ra)-1 = Ra-i. Thus, the mappings La and Ra are diffeomorphisms. A vector field X on a Lie group G is said to be left invariant if La*X = X for any a E G. The left invariant vector fields form a subspace of the vector space of vector fields on G. Denote, as usual, the identity element of the Lie group G by e. Let v be an arbitrary element of Te (G) and define the vector field -Y,, by

Xva = La* (v).

(2.35)

The vector field X, defined by (2.35) is left invariant. It is not difficult to show that the mapping v H X, from Te (G) to the vector space of left invariant vector fields on G is linear and injective. Let us show that any left invariant vector space on G can be obtained by the above procedure. Let X be a left invariant vector field on

G, then Xa = La*(Xe) and, hence, X = X for v = Xe E Te(G). In other words, if the mapping v H X is surjective, then it is an isomorphism.

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It can be shown that for any v E Te (G), the vector field Xv is smooth; hence, any left invariant vector field on G is smooth. From (2.9) it follows that the commutator of two left invariant vector fields is left invariant. Therefore, the vector space of left invariant vector fields on G is a Lie algebra which is a subalgebra of the Lie algebra of smooth vector fields on G. This Lie algebra is called the Lie algebra of the Lie group G and is denoted by g. The isomorphism of Te (G) and the space of left invariant fields on G becomes an isomorphism of Lie algebras if we introduce in Te (G) the structure of a Lie algebra by [v, u] = [Xv, Xu)e.

(2.36)

Thus, we can identify the Lie algebra g with Te (G). Actually, it is the interpretation of the Lie algebra of a Lie group which is used in the present book. EXAMPLE 2.20 Consider the case of the Lie group GL(m, IR). Us-

ing the standard coordinate functions on GL(m, R), defined by

(2.32), we can write an arbitrary tangent vector v at a point b E GL(m, IR) in the form v = vxi (91 a9xi )b,

where

vxi = v(9xi) For any a E GL(m, IR) we obtain i i k La9i =ak9 i.

(2.37) (2.38)

By definition, for any f E '(G) we have (La*(V))(f) = v(Laf); therefore, using (2.37) and (2.38), we obtain the relation (L0.(v))xi

= aikvkj.

From this relation we obtain the equality (X ,)'j (a) = axkvk j,

which implies (Xx,)x3 = gxkvki.

(2.39)

Recall that GL(m, IR) is an open submanifold of the vector space Mat(m, IR), provided with the standard differentiable struc-

ture. According to example 2.10, we can identify the tangent

.

2.7 Lie groups

103

space Trm (GL(m, IR)) with Mat(m, R). Here, a tangent vector v = vi j (a/agi j), is identified with the matrix (v), which will also be denoted by v. Hence, for any matrix v E Mat(m, IR) there is defined a left invariant vector field on GL(m, I1) which coincides with v at the identity element of GL(m, II8). Using (2.37) and (2.39), we see that Xv(Xu(91j)) = Xv(940 kj) = gi1v1kukj,

and from this relation we obtain

{Xv,Xu]'j(e) = Vkukj - Ukvkj. Therefore, the Lie algebra of the general linear group GL(m, II8) can be identified with the Lie algebra g((m, R). A representation of a Lie group G in a real or complex vector space V is defined as a smooth homomorphism from G to the Lie group GL(V) or GL(V)R respectively. In the former case we call the corresponding representation real, while in the latter case we say that the representation is complex. The adjoint representation Ad : a E G H Ad(a) E GL(g) of the Lie group G is defined by the relation Ad(a)v - (La o Ra-,)*e(v)

EXAMPLE 2.21 Continue the consideration of the Lie group GL(m, IR), that we started in example 2.20. Using the argument similar to those given there for left translations, for any a E GL(m,R) and v E Tb(GL(m, IR)) we have the relation (Ra.(V))ij=vikakj,

where vi j are the coordinates of v with respect to the basis (alagij)b. From this relation we immediately obtain Ad(a)v = ava-1. (2.40) This is the well-known relation for the adjoint representation of matrix Lie groups.

Any left invariant vector field on a Lie group G is complete. Therefore, for any v E g and any t E R there is the diffeomorphism

P t G -+ G. Define the exponential mapping from g to G by expG : v E 9 H exp(v) = PX° (e) E G.

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Basic notions of differential geometry

When it is clear which Lie group is being considered, we write simply exp for expG. The exponential mapping has the properties exp(tv) exp(sv) = exp((t + s)v), (exp(v))-1 = exp(-v) for all t, s E 118 and v E g. Note also that

aexp(v)a-1 = exp(Ad(a)v)

(2.41)

for any a E G and v E g. Note that the exponential mapping allows one to establish the relation between representations of the group G and representations of the Lie algebra g. Indeed, let PG be a representation of the Lie group G in a vector space V. It can be shown that the mapping p9 defined by d

P9(x)v = dtPG(eXP(tx))vLO,

is a representation of the Lie algebra g in V. In particular, we have the following relation between the adjoint representations of a Lie group and the corresponding Lie algebra:

ad(v)u = dt Ad(exp(tv))ut-o

(2.42)

EXAMPLE 2.22 Consider the Lie group GL(m,R). Recall the def-

inition of the exponential function of the matrix argument. The exponential function a E Mat(m, R) H ea E Mat(m, R) is defined by

The series in this definition is absolutely convergent with respect

to any norm in Mat(m,118). It can be shown that for any a E Mat(m,118) the function t E 118 H eta E Mat(m, R) is infinitely differentiable and satisfies the differential equation delta

= etaa.

(2.43)

Let A be an integral curve of the left invariant vector field Xv, where v is an arbitrary element of the Lie algebra gl(m,118). As follows from the definition of an integral curve and from (2.39),

2.7 Lie groups

105

the matrix valued function t E Ilk H A(t) E GL(m, R) satisfies the equation dA(t)

= )(t)v.

In accordance with (2.43), the general solution of this equation is A(t) = where a is an arbitrary element of GL(m, IR). Therefore, we have -x (a) = aety, aety,

hence,

exp(v) = ev. (2.44) In other words, in the case under consideration the exponential mapping coincides with the exponential function. Note that relation (2.41) takes the form aeva -i = eaves-1

Proceed now to the case of complex Lie groups. It is natural to require that the Lie algebra of a complex Lie group G be a complex Lie algebra g. Moreover, it is desirable that the Lie algebra of the

real Lie group GR be the realification of g. So, let us denote the Lie algebra of GR by OR and try to find the corresponding complex

Lie algebra g. Since the group operation in a complex Lie group G is holomorphic, we have for all a E G the following relations: La* o jG = jG o La*, (2.45) Ra* 0 JG = jG o Ra*. (2.46) The restriction J - Je of the complex structure JG to TR,(G) = Te(GR) generates a complex structure on the Lie algebra gR. It follows from (2.45) and (2.46) that Ad(a) o J = J o Ad(a) for any a E G. From this relation and from (2.42) one sees that ad(v) o J = J o ad(v) for any v E OR. This equality can be written in the form [v, Jul = J([v, u]). Hence, J is a Lie complex structure on OR, and OR has the struc-

ture of a complex Lie algebra g - gj. It is clear that the Lie algebra OR is the realification of the Lie algebra g. Therefore, g is

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Basic notions of differential geometry

the required complex Lie algebra. It is the Lie algebra which we call the (complex) Lie algebra of a (complex) Lie group G. There exists another, more convenient, realisation of the Lie algebra of a complex Lie group. Consider the complex Lie algebra (OR)' = (TR,(G))C = T, (G). Write the direct sum decomposition (2.17) in the form (OR)' = 0(1'0) ® g(1'0)

where 9(110) = T(1'0)(G) and g(°,1) =

T(°'1)(G). In fact, g(1,0) and

g(o,l>

are ideals of (OR)c, such that the former is isomorphic to g, while the latter is isomorphic to g. Here g is the Lie algebra which is obtained from OR with the help of the Lie complex structure

-J.

Thus, we can identify the Lie algebra of a complex Lie group G with the space T,0,0) (G). Here any element of T(1"0) (G) generates a left invariant complex vector field of type (1,0) on G, given by

the relation of form (2.35); and the Lie algebra operation in g is related to the commutator of the corresponding left invariant vector fields by (2.36). Note that any left invariant vector field on G of type (1,0) is a holomorphic vector field.

EXAMPLE 2.23 Repeating the arguments used in example 2.20, we see that the Lie algebra of the complex Lie group GL(m, C) can be identified with the Lie algebra g[(m, Q. Here the element of (GL (m, C)) corresponding to the element (vii) of 91(m, C) is vzj (a/C7gi j )Im . The adjoint representation and the exponential mapping are described by relations (2.40) and (2.44).

Let G be a real Lie group. Suppose that the Lie algebra g has a Lie complex structure J, so that we can define the complex Lie algebra g. Define the almost complex structure on G by Ja (v) =

v E Ta(G).

This almost complex structure is integrable. Hence, it is possible to provide G with the structure of a complex manifold G. It can be shown that the group operations in G are holomorphic and, therefore, that G has the structure of a complex Lie group.

2.7 Lie groups

107

2.7.3 Lie subgroups First consider the case of real Lie groups. A subgroup H of a Lie group G is called a Lie subgroup of G if it is a Lie group and a submanifold of G. In other words, a subgroup H of a Lie group G is called a Lie subgroup of G if it is endowed with a smooth differentiable structure such that the group operation in H is smooth and the inclusion mapping of H into G is an immersion.

Let H be a Lie subgroup of a Lie group G and let t be the inclusion mapping of H into G. It can be shown that t,(I) is a subalgebra of the Lie algebra g which is isomorphic to t , and it can be proved that for any subalgebra 6 of the Lie algebra g there exists a unique connected Lie subgroup H of G such that c,(Cl) _ 0, with t being the inclusion mapping of H into G. In other words, there is a bijective correspondence between the Lie subgroups of a Lie group and the subalgebras of its Lie algebra. It is customary to identify the Lie algebra 0 of the Lie subgroup H of a Lie group G with the corresponding subalgebra t,(13) of the Lie algebra g. In particular, we say that 0 is a subalgebra of g. Not every subgroup of a Lie group can be considered as a Lie subgroup. There are a few useful criteria for a subgroup of a Lie group to be a Lie subgroup. Let us discuss them briefly. Recall that a subset of a manifold may have different differentiable structures turning it into a submanifold. This is not the case for subgroups of a Lie group. If a subgroup H of a Lie group G has the structure of a submanifold of G, then this structure is unique, and H, with respect to this structure, is a Lie subgroup of G. From this fact it follows that a subgroup of a Lie group is a Lie subgroup if and only if it can be provided with the structure of a submanifold; and the statement that a subgroup of a Lie group is a Lie subgroup is not ambiguous. In other words, we do not need to explain which differentiable structure we have in mind. The corresponding differentiable structure, if it exists, is unique. Any open subgroup H of a Lie group G can be considered as an open submanifold of G; therefore, it is a Lie subgroup of G. Moreover, any closed subgroup H of a Lie group G is a Lie subgroup of G. Note that any open subgroup of a Lie group is also its closed subgroup. It can also be shown that a closed Lie subgroup H of a Lie group

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G is an embedded submanifold of G. Therefore, the topology of a Lie subgroup H of a Lie group G coincides with the induced topology if and only if H is a closed subgroup. Proceed now to the case of complex Lie groups. It is natural to call a subgroup H of a complex Lie group G a Lie subgroup of G if H is a complex Lie group and a submanifold of G. Let G be a complex Lie group and let H be a Lie subgroup of a real Lie group GR such that J(Ij) = Cl. The restriction of the natural Lie complex structure J on OR to 1) gives a Lie complex structure on Cl. Hence, the Lie group H has the natural structure of a complex Lie group H. It can be shown that the inclusion mapping of k into G is holomorphic. Therefore, the Lie group H is a Lie subgroup of G. The complex Lie algebra h can be considered as a subalgebra of the complex Lie algebra g. On the other hand, any subalgebra of g, considered as a subalgebra of OR, is invariant with respect to the action of the operator J. Thus, we have a bijective correspondence between the Lie subgroups of a complex Lie group and the subalgebras of its Lie algebra. Let H be a Lie subgroup of a real or complex Lie group G. An

element v E g belongs to h if and only if expG(ty) E H for all t E R.

A Lie subgroup H of a Lie group G is an invariant subgroup of G if and only if the Lie algebra I) is an ideal of the Lie algebra g. EXAMPLE 2.24 The set of elements a of the Lie group GL(m, R),

satisfying the requirement det a = 1, is a closed subgroup of GL(m, R). Hence, this subgroup has the natural structure of a Lie group denoted by SL(m, l1) and is called the real special linear group. The complex special linear group SL(m, C) is defined similarly. Using the well-known relation det en = etr a E Mat (m, IK), we conclude that the Lie algebra of SL(m, K) is the special linear algebra sI(m, K).

EXAMPLE 2.25 Let B be a bilinear form on an m-dimensional vector space V over a field K. Denote by GLB(V) the subset of GL(V) formed by the nondegenerate linear operators A satisfying the relation B(Av, Au) = B(v, u)

2.7 Lie groups

109

for all v, u E V. It can easily be shown that GLB (V) is a closed subgroup of GL(V). Hence, it has the natural structure of a Lie algebra, and the Lie algebra of GLB(V) coincides with the Lie algebra g B (V) defined in example 1.8.

Let {ei } be a basis of V. It is clear that A E GLB (V) if and only if the matrix a of A with respect to lei} satisfies the relation atba = b, (2.47) where b is the matrix of the bilinear form B with respect to {ei}. This relation gives the isomorphism of GLB(V) with some subgroup of the group GL(m,1K). Consideration of nondegenerate bilinear forms having definite symmetry leads to the following Lie subgroups of GL(m,1K). Note that for a nondegenerate bilinear form B it follows from (2.47) that (det a)2 = 1. Thus, either det a = +1, or det a = -1. The pseudo-orthogonal group O(k, 1) is defined as the subgroup

formed by the elements a E GL(m,118), m = k + 1, satisfying relation (2.47), with b = Ik,t, where Ik,t is defined by (1.5). The Lie groups O(k, 1) and O(l, k) are obviously isomorphic. For the Lie group O(m, 0) we use the notation O(m). The Lie group 0(m)

is called the real orthogonal group. The condition det a = +1 singles out the Lie subgroup of 0(k,1) called the special pseudoorthogonal group and denoted SO(k,l). The Lie group SO(m,0) is called the real special orthogonal group and is denoted by SO(m). The Lie algebra of O(k, 1) and SO(k, 1) is the pseudo-orthogonal Lie algebra o (k, 1).

The complex orthogonal group O(m, C) is, by definition, a Lie subgroup of GL(m, G) specified by condition (2.47) with b = Im. The complex special orthogonal group is a Lie subgroup of O(m, C)

singled out by the condition det a = +1. This group is denoted by SO(m, C). The Lie algebra of the Lie groups O(m, C) and SO(m, C) is the complex orthogonal algebra o(m, C). The real symplectic group Sp(n, R) is defined as the Lie subgroup of the Lie group GL(2n,118) consisting of the elements which satisfy relation (2.47) with b = Jn. Here the matrix Jn is the matrix defined by (1.6). The Lie algebra of Sp(n, R) is the real symplectic algebra sp(n, IR). The complex symplectic group Sp(n, C) having sp (n, C) as its Lie algebra. is defined similarly. Note that for any a E Sp(n, IK) we have det a = +1.

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Basic notions of differential geometry

EXAMPLE 2.26 Let V be an m-dimensional complex vector space endowed with a sesquilinear form B. Consider the subset GLB (V)

of the Lie group GL(V) formed by the elements A satisfying the relation B(Av, Au) = (v, u)

(2.48)

for all v, u E V. This subset, considered as a subset of the real Lie group GL(V)R, is a Lie subgroup. Let {e1} be a basis of V and let b be a matrix of B with respect to {ei}. It is obvious that relation (2.48) is equivalent to the matrix relation

atba = b,

(2.49)

where a is the matrix of A with respect to {e1}. This relation defines a subalgebra of GL(m, C)R which is isomorphic to GLB(V).

Consider the case of a nondegenerate hermitian form. In this case relation (2.48) gives a Lie group isomorphic to the unitary group U(m) defined by (2.49) with b = I,,,.. From (2.49) we also obtain I det al2 = 1. The Lie subgroup of U(m) specified by the condition det a = 1 is called the special unitary group and is denoted by SU(m).

L. 7.4 Maurer-Cartan form of a Lie group Let M be a manifold and let V be a real vector space. A mapping f from M to V is called a function on M taking values in V, or a vector valued function on M. A function on M taking values in V is said to be smooth if it is smooth as a mapping from the manifold M to the vector space V, that is provided with the standard differentiable structure. Let {er} be a basis for the vector space V. For any point p E M we can write f (p) = er f r(p), thus defining a set { f r} of functions on M. In this case we write f = er f r. The vector valued function f is smooth if and only if all the functions f r are smooth. For any vector field X on M we define the action of X on a vector valued function f as

X(f) = erX(f') Let M be a smooth manifold; a totally skew-symmetric '(M)linear mapping from X(M) x x 1(M) to the space of functions k

2.7 Lie groups

111

on M taking values in a real vector space V is called a k-form on M taking values in V, or a vector valued k-form on M.

Let w be a k-form on a manifold M taking values in a real vector space V. Writing the relation w(Xi, ... , X, =_ e, wr Xj,... , Xk , where {er} is a basis for V, we define a set {wr} of ordinary k-forms on M. In such a situation we write w = erwr.

The exterior derivative of a k-form w taking values in V is defined as

dw - erdwr, and its behaviour under a smooth mapping cp from M to a manifold N is governed by the relation cp*w = er(p*wr.

Here we have (2.50)

dco*w = cp*dw.

It is easy to show that the above definitions do not depend on the choice of a basis for V. It is interesting to consider the case of the forms taking values not just in a vector space but also in a Lie algebra. In this case we can define the commutator of forms in the following way. Let w and rl be, respectively, a k-form and an 1-form on a manifold M, taking values in a real Lie algebra g. The commutator of w and rl is defined by [w, rJ] = [erwr, esr)s] _ [er,

es]wr A q',

where {er} is a basis for g. It is clear that this definition does not depend on the choice of a basis for g, and

[w,ril = -(-1)kt[rl,w], d[w, 77] = [dw, ri] + (-1)k[w, di71.

Furthermore, for any smooth mapping cp from M to a manifold N we have W* P, rl] = [o*w, W*rll.

(2.51)

When w and i are 1-forms, we obtain

[w777](X,I') = [w(X),rl(I')] - [w(I'),r!(X)]

(2.52)

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112

The Maurer-Cartan form 9 of a Lie group G is a 1-form on G, taking values in g, and defined by the relation

9(v) -

v E Ta(G).

(2.53)

Let us show that the form 9 is left invariant. Indeed, for any v E

Tb(G)and aEGwe have (La9)(v) = 9(La*b(V)) = L(ab)-1=ab(La*b(V)) = 9(v).

Therefore,

Lag = 9 (2.54) for any a E g. The behaviour of 9 under the right translations is described by the formula Rag = Ad(a-1) o 9. (2.55) Further, for any v E g we have v,

(2.56)

where v on the right-hand side is understood as a g-valued function on G taking the value v at any point of G. Note that equality (2.56)

can be taken as an alternative definition of the Maurer-Cartan form.

It can be shown that relations (2.14) and (2.15) are valid for the forms taking values in a vector space. In particular, from (2.15) for the Maurer-Cartan form 9 of a Lie group G we obtain d9(X,,,X.a)

= -0(X[,]) = -[v,u].

On the other hand, using (2.52), we arrive at the equality [9,0](Xv,Xa) = 2[9(X,,),0(Xu)] = 2[v,u]. It is not difficult to show that these equalities imply

d9+

2[0,0] = 0.

(2.57)

Another interesting case is the case of matrix valued differential forms. Let w be a k-form taking values in the vector space Mat(n, R). Such a differential form is called a matrix valued differential form. We can associate with w the set {wrs}r,s-1,...,.a. consisting of n2 ordinary k-forms defined by (2.58) wrs(X1 i ... , Xk) = (w(X1, ... , Xk))rs.

On the other hand, reversing equality (2.58), we associate with a set {wrs}r,s-1,.,.,, of ordinary k-forms a matrix valued k-form.

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113

The exterior derivative of a matrix valued k-form w is defined by the relation (dw)''3 - d(w'.), while the exterior product of two matrix valued differential forms w and q is given by (w A 77)rs - wrt A its.

If we consider Mat (n, R) as a Lie algebra, that is, gl(n, R), we can write [w, 77] = 2w A 77.

(2.59)

EXAMPLE 2.27 Consider the general linear group GL(m, IR). The standard coordinate functions gx, define a matrix valued function on GL(n, I18) which is denoted by g. Following example 2.20, we identify the Lie algebra of GL(m, IR) with the Lie algebra g((m, ]R). Therefore, the Maurer-Cartan form in the case in question can be

considered as a matrix valued 1-form. Using relation (2.39), for any v E g[(m, IR) we obtain d9(Xv) = gv. Define the matrix valued function g-1 on GL(m, IR) by

g-1(a) - a-1 Then, we can write 9-1d9(Xv) = v.

Comparing this equality with the alternative definition of the Maurer-Cartan form (2.56), we conclude that in our case 9 = g-1dg.

Taking the exterior derivative of the obvious equality

g-1g = I., we arrive at the relation dg-1 = -9-1(d9)9-1

Using this relation, we obtain dO = dg-1 A dg = -g-1dg A g-1dg. Hence, we have the equality

d9+0A0=0, which, by virtue of (2.59), coincides with (2.57).

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Now let G be a complex Lie group. The real Lie group GR has the Maurer-Cartan form OR taking values in OR. Note that (2.60) OR(JG(X)) = Je o OR (X)

for any vector field X on GR. The Maurer-Cartan form 0 of the complex Lie group G is defined by 0(X) _ (0R(X))(1'0). It follows from this definition that 0 takes values in the Lie algebra g of G. Using (2.60), we obtain 0(X) = OP.(X(l,O));

hence, 0 is a 1-form of type (1, 0). Moreover, it can be shown that it is a holomorphic 1-form. Actually, we can define 0 as a unique 1-form of type (1, 0) sat-

isfying (2.56), where X, is the left invariant vector field on G corresponding to the element v E g = T(")(G). It can be shown that in the case under consideration we still have relations (2.54), (2.55) and (2.57) while, instead of (2.53), we obtain 0(v) = La- *a(Pa v), (2.61)

where v E Ta (G) and the linear operator PQ projects v to its (1,0)-component; see (2.18).

EXAMPLE 2.28 Repeating the arguments of example 2.27, we conclude that the Maurer-Cartan form of the Lie group GL(m, C) is given by

0 = g-'dg, (2.62) where g is the matrix valued function formed by the standard coordinate functions g=3 on GL(m, C).

Suppose now that H is a Lie subgroup of a real or complex Lie group G. Let 9G be a Maurer-Cartan form of G and let t be the inclusion mapping of H into G. It is not difficult to show that the 1-form t*9a takes values in the Lie algebra tj of H and, moreover, that this form coincides with the Maurer-Cartan form of H.

EXAMPLE 2.29 Let G be a Lie subgroup of the general linear group GL(m, IK). It is clear that the Maurer-Cartan form of G is given by relation (2.62), where g is now the matrix valued function on G formed by the restrictions of the standard coordinate functions on GL(m, ]K) to G.

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115

2.7.5 Lie transformation groups Let G be a Lie group and M a manifold. A smooth mapping 4): M x G -+ M, satisfying the conditions

(RA1) 4)(p,e)=p forallpEM, (RA2) (4) (p, a), b) = 4>(p, ab) for all a, b E G and p E M, is called a right action of the group G on the manifold M. Here the manifold M is often called a right G-manifold. It is customary to write p a for fi(p, a). For the case where G is a complex Lie group and M is a complex manifold, we require 4) to be a holomorphic mapping. Sometimes in such a situation we also say that 4> is a holomorphic action.

EXAMPLE 2.30 Let X be a complete vector field on a manifold M. The flow 4X , induced by X is a right action of R considered as an abelian group with respect to the addition operation.

A right action of a Lie group G on the manifold M is said to be effective if, for any a E G such that a e, there is an element

pEMsuch that

p.

Let M and N be two right G-manifolds. A mapping cp : M -+ N is called equivariant if cp(a p) = a V(p) for all a E G and p E M. Let 4> be a right action of a Lie group G on the manifold M. For each a E G define the smooth mapping 4>a : M - M by 4>a(p) = 4>(p, a).

From the definition of a right action we have Rb o Ra = Ra.b. Re = idM, These equalities imply that Ra-= = (Ra,)-'. Hence, for any a E G the mapping Ra is a diffeomorphism. Let G be a group, M a right G-manifold, and S a subset of M. Introduce the notation

pES, aEG}. The set p G, p E M, is called the orbit of the point p. Two orbits are either coincident or disjoint, and M is the union of the orbits. The set of orbits is denoted by M/G. There is a natural surjective mapping -7r from M to M/G. This mapping sends a point p E M to the corresponding orbit p G. The mapping it is

Basic notions of differential geometry

116

called the canonical projection. We shall always endow M/G with the quotient topology with respect to the canonical projection 7r. Hence, 7r is a continuous mapping. It can be shown that it is also an open mapping. Moreover, the topological space M/G is second countable, but it need not be Hausdorff.

EXAMPLE 2.31 Let V be a vector space over a field K The set IP(V) of all one-dimensional subspaces of V is called the projective space of V. The projective space IP(Km+l) is denoted by 11Pt. Choose a basis {ea}a=l of V. Any point p of P(V) is uniquely characterised by any nonzero vector v belonging to the corresponding one-dimensional subspace. The coordinates vl,... , vn of the vector v are called the homogeneous coordinates of p. These coordinates are defined up to multiplication by a nonzero element of K Due to this fact it is customary to denote the point p by (v1 : v2 :...: v"). Thus, we can define the projective space as the orbit space of the right action of the Lie group K" on V defined as v a H av. The orbit space here is a Hausdorff topological space. It is possible to define a natural differentiable structure on IP(V). Consider, for example, the projective space ]Ipm. For each a = 1,. .. , m + 1, denote by Ua the set of all one-dimensional subspaces intersecting the hyperplane H. - {(al,...,am+1) E ]( +1 I as = 1}. Let p E Ua and let a point a = (al, ... , am+l) belong to the subspace p. Define the mapping cpa : Ua -+ IKm by co, (p) _=

(a'/a',...

+1 a ,aa/a a ,...,a m}1 /aa). ,aa-/a 1

It is clear that this definition does not depend on the choice of the point a in p. It can be shown that {(Ua, cpa)} is an atlas of RPt.

Let G be a Lie group and let H be a Lie subgroup of G. The

mapping (a, b) E G x H H ab E G is a right action of H on G. It appears that the space of orbits G/H is Hausdorff if and only if H is a closed subgroup. Moreover, in this case there is a unique smooth differentiable structure on G/H such that the natural projection it : G -+ G/H is a smooth mapping. We shall always consider G/H as a smooth manifold with respect to this differentiable structure. In the case where G is a complex manifold, the manifold G/H has the unique structure of a complex manifold

with respect to which the natural projection it is holomorphic.

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117

A left action of a Lie group G on a manifold M is specified by a mapping ' : G x M -4 M satisfying the conditions (LA1) '(e,p) = p for all p E M; (LA2) IQ (a, 1Y (b, p)) = T (ab, p) for all a, b E G and p E M.

Here M is called a left G-manifold.. The left action ' defines the set of diffeomorphisms T: p E M H a- p E M. The corresponding space of orbits is denoted here by G\M. Any right action of a Lie group G on a manifold M generates a left action T of G on M defined by T (a, p) =_ -ID (p, a-')

Actually, any definition or a statement which uses a right action of a Lie group on a manifold has its analogue for a left action. Let M be a right (left) G-manifold. The corresponding action of G on M is called transitive if for any p, q E M there is a E G such that p = q - a (p = a - q). If an action is transitive, then there is just one orbit. A manifold endowed with a transitive action of a Lie group G is called a homogeneous space of G. Let H be a closed subgroup of a Lie group G. It can be shown that the mapping T : G x G/H -+ G/H, defined by IQ (a, bH) - abH,

(2.63)

is a left action of G on G/H. This action is obviously transitive, therefore G/H is a homogeneous space of G. In the case where G is a complex manifold, the action 4' defined by (2.63) is holomorphic.

Let ' be a left action of a Lie group G on a manifold M, and let p E M. The set

GP- {aECI a subgroup of G called the isotropy subgroup of IF at p. Denote by TP the mapping from G to M, defined by 'P(a) = a - p.

The mapping TP is smooth and certainly continuous. Hence, the subgroup GP = TP 1(p) is a closed subgroup and G/GP has a natural smooth differentiable structure. Define a surjective mapping

OP:G/GP-*G-pby /P(aGp) Show that this mapping is injective. Indeed, suppose that for some

a, b E G we have 'P(aGP) = O,,(bGP). Here a - p = b - p, and,

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Basic notions of differential geometry

therefore, a-lb p = p. From this fact it follows that a-lb E GP, and aGP = bGP. Thus, the mapping op is a bijection. It can be shown that 4/IP, considered as a mapping from G/Gp to M, is an immersion. This means that the orbit G p has the structure of a submanifold of M which is diffeomorphic to G/Gp. Note that the mapping ?/gyp is equivariant.

If ' is a transitive action, then it can be shown that the map-

, is a diffeomorphism. It is important here that we consider only manifolds that are second countable topological spaces. In other words, any homogeneous space of a Lie group G is diffeoping

morphic to a manifold of the type G/H, where H is a closed subgroup of G. In the case where G is a complex Lie group, the corresponding diffeomorphism is a holomorphic mapping. Actually, we can consider actions of Lie groups on arbitrary sets. For example, a right action of a Lie group G on a set M is defined as a mapping 4P : M x G-* M satisfying conditions (RA1) and (RA2). If IQ is a transitive action, then for any p E M the mapping ,op is bijective and we can identify the set M with G/Gp, thus equipping M with the structure of a manifold. Let M be a right G-manifold. Construct a mapping from the Lie algebra g to the Lie algebra of smooth vector fields on M, assigning to an element v E g the vector field X M acting on a

function f E '(M) as

X p(f) =

(2.64) t=o

It can be shown that for any v, u E g we have [XM' XMJ = X[MuJ.

Thus the mapping v H X v" is a homomorphism of Lie algebras. In the case of a left action of a Lie group G on a manifold M,

we define a mapping from the Lie algebra g to the Lie algebra of smooth vector fields on M, assigning to an element v E g the vector field X m given by

X P (f) = dt f (eXP(ty) . p)

t=o

It can be shown that in this case for any v, u E g we have [XM 'XUM] _ -X[vMU] V

2.7 Lie groups

119

Thus the mapping v H X M is an antihomomorphism of Lie algebras. A right action of a Lie group G on a manifold M is said to be free if for any fixed p E M the equality p a = p is valid only for a = e. A necessary condition for a right action of a Lie group G to be free is the injectivity of the mapping v E g XVP E TP(M) for any p E M.

EXAMPLE 2.32 Let V be a vector space and let A E GL(V). For any basis lei} of V, the set {Aei} is also a basis of V. Thus, we have a left action of the Lie group GL(V) on the set of all bases

of V. It is obvious that this action is free and transitive. Using this fact we identify the set of all bases of V with the Lie group GL(V), equipping it with the structure of a manifold. EXAMPLE 2.33 Let V be an n-dimensional vector space. The set of all k-dimensional subspaces of V is denoted by Gk (V ). For any A E GL(V) and W E (Gk (V), the set AW is a k-dimensional subspace of V. Therefore, we have a left action of GL(V) on Gk (V),

which is obviously transitive. Let lei} be a basis of V. Consider the linear span of the vectors ea, a = 1, . . , k. It is a k-dimensional subspace of V. This subspace is invariant with respect to the subgroup H of GL(V), formed by the linear operators A E GL(V) whose matrix representation a with respect to the basis lei} has the block form .

a=(\ Xl 0

Y

X2

Here X1 and X2 are nondegenerate k x k and (n - k) x (n - k) ma-

trices, and Y is an arbitrary k x (n - k) matrix. Using this fact, we identify Gk (V) with the homogeneous manifold H\GL(V), and consider it as a manifold. Any such a manifold is called a Grassmann manifold or a Grassmannian. The Grassmann manifold Gk (V) is k(n - k) -dimensional and compact. The Grassmann manifold G1(V) is the projective space IP(V). Consider another representation of a Grassmann manifold. Let

V be an n-dimensional vector space over a field K. Fix a basis lei} of V. Any basis { fa}Q=1 of a k-dimensional subspace of V is uniquely characterised by the coordinates f is of the vectors fa with respect to the basis lei}. The nxk matrix f - (fia) is of

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120

rank k. It is clear that we have a bijective correspondence between the bases of k-dimensional subspaces of V and n x k matrices of

rank k. The set of such matrices is an open submanifold of the manifold of nxk matrices. Two bases {fa} and {fa} of the same k-dimensional subspace V lead to the matrices f and f' connected by the relation f' = f a, where a E GL(k,1K). Thus, different matrices corresponding to the same subspace are connected by a right action of the Lie group GL(k,K). The orbits of this action are in a bijective correspondence with the elements of Gk (V). It can be shown that in actual fact the Grassmann manifold Gk (V) is diffeomorphic to the manifold obtained by factorising the manifold of all n x k matrices of rank k by the right action of the Lie group GL(k,1K).

The Grassmann manifold Gk(1Kn) is denoted by KGk,n-k. The Grassmann manifold KG1'm is the projective space XFm.

Exercises 2.13 Consider the mapping 0 from C x C x Cx C C3 to the Lie group SL(2, C) defined as

z/i(a,b,c) _ (0 1) (b 1 10) (0

1/c

Prove that ('((CxCx(Cx ), 0-1) is a chart on SL(2, C). Denote the corresponding coordinate functions by a, 3 and -y. Show that the functions g`j are connected with the functions a, ,3 and y by

0921 922)(0 1) \,a

O)

0

1/'Y)

Find the expression for the Maurer-Cartan form in terms of the coordinate functions a, 3 and y. 2.14 Show that the projective space RIP' is diffeomorphic to the sphere S1. 2.15 Prove that the projective space CIP1 and the sphere S2 are diffeomorphic as complex manifolds.

2.8 Smooth fibre bundles

121

2.8 Smooth fibre bundles 2.8.1 Definition of a fibre bundle Let E and M be manifolds and let it be a smooth mapping from E to M. The triple - (E, 7r, M) is called a bundle with the bundle

manifold E, the base manifold M and the bundle projection it. The counter image Fp - 7r-1(p), p E M, is called the fibre over the point p. The fibre bundle (E, 7r, M) is also denoted by E - M, or simply by E -* M. Let = (E, 7r, M) and (E', 7r', M') be two bundles. A smooth mapping cp : E -+ E' is called fibre preserving if it maps each fibre of Fr = 7r (p), p E M, of the bundle into some fibre F,, of the bundle '. Any such a mapping, with the help of the relation p' = &(p), defines the mapping Eli : M --4 M' satisfying the relation '007r= it'0(0. If the mapping 0 is smooth, the mapping cp is called a bundle

morphism. In this case we also write cp : -4 '. A morphism cp : -+ ' is called an isomorphism if the mapping cp : E -+ E' and the corresponding mapping V) : M -* M' are diffeomorphisms. In such a case the inverse mapping cp-1 : E -+ E' can be defined, and this mapping is also a bundle morphism. The bundles 6 and are called isomorphic if there exists an isomorphism cp :

-+ '.

In the case of M = M' and

= idM, any fibre preserving

mapping cp : E -+ E' is a morphism called a morphism over M.

In this case we have it = (p o 7r.

A morphism over M, being an isomorphism, is called an isomorphism over M. Two smooth bundles and ' with the same base manifold are called isomorphic if there exists an isomorphism over M

Let U be an open subset of the base manifold of a bundle _ (E, it, M). It is clear that Jv - (7r-' (U), 7rl.,-1 (u), U) is a bundle. This bundle is called the restriction of the bundle to U. In general, a fibre of a bundle cannot be provided with the structure of a submanifold of the bundle manifold. A bundle (E, it, M)

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Basic notions of differential geometry

isomorphic to the bundle (M x F, prM, M), where F is a manifold, is called a trivial bundle. It is clear that the fibres of a trivial bundle are embedded submanifolds of the bundle manifold. Furthermore, they are diffeomorphic one to another. Such a situation also occurs in the case of a fibre bundle. Let = (E, 7r, M) be a bundle and let F be a manifold. The bundle is said to be a fibre bundle with the typical fibre F if for any point p E M there exists a neighbourhood U of p such that I u is isomorphic to the trivial bundle (U x F, pru, U). In other words, there exists a diffeomorphism io : it-1(U) --* U x F such that pru o o =

Here the pair (U, Eli) is called a bundle chart on E. A set {U«,'0«}«EA of bundle charts on E is called a bundle atlas on E if UaEA Ua = M. A fibre bundle is also called a locally trivial bundle.

A trivial bundle is a fibre bundle which has an atlas consisting of just one chart. Let = (E, ir, M) be a fibre bundle and let {U-, z/i«LEA be a bundle atlas on E. For each p E U« the mapping O«p = prF °VG« I

n-1 (p)

is a diffeomorphism from the fibre 7r-1(p) to the typical fibre F. Let (Ua, 2/ia) and (Up, ?PQ) be two elements of the bundle atlas {(Uc,i/ia)} such that Ua n UQ 0 0. For any point p E Ua n Up we define a diffeomorphism of the typical fibre /a,Q(p) : F -* F by ?P.,3(P) _«p O 'OOP .

(2.65)

From this definition it follows that «« = idF. Furthermore, it is clear that 0.0 (p) = 0., (p) ° V)"3 (p) (2.66) for any a, /3, ry E A such that Ua n UQ n U7 0, and p E U« n U,Q n

U. Suppose that F is a complex manifold. If all diffeomorphisms 1lia i (p) are holomorphic mappings, then the fibre bundle (E, it, M) is called a complex fibre bundle. The fibre bundle (E, ir, M) is called holomorphic if E and M are complex manifolds and the mappings it, 0a and ?/i« 1, a E A are holomorphic. The group of diffeomorphisms of a differential manifold cannot be endowed with the structure of a finite-dimensional smooth

2.8 Smooth fibre bundles

123

manifold; therefore, we can say nothing about the differentiability of the mappings 7pa,Q. To remain in the framework of the differential geometry of finite-dimensional manifolds, let us suppose that

a left action of a Lie group G on the typical fibre F is given, and that there exist smooth mappings g,,o : Ua n U -+ G such that (2.67) = Lg-0 (p) for any p E Ua n U. If the action LF of the Lie group G on F is V).,3 (p)

effective, then from (2.66) we obtain p E Ua n U.y n U. g.O(p) = ga7(p)gO(p),

(2.68)

If the left action LF is not effective, we will suppose that the mappings gaQ, a, , 3 E A can be chosen in such a way that (2.68) is valid. The Lie group G is said to be the structure group of the fibre bundle , and we say that is a fibre G-bundle. The mappings ga(3

are called the transition functions of the bundle atlas {(U,,, la)}. It can be shown that for any fibre G-bundle = (E, 7r, M) and any open subset U of M the bundle Iu has a natural structure of a fibre G-bundle. Indeed, let {(UU, 'Oa)}cEA be a bundle atlas on E. Denote by 13 the set of all 3 E A such that U n UQ # 0. For any Q E 8 the pair (Va, XQ), where Vp = U n Up, X0 = 7pal is a bundle chart on 7r-1(U), and the set {(V13, XQ)}13E/3 is a bundle atlas on 7r-1(U).

EXAMPLE 2.34 Let M be an m-dimensional manifold. Denote by T (M) the union of all tangent spaces to M, i.e.,

T(M)

U Tp(M). pEM

Define the projection 7r by the relation v E Tp(M). 7r (v) = p, Let (U,,, Oa)aEA be an atlas of the manifold M. Denote the coordinate functions corresponding to the chart (Ua, (pa) by xa. Consider the set of mappings /yea : 7r-1(Ua) -4 Ua x Jjm, v E 7r-1 (Ua). 0. (V) = (ir( /v), (dx' (v), ... , dxa (v))), It can be shown that there exists a unique differentiable structure on T(M), such that the mappings 0,, are diffeomorphisms. This differentiable structure does not depend on the choice of an atlas

Basic notions of differential geometry

124

on M, and the projection 7r is smooth with respect to it. Thus, we see that (T(M), 7r, M) is a fibre bundle called the tangent bundle of M. The typical fibre of the tangent bundle T(M) is the space W'. It is clear that the diffeomorphisms 0,,,3(p), p E u,, fl U, are linear transformations of W, defined by the matrices ax,

(aaa)Zj(p)

1t')

Q

Therefore, the structure group of the fibre bundle T(M) is GL(m, III).

Similar arguments show that for any complex manifold the manifold

T(1'0) (M) = U PEM

has the natural structure of a holomorphic fibre bundle with the typical fibre Cm and the structure group GL(m, Q. Let = (E, ir, M) be a bundle. A mapping s : M -* E is called a section of if

7ros=idM. In particular, the sections of the tangent bundle T(M) -4 M are vector fields on M. Note that not each smooth fibre bundle has smooth sections. A section of a fibre bundle Iu is called a local section of . Let

{Ua}aEA be an open cover of M and let sa, a E A, be sections of the bundles Jv. ; in such a situation we say that the set of the sections {sa}aEA is a family of local sections of covering M. It is clear that for any fibre bundle there is a family of smooth local sections covering its base.

2.8.2 Principal fibre bundles and connections Let (P, ir, M) be a fibre G-bundle. Suppose that the typical fibre of coincides with the Lie group G which acts on itself by left translations. In such a situation (P, ir, M) is called a principal fibre G-bundle. EXAMPLE 2.35 Let G be a Lie group and let H be a closed Lie subgroup of G. The triple (G, 7r, G/H), where 7r is the canonical

2.8 Smooth fibre bundles

125

projection, can be naturally considered as a fibre H-bundle. If G is a complex Lie group, then (G, ir, H) is a holomorphic fibre H-bundle. Let (P, 7r, M) be a principal fibre G-bundle. Define a right action of the Lie group G on P in the following way. Let p E P; suppose that q - 7r(p) E Ua, where (U., V,,,) is a chart on the fibre bundle

P. For any aE C we put RP(p) (p) = tbaq o R. o Y'aq(p).

(2.69)

It can easily be shown that this definition does not depend on the choice of a chart. Indeed, let p E U,Q, where (U0, ,0,3) is another chart on P -4 M. From (2.65) and (2.67) we have Paq = Lg«p(q) °

Substituting this relation into (2.69), we obtain Ra (p) = '+b0q ° (Lga,o (q)) -1 o Ra o Lsaa (q) 01/) q (p)

Using (2.34) now, we come to the equality 'Pcq ° R. ° Puq (p) _ 0Qe °

R. °'Qq (p)

-

Each fibre of the fibre bundle P 4 M is invariant with respect to the transformations defined by (2.69), which can be written as 7roR9 =7r, g E G.

It is clear that RP is a free action, and the orbits of this action coincide with the fibres of the fibre bundle P -4 M.

Let P -4 M be a principal G-bundle and let P 1; M' be a principal G'-bundle. A smooth mapping co : P -4 P is called a principal bundle morphism if there exists such a group homomorphism co' : G -* G' that cp(p

a) = V(p) V (a) for any p E P,

a E G. Any principal bundle morphism is a fibre preserving mapping; in other words, it is a bundle morphism. In the case where

P = P', M = M', cp' = idG, and the principal bundle morphism co : P -+ P is an isomorphism over M, we call V a principal bundle isomorphism.

Let P 4 M be a principal G-bundle. For an arbitrary p E P, denote by Vp a tangent space to the fibre through p. Since the fibres

of P are diffeomorphic to the Lie group G, then dim V. = dim G. A smooth distribution f on P is called a connection on P, if

(Cl) Tp(P)=Vp®7-lp,for any peP;

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Basic notions of differential geometry

(C2) flp.a = RQ (7-lp) for any a E G, and p E P. A tangent vector v E Tp(P) is called vertical (horizontal) if v E Vp

(v E 7-lp). A vector field X on P is called vertical (horizontal) if for any p E P the vector Xp is vertical (horizontal). For any v E Tp(P) there is a unique expansion v = vV +v71 , where vv E Vp, V" E 7-lp.

With a connection 7-l on a principal fibre G-bundle P 4 M we can associate a 1-form w taking values in g in the following way. Let v E Tp(P); there is a unique element v E g, such that

vv=X

p,

where the vector field Xv is defined by relation (2.64). Define the form w by

w(v) - v. A vector v E Tp(P) is horizontal if and only if w(v) = 0. The form w is called the connection form of the connection R. The connection form w has the properties

w(X') = v

(2.70)

Ra *w = Ad(a-1) o w

(2.71)

for any v E g, and for any a E G. Conversely, any g-valued 1-form on a principal fibre G-bundle

P 4 M satisfying conditions (2.70) and (2.71) is a connection form of a unique connection on P 4 M. Here the corresponding subspaces lip, p E P, are defined by Rp - {v E 7p(P) I w(v) = 0}. Let (P, ir, M) be a principal fibre G-bundle; consider some set {sa}aEA of local sections sa : Ua -* i-1(Ua) covering M. The mappings ?/la : 7-1(Ua) -+ Ua x G, defined with the help of the relation

,0,'(q, a) = sa (q) - a, are diffeomorphisms satisfying the condition pruu o?/Ia = ir.

2.8 Smooth fibre bundles

127

It is clear that the set {(U., Y'a)}aEA is a bundle atlas on P. The transition functions in this case have the form 9.,3 (P) = 9.(sQ(p)),

p E Ua fl Ua,

where the mappings ga : 7r-1(Ua) - G are defined by

9a = prc ooa. Thus, any set {sa}aEA of local sections sa : Ua -4 i-1(Ua) of a principal fibre G-bundle P -4 M covering M generates a bundle atlas on P. On the other hand, let {UU, a}aEA be a bundle atlas on P. The set {sa}aEA of local sections defined by sa (q) = 0a 1(q, e), generates the bundle atlas {Ua, Y'a}.

q E U.

Now let w be a connection form of some connection on P. On i-1(Ua) we obtain w = Ad(9-1) o it*wa + 9*8, (2.72) where 0 is the Maurer-Cartan form of G and wa - saw. It can be also shown that wa = Ad(gap) o w,3 + ga,30

2.73)

on ?f-1(Ua fl Up).

On the other hand, if there is given a set of g-valued 1-forms {wa}aEA satisfying (2.73), we can construct a unique g-valued 1form w satisfying (2.72). In other words, a set of 9-valued 1-forms {wa}aEA satisfying (2.73) define a unique connection on P. Let rl be a k-form on a principal fibre bundle P -4 M provided with a connection R. The horizontal component qn of 77 is defined by

77'(X1i...,Xk) = r7(Xx,...7Xk ) for all X1, ... , Xk E 1(P). It is clear that ix is a k-form on P. The horizontal component of a form taking values in a vector space can be defined similarly. Here, for the connection form w corresponding to the connection 71, we have ww = 0 but, in general, (dw)w 0. The g-valued 2-form

ci - (dw) u

128

Basic notions of differential geometry

is called the curvature form of the connection W. It can be shown

that S2 = dw + 2 [w, w].

(2.74)

Using a set {sa} of local sections covering M, we obtain the relations fl = Ad(g;') o it*S2a, (2.75) where

is = s*i. The 2-forms fZa are connected with the 1-forms wa by

Qa = dwa +

1

2

[wa) wa11

and

Q. = Ad(g.0) o SZQ on 7r-1(Ua fl UO).

Note also that the following relation is valid:

(dc) = 0; this is called the Bianchi identity.

3

Differential geometry of Toda-type systems

3.1 More about semisimple Lie algebras In the next four sections we present some additional information on semisimple Lie algebras which will be needed in our consideration of the Toda-type systems. Here we mainly follow the remarkable books by Helgason (1978); Bourbaki (1975); Gorbatsevich, Onishchik & Vinberg (1994) and the original papers of Dynkin (1975a,b) and Kostant (1959).

3.1.1 Groups of automorphisms The group Aut(A) of automorphisms of an algebra A is a closed subgroup of the Lie group GL(A). Therefore, we can consider this group as a Lie group. It can be shown that the Lie algebra of the group Aut(A) coincides with the Lie algebra Der(A) of derivations of A.

Let g be a Lie algebra. The mapping x E g H ad(x) E Der(g) is a homeomorphism of g onto some subalgebra of Der(g) denoted

by ad(g). The corresponding connected Lie subgroup of Aut(g) is called the group of inner automorphisms of g and is denoted by Int(g). The elements of Int(g) are called inner automorphisms of g. Sometimes Int(g) is called the adjoint group of g. An inner automorphism can be represented as a product of a finite number of automorphisms of the form exp(ad(x)), x E g. Since ad(g) is an ideal of Der(g), the group Int(g) is an invariant subgroup of Aut(g). Elements or subsets of g connected by an inner automorphism are called conjugated in g.

Let g be a Lie algebra and let G be a connected Lie group having g as its Lie algebra. For any a E C the mapping con : 9 -+ g defined by cpa(v) - Ad(a)v 129

130

Differential geometry of Toda-type systems

is an automorphism of g. Any element of G can be represented as a product of a finite number of elements of the form exp(x), x E g. Using the relation Ad(exp(x))y = exp(ad(x))y, we see that Wexp(x) = exp(ad(x)).

Therefore, for any a E G the automorphism Pa is an inner automorphism.

On the other hand, since any inner automorphism of g can be represented as a product of a finite number of inner automorphisms of the form exp(ad(x)), x E g, we conclude that any inner automorphism of g has the form cpa for some a E G. Thus, we have Ad(G) = Int(g). From relation (2.41) it follows that the centre Z(G) of the group G coincides with the kernel of the adjoint representation of G. Therefore, the adjoint group of g is isomorphic to the group G/Z(G). Any derivation of a semisimple Lie algebra is an inner derivation. Hence, for any semisimple Lie algebra g, the component of Aut(g), containing the identity element, coincides with Int(g). Let g be a complex semisimple Lie algebra, E a Cartan subalgebra of g, and A a root system of g with respect to I). Consider a base 11 of A. Denote by Aut(II) a subgroup of a symmetric group of II, consisting of the permutations which do not change the Cartan matrix. The elements of Aut(II) are naturally identified with the symmetry transformations of the corresponding Dynkin diagram. It can be shown that the group Aut(II) is isomorphic to the subgroup of Aut(o) consisting of the automorphisms which transform II onto itself; this subgroup is also denoted by Aut(II). Actually, the group Aut(o) is the semidirect product of the Weyl group W(O) and the group Aut(II).

Let a E Aut(II) C Aut(o); the mapping (a-')t is an element of the group Aut(O'). It can be proved that there exists a unique automorphism W, of g such that W (a") _ (a-' )t (av) for any a E II. The automorphism co, is called the automorphism of g induced by a. Hence, the group Aut(II) can be considered as a subgroup of Aut(g). Furthermore, one can easily see that any automorphism of g can be uniquely represented in the form 0 o cpo, where '& E

3.1 More about semisimple Lie algebras

131

Int(g) and o, E Aut(II). Therefore, for any complex semisimple Lie algebra, the group Aut(g) has the representation Aut(g) = Int(g) A Aut(II).

EXAMPLE 3.1 Considering the Dynkin diagrams of the complex simple Lie algebras (table 1.2), we conclude that the group Aut(II)

is Z2 for the Lie algebras of types Ar (r > 1), Dr (r > 4), E6; and it is S3 for the Lie algebras of type D4. For the remaining simple Lie algebras, the group Aut(II) is trivial; hence, for these Lie algebras the group of automorphisms coincides with the group of inner automorphisms.

Consider now the connection between automorphisms of Lie groups and Lie algebras. Let E be an automorphism of a real Lie group G; then the mapping or = E*e

(3.1)

is an automorphism of the corresponding Lie algebra g. On the other hand, if the group G is simply connected, then, for any automorphism a of the Lie algebra g, there is a unique automorphism E of the group G such that E*e = Q. It follows from the definition of the adjoint representation of a Lie group that or o Ad(a) = Ad(E(a)) o o,.

(3.2)

Furthermore, it can easily be shown that for the Maurer-Cartan form 0 of G we have E*B=QOB. In the case of a complex Lie group G, it is natural to consider only holomorphic or antiholomorphic automorphisms of G. For a holomorphic automorphism E, relation (3.2) is valid. If E is an antiholomorphic automorphism of G, then E*e (T(1'0)(G)) = T(°'1)(G).

In this case we define the corresponding mapping Q : g -+ 9 by (3.3) a(x) = E*e(x). The mapping o, is now an antilinear automorphism of g. Definition (3.3) leads to the equality

E*O(x) = v o O(x),

(3.4)

132

Differential geometry of Toda-type systems

which is valid for any x E Ta(G), a E G. Note that relation (3.2) is valid both for real and complex Lie groups with the mapping or defined either by (3.1) or by (3.3).

3.1.2 Regular subalgebras and subgroups In this subsection g is a complex semisimple Lie algebra and 0 is some fixed Cartan subalgebra of g. A subalgebra f of g is called regular with respect to fl, if [F), f] C f

In other words, f is regular with respect to h if 1 is contained in the normaliser NB (f ). A Lie group G is called semisimple if its Lie

algebra g is semisimple. A subgroup F of a complex connected semisimple Lie group G is called regular with respect to a Cartan subgroup H if NG(F) contains H. All regular subalgebras can be described in the following way. Let A be a root system of g with respect to Il. A subsystem r C A is called closed if for any a, ,C3 E F, such that a +,3 E A, one has

a + ,Q E F. A subsystem F E A is said to be symmetric if the inclusion a E F implies that -a E F. Let F be a closed subsystem of A and let t be a subalgebra of 0 containing the element a" for any a E F fl (-F). In this case

f=t®®go,

(3.5)

«Er

is a subalgebra of g, which is regular with respect to Cl. We denote such a subalgebra by f (t, r). On the other hand, any subalgebra f

of g which is regular with respect to 0 can be represented in the form of (3.5). A subalgebra f (t, r) is semisimple if and only if r is symmetric and t is generated by the elements a", a E F. In such a case the subalgebra t is a Cartan subalgebra of f (t, r), and r is the root system of f (t, r) with respect to t. Since a semisimple subalgebra of type f (t, r) is determined only by the subsystem r, we use the notation f (r) for it. Consider the semisimple regular subalgebras in more detail. Let ' be a subsystem of A; denote by [W] the set of the elements of A which can be represented as linear combinations of elements of ' with integer coefficients. It is clear that if 11 is a base of A, then [II] = A. For any subsystem' of A, the set [W] is a closed symmetric subsystem of A which defines a semisimple regular subalgebra of g.

3.1 More about semisimple Lie algebras

133

A subsystem' of A is called a ir-system if T is linearly indeA for all a,,3 E T. Any 7r-system W is a base of the root system f - [W] of the semisimple regular subalgebra f ([]). On the other hand, any base of a closed symmetric subsystem of A is a ir-system in A. Thus, the problem of the classification of the semisimple regular subalgebras is reduced to the problem of the enumeration of all ir-systems in A. If two 7rsystems are connected by a transformation of Aut(o), then the

pendent and a -,3

corresponding subalgebras are connected by an element of Aut(g). If two 7r-systems are connected by a transformation of the Weyl group W (A), then the corresponding subalgebras are conjugated. Note that any subsystem of a 7r-system is also a 7r-system. Moreover, if g is of rank r, then any 7r-system is a subsystem of a 7rsystem consisting of r elements. Hence, to find all 7r-systems, it suffices to find all 7r-systems consisting of r elements. Let II = {al, ... , a,.} be a base of A; add to 11 the minimal root ao. The resulting subsystem of A is called the extended system of simple roots of g, and the corresponding Dynkin diagram is called the extended Dynkin diagram of g. The extended Dynkin diagrams for simple Lie algebras and their standard notations are listed in

table 3.1. The vertices corresponding to the minimal roots are blackened. Note that the Dynkin diagrams of table 3.1 correspond to the so-called untwisted affine Lie algebras. Now let T be a 7r-system in A. As we noted above,' is a base of the root system of the semisimple Lie algebra f (['1 ]). In particular,

any irreducible component of ' is a base of the root system of the corresponding simple subalgebra of f ([WY]). Complement T by the minimal root corresponding to some irreducible component of

T, and then remove any root belonging to this component. As a result we obtain another 7r-system 'Y' in A, and we say that the ir-system 'Y' is obtained from the ir-system J by an elementary transformation. It appears that any ir-system in A consisting of r elements, with r being the rank of g, can be obtained from some base II of A by a chain of elementary transformations. There exists an important class of nonsemisimple regular subalgebras and subgroups. A Lie subalgebra b of a Lie algebra g is called a Borel subalgebra if b is a maximal solvable subalgebra of g. A connected subgroup B of a real or complex Lie group G is said to be a Borel subgroup of G if the Lie algebra b of B is a

134

Differential geometry of Toda-type systems Table 3.1.

Ail)

G21)

ATl>

F(1)

Br(l)

o-L

C(1) moo-D;.1)

-o---

E(6l)

o-o

Ell)

0

Egl)

0-o

0

0

c

0

oo c

c

0

0

0

Borel subalgebra of the Lie algebra g of G. According to the BorelMorozov theorem, all Borel subalgebras of a complex Lie algebra and all Borel subgroups of a complex Lie group are conjugated to

each other. The Borel subalgebras of a complex Lie algebra and Borel subgroups of a complex Lie group are regular subalgebras and regular subgroups, respectively. For any Borel subgroup B of a complex connected Lie group G, the homogeneous space G/B is a simply connected projective manifold. The root decomposition (1.15) of g implies that

g=n-®f)®n+,

(3.6)

where nt = ® go' aEA±

are nilpotent subalgebras of g. The corresponding connected nilpo-

tent Lie subgroups of G are denoted by N. For the connected subgroup generated by Ij we use the notation H. It can be shown that the subalgebras b f - lj E) n±

(3.7)

are Borel subalgebras of g. The Borel subalgebra b - is called opposite to the Borel subalgebra b+. Using the notation introduced for

3.1 More about semisimple Lie algebras

135

regular subalgebras, we can write b+ = f (1), A±). The subalgebras b+ generate Borel subgroups B+.

EXAMPLE 3.2 Let g be one of the matrix complex simple Lie algebras .s[(m, C), 5(m, C) or sp(m/2, (C) considered in section 1.3. Recall that a Cartan subalgebra Cl for all these Lie algebras can be chosen as fj = g n ct(m, C), where cZ(m, C) is the Lie algebra of all diagonal complex m x m matrices. In this case, using the bases of the corresponding root systems defined in section 1.3, we find that nt = g n n± (m, C), where n+ (m, C) and n_ (m, C) are the Lie algebras of all strictly upper and lower triangular complex m x m matrices respectively. Therefore, the Borel subalgebras b+ coincide with g n t± (m, C), where t+ (m, C) and t_ (m, C) are the Lie algebras of all upper and lower triangular complex m x m matrices respectively.

Let us consider the corresponding complex Lie groups. Denote by SO(m, C) the complex Lie group consisting of complex m x m matrices which satisfy condition (2.47) with b = I7z and have unit determinant. Similarly, for an even m denote by §'p- (m/2, C) the complex Lie group formed by all complex _m x m matrices satisfying

(2.47) with b = Jm/2. The Lie groups SO(m, C) and Sp(m/2,C) are isomorphic to the Lie groups SO(m, C) and Sp(m/2, C) respectively. The Lie algebras of SO(m, C) and Sp(m/2, C) are 5(m, C) and sp(m/2, C).

_

Now let G be one of the complex Lie groups SL(m, C), SO(m, C)

or Sp(m/2, C). The subgroup H in such a case coincides with G n D(m, C), where D(m, C) is the Lie group of all nonsingular diagonal complex m x m matrices. Further, we have N± = G n N, (m, C), where N+ (m, C) and N_ (m, C) are the complex Lie groups of all complex upper or lower triangular m x m matrices with unit diagonal elements. Finally, the Borel subgroups Bt are given by the intersection G n Tt (m, C) with T+ (m, C) and T_ (m, C) being the complex Lie groups formed by upper and lower triangular nondegenerate complex m x m matrices respectively.

By definition, a parabolic subalgebra of g is a subalgebra of g which contains some Borel subalgebra of g. A subgroup P of a Lie group G is called a parabolic subgroup if it contains a Borel sub-

Differential geometry of Toda-type systems

136

group of G. The parabolic subalgebras and parabolic subgroups are regular subalgebras and regular subgroups respectively. Let II = {al, ... , a,.} be a system of simple roots of A and let ' be a subsystem of H. Introduce the notation

P±w = b ® ® g°.

(3.8)

c,E[wlnr:F

The subalgebras pfw are parabolic subalgebras and any parabolic subalgebra of g, up to a transformation of the group Aut(g), can be obtained in such a way. Thus, we have a transparent classification of the parabolic subalgebras of any complex semisimple Lie algebra. One usually writes pfi...... ik for the parabolic subalgebra corresponding to the subsystem ' = {ail , ... , ai, }. The parabolic subalgebras p±(n_w) are denoted by p'w. Therefore, it is natural to write p'l i,, for the parabolic subalgebras corresponding to the subsystem of H which contains all simple roots except the roots ail , ... , c In particular, pi is the parabolic subalgebra corresponding to a subsystem containing all simple roots except the .

root ai. Any maximal parabolic subalgebra of g is not semisimple and,

vice versa, any nonsemisimple maximal subalgebra of g is a parabolic subalgebra. For the parabolic subgroups we use the same notation as for the parabolic subalgebras with the change of p to P. For example, the parabolic subgroups corresponding to the parabolic subalgebras are denoted by

Again let T be a subsystem of the system of simple roots H. Below we use the following notations:

nfw = f(0, A' - [W]), hw = f(h,

(3.9) (3.10)

When it is obvious which subsystem T is under consideration, we write simply n f and Cl. It is clear that (3.11)

Actually, nt0 = n±, 40 = C , and we come to (3.6). In the general case ii C nt and Clw D Cl. From such point of view, it is also natural to introduce the notations btw = 04, ® ntw = ptw

(3.12)

3.1 More about semisimple Lie algebras

137

For the Lie subgroups corresponding to the subalgebras nt, bt and h we use the notations N+, B+ and H respectively. EXAMPLE 3.3 Consider the Lie algebra 5C(m, Q. Let {nc,}a±i be a fixed set of positive integers such that Ek+' na = m. It is obvious that the subalgebra of 5((m, (C), formed by the matrices a having the following block form:

a=

X1

*

...

0

x2

...

0

0

xk+1

where xc, are complex nc, x na matrices such that Ek+1 tr xc, = 0, is a parabolic subalgebra of sl(m, Q. Using the Cartan subalgebra of 5[(m, C) and the base of the corresponding root system intro-

duced in section 1.3.1, we can easily see that one has here the parabolic subalgebra pit ik with it = E1=1 na. The corresponding parabolic subgroups of SL(m, (C) consist of the matrices of the block form (3.13) 0

0

where X,, are arbitrary complex nc, x na matrices such that na+i det X,,

= 1.

Any parabolic subgroup of a complex connected semisimple Lie

group is connected and the homogeneous space G/P is a simply connected projective manifold. In particular, G/ P is a compact manifold. The homogeneous space G/P is called a flag manifold or, quite rarely, a parabolic space. The flag manifolds corresponding to the parabolic subgroups Pti...... ik are denoted by F 1... j ; while for the flag manifolds corresponding to the parabolic subgroup Pail i, we use the notation Fi1...,ik EXAMPLE 3.4 The relation of the notion of a flag manifold to the notion of a flag is explained as follows. Let V be an m-dimensional

vector space and let i1, ... , ik be a set of integers such that 0 < it < ... < ik < m. A family {U} of subspaces of V such that

138

Differential geometry of Toda-type systems

dimV = il, l = 1, ... , k, and V1 C ... C Vk is called a flag of type (i1, ... , 2k) in V. A flag of type (1,... , m - 1) is called the ik) in V is denoted by full flag. The set of all flags of type Fi1 _.,ik (V ). It is obvious that Fi (V) is the Grassmann manifold Gi (V) defined in example 2.33. Consider the case of the complex m-dimensional vector space V. It is clear that there is defined a natural left action of the Lie

group SL(V) on the set Fi...... ik(V). It can be verified that this action is transitive. Let lei} be a basis of V; consider the flag formed by the subspaces v, = ® Ci, 1 = 1, ... , k. 1
The corresponding isotropy subgroup consists of the elements A E SL(V) whose matrix representation a with respect to the basis {ei} has the block form (3.13), where Xa, a = 1, . , k + 1, . .

are n(xna matrices with na = i, - i,Y_1 (io - 0, ik±1 - m). Thus, we see that the set Fi,,_..,ik(V) can be identified with the homogeneous space P\SL(m, C), where P is a parabolic subgroup of SL(m, C) formed by the matrices of form (3.13). This allows us to consider Fil,..,,ik(V) as a complex manifold. Note that the mapping a E SL(m, cC) H a-1 E SL(m, cC) provides identification of the homogeneous spaces P\SL(m,C) and SL(m,(C)/P. Taking

into account that P =

we conclude that the manifold

of the Lie group SL(m, C). Note here that the flag manifold Fi' of SL(m, C) (CGi'm'-i; is diffeomorphic to the Grassmann manifold in particuFi1,...,ik (V) is diffeomorphic to the flag manifold

lar, the flag manifold Fl of SL(m, C) can be identified with the projective space apm-1

3.1.3 7G-graded Lie algebras The notion of a gradation of an algebra is important both for the elucidation of its structure and for the classification of algebras. Let us begin with the definition of a graded vector space. Let V be a vector space over a field 1K and let M be an abelian group. An M-gradation of V is a family {Vm}mEM of subspaces of

V such that V = ® Vm. mEM

3.1 More about semisimple Lie algebras

139

A vector space V is said to be M-graded if it is equipped with an M-gradation. An element of an M-graded vector space V is called homogeneous of degree m E M if it is an element of Vm. An algebra A is called M-graded if it is an M-graded vector space and AmAn C Am+n, m, n E M. In other words, the product of any element of the subspace A. with any element of the subspace An belongs to the subspace Am+n. Note that the subspace AO is a subalgebra of A. Let an algebra A be endowed with an M-gradation

A= ®Am

(3.14)

mEM

and let (p be an automorphism of A. The representation

A = ® cy(Am)

(3.15)

mEM

defines another M-gradation of A. The M-gradations (3.14) and (3.15) are said to be connected by the automorphism co. Two Mgradations of A, connected by an inner automorphism of A, are called conjugated.

EXAMPLE 3.5 Let 0 be a root system of a semisimple complex Lie algebra with respect to some Cartan subalgebra f and let H = {al, ... , a..} be a system of simple roots. Associate with a

root a = F_'=, miai E 0 the element m = (ml,... , mr) E Zr, and denote the root space g" by gm. If for m = (ml, ... , mr) E Z one has Ei=, mia; V A, put gm - {O}. Also denote the Cartan subalgebra h by g(q.,.,0). It is easy to verify that the decomposition

g= (D on mEZ' is a Z'-gradation of g. According to a general definition, a Z-gradation of a Lie algebra g is a decomposition of g into a direct sum of subspaces gm

g= ®gm mEZ such that [

m, 9k]

C gm+k

140

Differential geometry of Toda-type systems

For any Z-gradation of g we can define the derivation D of g by

Dx = E mx,n. mE7L

We restrict ourselves to the case of semisimple Lie algebras. Since

in this case any derivation of g is an inner derivation, and the centre of g is trivial, then there exists a unique element q E g such

that Dx = [q, x] for all x E g. In other words, for any x E [q, x] = mx.

we have

The element q is called the grading operator of the Z-gradation under consideration. Thus, for a semisimple Lie algebra, any Zgradation may be defined with the help of the corresponding grading operator. Suppose now that g is a complex semisimple Lie algebra. It is clear that for any grading operator q, the linear operator ad(q) is semisimple and satisfies the relation

exp(27rfJ ad(q)) = idg. (3.16) On the other hand, it is clear that any semisimple element q of g satisfying (3.16) can be considered as the grading operator of some Z-gradation of g.

Since any grading operator q is semisimple, we can suppose without loss of generality that q belongs to some Cartan subalgebra Cj of g. Let H = jai.... , ar} be a base of the root system A of g with respect to fj. From (3.16) it follows that (ai, q), i = 1, ... , r, are integers. Furthermore, the element q belongs to the closure of some Weyl chamber C, and if II is the base of A corresponding to this Weyl chamber, then all the numbers ni - ai(q) are nonnegative. The numbers ni do not depend on the choice of a Weyl chamber whose closure contains q. The corresponding Dynkin diagram, with the vertices labelled by the numbers ni, is called the characteristic of q.

On the other hand, choosing an arbitrary set of nonnegative integers {n,} 1, we can construct the element q E h as

q = E (k-')ijnjhi. i,j=1

(3.17)

3.1 More about semisimple Lie algebras

141

Here k-1 is the inverse of the Cartan matrix k of g. It is clear that the element q, defined by (3.17), is semisimple and satisfies (3.16). In other words, q is a grading operator. Moreover, for this element we have (ai, q) = ni. Thus, the Dynkin diagram, with the vertices labelled by arbitrary nonnegative integers, is the characteristic of some grading operator. Two grading operators are connected by an automorphism of g if and only if they have the same characteristics. Two grading operators having the same characteristics, may not be conjugated. The number of the classes of conjugated grading operators coincides with the order of the symmetry group of the Dynkin diagram divided by the order of the symmetry group of the characteristic.

Note that the subspace 9m is the sum of the root spaces 9" corresponding to the roots a = E1
subspace gm with m > 0 (m < 0). Further, if g° C gm, then 9-" C 9-m; hence, we have dim 9m = dimg_m.

Let a Z-gradation of a complex semisimple Lie algebra g be given. Find a Cartan subalgebra h and a base H = {al,... , a.} of the root system 0 of g with respect to C1 such that for the corresponding grading operator q we have (ai, q) > 0. Introduce the notation T - {ai E 111 (ai, q) = 0}.

In accordance with (3.8), the subsystem ' defines a parabolic subalgebra of g. Here, for the subalgebras n± (IF), bt(T) and CA(W) defined by (3.9), (3.12) and (3.10), we have n+ = ® 9m, m<0

(3.18)

m>0

6+ _

(3.19)

M
=9o.

(3.20)

For any complex semisimple Lie algebra one distinguished Zgradation arises when we choose all the numbers ni equal to 1. In

142

Differential geometry of Toda-type systems

this case the corresponding grading operator has the form r

r

q = E kihi, i=1

ki = E(k-' )ij.

(3.21)

j=1

Such a gradation is called the canonical or principal gradation. For the principal gradation nt = n±, b± = b± and fl = 0. Hence, the subalgebra go is in this case abelian and the subspaces gt1 coincide with the linear spans of the Chevalley generators x ±i,

3.1.4 s((2,C)-subalgebras

In this section we discuss the embeddings of the Lie algebra s((2, C) into a complex semisimple Lie algebra and the correspond-

ing Z-gradations. By an embedding of sl(2, C) into g we mean a nontrivial homomorphism from s[(2, C) into g. Let h, xt be the elements of the standard basis of s((2, C) introduced in example 1.11. Note that h is a Cartan generator, while x± are Chevalley generators of sE(2, C). The images of the elements h and x± under the homomorphism defining the embedding of s1(2, C) under consideration are usually denoted by the same letters, optionally endowed with an index distinguishing different embeddings. The image of the whole sl(2, C) is called an s((2, C)-subalgebra of g. It can be shown that two sI(2, C)-subalgebras of g are connected by an automorphism of g if and only if the corresponding Cartan generators are connected by an automorphism of g. Here s((2, C)subalgebras are conjugated in g if and only if their Cartan generators are conjugated in g. EXAMPLE 3.6 Let

Cl

be a Cartan subalgebra of a complex

semisimple Lie algebra g and let a be a root of g with respect to Cl. It can be shown that one can choose the elements xta E g in such a way that K(xQ, x_.) = 2/(a, a), where K is the Killing form of g, and ( , ) is the bilinear form on induced by the restriction of K to C7. In this case we have av, jav, xfa] = ±2x±,,,. [xa, XaJ =

Hence, the elements h - a", xt - xta form a basis of some sl(2, C)-subalgebra of g.

3.1 More about semisimple Lie algebras

143

For a given embedding of s1(2, C) into g, the adjoint representation of g defines the representation of the Lie algebra sl(2, cC) in g. From the properties of the finite-dimensional representations of sl(2, C) it follows that the element h of g must be semisimple, and

the elements xt E g must be nilpotent. Moreover, it is clear that exp(27rv'r-_1_ad(h)) = id9.

(3.22)

Therefore, the element h can be used as the grading operator defining some Z-gradation on g. As above, without any loss of generality, one can suppose that h belongs to some Cartan subalgebra i of g, and we can choose a base lI = {al, ... , ar} of the root system of g with respect to El in such a way that the numbers (ai, h), i = 1, ... , r, are nonnegative integers. It can be shown that the numbers (ai, h) can be equal only to 0, 1 and 2. In fact, it is more convenient for our purposes to define the grading operator q, connected with the given embedding of s1(2, C) into g, by the relation h = 2q. It is clear that this definition necessarily leads to Z/2-gradations of g as well. If, instead of (3.22), one has the relation exp(7rV_-1 ad (h)) = id9,

we call the corresponding embedding integral; otherwise we deal with a semi-integral embedding. For an integral embedding the numbers (ai, h) are equal to 0 or 2. Note here that not every element h E 1), even one satisfying the requirement (ai, h) = 0, 1 or 2, can be considered as the corresponding element of some embedding of s((2, C) into g, and we do not have here a direct relation to parabolic subalgebras, as for the case of a general Z-gradation. Note also that the properties of the finite-dimensional represen-

tations of sl(2, C) imply that if one considers a Z-gradation or a Z/2-gradation associated with an embedding of s((2, C) into g, then dim go > dim g±l . Specifying in a certain way an sl(2, C)-subalgebra of a complex

semisimple Lie algebra g, one can parametrise g in accordance with the arising representation of sl(2, C). Here all the elements of the algebra g can be collected into multiplets corresponding to finite-dimensional irreducible representations of sl(2, C). In order

Differential geometry of Toda-type systems

144

to parametrise the elements of g, one usually uses one of the two bases. The first is the root basis which is universal for all semisimple Lie algebras but is not very suitable for physical applications

with tensor calculations, in particular for standard methods in atomic physics. The second basis uses rather cumbersome tensor notations which are restricted in their applications to the classical series and are quite tedious. Grouping of the elements of Lie algebras g based on the consideration of some embedding of the Lie algebra s[(2, cC) takes, in a sense, some intermediate place since in the framework of this classification the generality of the root language is complemented by the visuality of the multiplet structure that physicists use and find convenient. We now proceed to the discussion of concrete s((2, (C)-subalgebras of complex semisimple Lie algebras. First consider the case of complex matrix semisimple Lie algebras. Here, as well as the representation of s((2, C) induced by the adjoint representation, we have one more representation of s((2, (C) realised by the corresponding matrices. The simplest case here is the special linear algebra 5C(m, C). Having an sr(2, (C) subalgebra of s((m, (C), we actually have a faithful m-dimensional representation of sI(2,SC). Recall that any finite-dimensional representation of s((2, (C) is a direct sum of irreducible representations, each of them being isomorphic to some of the representations considered in example 1.11. Any such representation is uniquely determined by a nonnegative integer n and has the dimension d = n+1. So we have the splitting

m = dl + where dl,

. . . ,

+ ds,

d, > d2... > ds > 1,

(3.23)

ds are the dimensions of the irreducible components

of the representation of s((2, Q. The case of s = m, when all dis are equal to 1 must be excluded because it corresponds to the trivial representation. On the other hand, any splitting of m of form (3.23) corresponds to a possible m-dimensional representation which is realised by matrices belonging to s((m, Q. Therefore, such a splitting corresponds to an embedding of s1(2, C) into s((m, (C). Two representations corresponding to the same splitting are connected by a transformation of SL(m, C), and any two representations of s1(2, C) corresponding to different splittings cannot be connected by an automorphism of sl(m, (C). Thus, the splittings of the integer m of form (3.23) and nonconjugated embeddings of

3.1 More about semisimple Lie algebras

145

sl(2, cC) into s((m, (C) are in bijective correspondence. A representation p of a Lie algebra g in a vector space V is said

to be orthogonal (symplectic) if there exists a symmetric (skewsymmetric) nondegenerate bilinear form B on V such that

B(v, p(x)u) + B(p(x)v, u) = 0

for all v, u E V and x E g. An irreducible representation of -C(2, (C) is orthogonal (symplectic) if and only if its dimension is odd (even). A representation of s((2, C) is orthogonal (symplectic) if and only if the multiplicities of its even-dimensional (odd-dimensional)

irreducible components are even. These properties of the representations of s1(2, C) imply that if one has an embedding of z((2, (C) into o(m, C) (sp(m/2, C)), then the corresponding splitting (3.23) contains any even (odd) summand an even number of times. On the other hand, any representation of sl(2, C) defined by such a splitting can be realised by orthogonal (symplectic) matrices. Moreover, any two representations corresponding to the same splitting are connected by an element of the group SO(m, C) (Sp(m/2, C)), except in the case of the orthogonal representations of sl(2, C) with all the dimensions of the irreducible components being even. In the last case, realised only when m is a multiple of 4, the corresponding representations fall into two classes. The representations belonging to the same class are connected by elements of SO(m, C), while the representations of different classes are connected by elements of O(m, (C). Thus, with only one exception, the nonconjugated embeddings of s((2, (C) into o(m, (C) (sp(m/2, C))

are in bijective correspondence with the splittings (3.23) of m, where any even (odd) summand is contained an even number of times. The embeddings of sl(2, (C) into o(m, C) corresponding to a splitting into even numbers only fall into two classes of conjugated embeddings connected by an `external' automorphism of o(m, (C). Thus, we have a complete classification of s[(2, C)-subalgebras

of complex matrix simple Lie algebras. In table 3.2 we present the results of this classification for the algebras of rank less than or equal to 3. Underlined numbers denote the dimensions of the irreducible components of the corresponding representation of sI(2, (C), while the ordinary numbers denote the multiplicities.

146

Differential geometry of Toda-type systems Table 3.2.

Al

2

3,2+1

sl(4,C)

A2 A3

o(3, C) o(4, C)

A1

3

Al x Al

o(5,C)

C2

o(6,(C)

A3 B3

s1(2, Q s1(3, Q

o(7,C)

4, 3+1, 2.2, 2+2. 1

3+1, 2.2 5, 3+2. 1, 2.2+1

5+1,2.3,3+3.1,2.2+2.1 7, 5+2.1, 2.3+1, 3+2.2, 3+4.1, 2.2+3.1

sp(1, C)

A 1

2

sp(2,C)

C2 C3

4,2.2,2+2.1 6,4+2,4+2.1,2.3,3.2,2.2+2.1,2+4.1

sp(3,C)

Unfortunately, the above consideration is applicable only to the case of the classical series of Lie algebras; moreover, it does not give a constructive procedure for finding the concrete form of the elements h, x± specifying a concrete embedding. A general method, which can be used for all simple Lie algebras, was developed by Dynkin (1957a). This method is based on the fact that an embedding of sl(2, (C) into a complex semisimple Lie algebra g is uniquely characterised by the characteristic of the corresponding element h E g. Namely, two sl(2, (C)-subalgebras of g are connected

by an automorphism of g if and only if the corresponding Cartan generators have the same characteristics. The number of classes of conjugated sl(2, C)-subalgebras corresponding to the same characteristic coincides with the order of the symmetry group of the Dynkin diagram divided by the order of the symmetry group of the characteristic. It is important for the method considered that for any complex semisimple Lie algebra g there is an embedding of s1(2, C) leading to the principal gradation. This embedding is defined by X :L

r

= E(2ki)112xti, i=1

r

h = 2 E kihi,

(3.24)

i=1

where hi and xti are Cartan and Chevalley generators of g and the numbers ki are defined by (3.21). This embedding is called the

3.1 More about semisimple Lie algebras

147

Table 3.3. k

k-1

2ki

A2

(-1 2)

3(1 2)

2

2

-1

0

(

A3

-1

2

-1

0

-1

2

B3

C2

C3

G2

2

-1

0

1

2

0

-1

-2

1 4

2

2

2

1

3

2

4

2

4

1

2

3

3

2

2

2

2

4 2

4 3

6 10 6

1

-1

2

3

2(2 2)

(-2 2)

4

2

-1

0

2

2

1

5

1

2

-1

2

2

8

0

-2

2

4 4

3

9

2

-1

(-3 2)

2

2

(3 2)

10

principal embedding ; its exhaustive investigation was performed

by Kostant (1959). For the principal embedding all the numbers (ai, h) are equal to 2. Hence, this embedding is defined up to conjugation. Note that for a principal embedding dimgo = dimg±l. In table 3.3 we give the necessary information about the principal embeddings of s1(2, C) into simple Lie algebras of rank less or equal to 3. For a principal sl(2, C)-subalgebra of a complex semisimple algebra g, the representation of s((2, C), obtained by reducing

the adjoint representation of g to s1(2, C), has the number of irreducible components equal to the rank of g. The dimensions of the irreducible components can be calculated as follows. Let

148

Differential geometry of Toda-type systems

II = {al, ... , ar} be a base of the root system A of g. Consider the element s - si usual, si = sa; , i = 1, .

.

.

sr of the Weyl group W (A), where, as

.

, r. The eigenvalues of s have the form

where li, i = 1, ... r, are positive integers and c is the Coxeter number of A. The numbers li are called the exponents of A. The dimensions ni of the irreducible components of the considered representation of s1(2, C) are ni = 21i + 1. Furthermore, in the case of complex simple Lie algebras, the multiplicity of a given irreducible representation is always unity, except the case of exp(27r

l i / c ),

the series Dr for an even r, where there are two representations of dimension 2r - 1. An sl(2, C)-subalgebra of g is called semiprincipal if it is not contained in any proper regular subalgebra of g. For any semiprincipal embedding, the labels of the Dynkin diagram, specifying the corresponding characteristic, are equal to 0 or 2 only. Any principal s((2, (C)-subalgebra of a complex semisimple Lie algebra is also

semiprincipal. For the Lie algebras of types Ar, Br, Cr, G2 and F4, any semiprincipal sl(2, (C) subalgebra is a principal sI(2, C)-

subalgebra. For a Lie algebra of type Dr there are [(r - 2)/2] nonconjugated semiprincipal sI(2, (C)-subalgebras which are not principal. For the Lie algebras of types Er, r = 6, 7, 8, there are [(r - 3)/2] such subalgebras. Note that the classification of the semiprincipal sl(2, C) subalgebras of complex semisimple Lie algebras is reduced to the classification of the semiprincipal subalgebras of complex simple Lie algebras in the following way. Let g be a complex semisimple Lie algebra and let 0=g1X...Xgk

be its representation as a direct product of simple idoals. Consider a set of homomorphisms ti : sl(2, C) -> gi, i = 1,... , k, which specify semiprincipal subalgebras of Lie algebras gk. The mapping t : s((2, (C) -* g, defined by

t(x) - tl (x) + ... + tk lx), for all x E s((2, C), specifies a semiprincipal sI(2, (C)-subalgebra of

g. It appears that any semiprincipal sl(2, C)-subalgebra of g can be obtained in this way. From the definition of semiprincipal st(2, C)-subalgebras it follows that any sI(2, C)-subalgebra g' of a complex semisimple Lie

3.1 More about semisimple Lie algebras 2al + a2

149

hi + 2h2 h2

hi + h2

-hl

hl

-hl - h2 -a2

-h2

-hi - 2h2 Fig. 3.1

algebra g is a semiprincipal s[(2, C)-subalgebra of any minimal regular subalgebra of g containing g'. This fact allows one to formulate the following constructive procedure for finding all sl(2, C)subalgebras of a complex semisimple Lie algebra g. First, enumerate all semisimple regular subalgebras of g. This can be done using

the method described in subsection 3.1.2. Then, for each of these subalgebras, consider all its semiprincipalsi(2, C)-subalgebras. For any such subalgebra find the characteristic of the corresponding Cartan generator. Finally, comparing the characteristics obtained, single out the nonconjugated subalgebras.

For a detailed and rather explicit consideration of s[(2, C)subalgebras of complex simple Lie algebras we refer to Dynkin (1957a) and Lorente & Gruber (1972). Here we consider only one example.

EXAMPLE 3.7 Consider the case of the Lie algebra sp(2, C). As follows from table 3.2, there are three noncor jugated s((2, C)subalgebras. The Lie algebra sp(2, C) is of type C2. The corresponding root system is 0 = {±al, ±a2 i ±(al +a2 ), ±(2a1 +a2) }. This root system and the corresponding dual root system are de-

picted in figure 3.1, where the notation hi - (ai)", i = 1, 2, is used and the Weyl chamber corresponding to the base H = {al, a2} is coloured gray. The analysis based on the usage of the extended Dynkin diagram shows that one has four 7r-systems: IF1 = {al, a2}, 'P2 = {a1 }, T3 = {a2} and T4 = {a2, 2a1 + a2},

Differential geometry of Toda-type systems

150

which are not connected by transformations of the Weyl group W(O). The first 7r-system corresponds to the principal embedding and, as follows from table 3.3, the corresponding Cartan generator here is 3h1 + 4h2. For the next three cases, the Cartan generators are hl, h2 and h1 + 2h2. To find the characteristics, consider for each case a transformation of W (A) which brings the Cartan gen-

erator to the closure of the Weyl chamber corresponding to the base H. It is clear that this procedure gives new Cartan generators 3h1 + 4h2, h1 + 2h2, h1 + h2, and h1 + 2h2. Hence, the 7r-systems T2 and'I14 give conjugated s1(2, C)-subalgebras; therefore, one can exclude the 7r-system I4 from the consideration. The labels of the characteristic are calculated using the Cartan matrix. The charac-

teristics and the grading subspaces corresponding to the grading operator q = h/2 are 2

2

oc= 90 = 1

, 9t1 = gfai ® g±a2, 9±2 = 9f(a1+a2), 9±3 = 9+(2a1+012)

cD 0

2

gt(al+a2) +a2) ® 9}(2a' 90 = ® 9a' ® 9-a', 9±1 = g±22 ® 0

1

90 = ® 9a2 ®

9-012,

9±1/2 =

9fai

®

g±(a1+a2)

9±1 = g±(2a1+a2)

As we have seen, any s[(2, (C)-subalgebra of a complex semisim-

ple Lie algebra g gives an integral or a semi-integral gradation of g. The grading operator here is h/2. On the other hand, let a complex semisimple Lie algebra g be equipped with an integral or semi-integral gradation and let q be the corresponding grading operator. For any element x E g+1 there is an s((2, C) subalgebra such that x+ = x, h E go and x_ E 9_1i see Delduc, Ragoucy & Sorba (1992); Feher et al. (1992). Writing the grading operator in the form q = h/2 + y, one can prove that the element y belongs to the subspace go - C1° - {x E go I [xt, x] = 0}.

(3.25)

Thus, any integral or semi-integral gradation of g gives a set of sl(2, C) x g((1, (C)-subalgebras of g specified by the choice made

3.2 Zero curvature representation of Toda-type systems

151

for the element x+.

Exercises Construct the generalised Cartan matrices corresponding to the extended Dynkin diagrams for the simple Lie algebras of rank 2 and 3. 3.2 Verify that the Cartan generators of the principal embedding of sr(2, C) into complex matrix simple Lie algebras are given 3.1

by r

sr(r+1,(C) : h=>i(r-i+1)hi, i=1

r-1

o(2r + 1,(C) :

r(r + 1)

h = I:i(2r - i + 1)hi + i=1

2

hr,

r-1

sp(r, (C)

:

h=

i(2r - i)hi + r2hr, i=1 r-2

o(2r, (C)

:

i(2r - i - 1)hi +

h=

r(r - 1)

2

(hr-1 + hr).

i=1

Study the embeddings of sr(2, C) into complex matrix simple Lie algebras of rank 2 and 3; namely, find the characteristics and the grading subspaces. 3.4 Show the multiplet structure of the complex simple Lie algebras of rank 2 with respect to the integral embeddings of sr(2,C). 3.5 Consider the sr(2, (C)-subalgebra of sf(m, C), such that the corresponding representation of sr(2, C) has the splitting into irreducible components of the form m = k d, or of the form 3.3

m = k d + 1. Find the structure of the grading subspaces.

3.2 Zero curvature representation of Toda-type systems 3.2.1 Gauge transformations In the physical literature, principal bundle isomorphisms are often

called gauge transformations. In other words, a gauge transformation of a principal fibre G-bundle P - M is a diffeomorphism

152

Differential geometry of Toda-type systems

p : P -4 P satisfying the relations 1rocp=1r, co o RP = RP o co

(3.26) (3.27)

for any a E G. The notion of a gauge transformation is very important in modern mathematical physics; therefore, we consider it in a more general framework than is really ideal for the problems considered in the book. Let cp be a gauge transformation of a principal fibre G-bundle

P 4 M. Consider a set of local sections sa : Ua -+ P, a E A, covering M. Recall that the set {sa} generates a bundle atlas {Ua,ba}aEA, where the mappings is : i-1(Ua) -* Ua x G are defined with the help of the relation (3.28) (q, a) = sa(q) a. Note also that the mappings ga : i-1(Ua) -+ G, defined by

ga = prG o7a ,

satisfy the evident relation (3.29)

ga (p . a) = ga (p)a

for any a E G. Using the mappings ga, we can also write 0,,, (p) = (ir(p),ga(p)),

p E 7r

-1(Ua).

(3.30)

Proposition 3.1 For any (q, a) E Ua x G one has a) = (q, a (q) a), where the mappings cpa : Ua -* G are defined by Va o coo

a 1(q,

(Pa =ga 0(o0Sa.

Proof Using relations (3.28) and (3.27), we obtain

b,,oV04a1(q,a) = Ya0 p (sa(q).a). Now (3.30) and (3.29) give "Y)a o cp o

0,'(q, a) = (ir o cp o sa (q), ga op o sa (q)a)

Taking into account (3.26) and the relation 7 0 Sc, = idm,

we obtain (3.31).

(3.31)

3.2 Zero curvature representation of Toda-type systems

153

From (3.28), (3.31) and (3.30) it follows directly that WI U. (p) = sa(ir(p)) - (ya

(7r(p))ga(p)).

(3.32)

Using this relation, we can recover the mapping cp from the mappings cpa, a E A. We introduce some notations which will be used for mappings from a set to a group. Let cp be a mapping from a set S to a group G. Denote by cp-1 the mapping from S to G defined by

o 1(p) = Further, for any two mappings cp and 0 from S to G we use the notation y for the mapping from S to G defined by (W')(p) = W(p)O(p).

Proposition 3.2 On Ua n Up one has 'po = goo, cagao = goa(Pagoa,

(3.33)

where the gao are the transition functions of the bundle atlas on P, generated by the set {sa}.

Proof Formulas (3.28) and (3.30) imply (3.34) p = sa(ir(p))-ga(p) for any p E ir-1(Ua). Recall that the transition functions gap are given by (3.35) gao = ga o so. Now, from the definition of the mappings ga and from (3.34) and (3.35) we obtain go(p) = goa(ir(p))ga(p),

p E .7r-1(Ua n Uo).

(3.36)

Putting p = sp(q), q E Ua n Up in (3.34) we obtain the following relation:

so(q) = sa(q)gao(q)

(3.37)

Now writing oo(q) =gooW oso(q) and taking (3.36) and (3.37) into account, we obtain (3.33).

Proposition 3.3 Any set of mappings cpa, a E A, satisfying (3.33) defines a gauge transformation cp of P via (3.32).

154

Differential geometry of Toda-type systems

Proof First, we should show that (3.32) defines the mapping cp in a correct way. In other words, for any p E 7r-1(Ua fl UQ) we must have s«(ir(p)) - (co (ir(p))g.(p)) = sa(lr(p)) - ((P,3(ir(p))g3(p))

It is easy to show that, due to (3.33), (3.36) and (3.37), this equality is valid. Now we should verify the validity of relations (3.26) and (3.27). In fact, (3.26) is obvious, while (3.27) follows from (3.29). Proposition 3.4 Let w be the connection form of some connection on P. The g-valued 1-form co*w defines some new connection on

P. Proof Show that for the form cp*w the corresponding analogues of relations (2.70) and (2.71) are valid. From (2.12), for any v E g and any f E J(P) we obtain (cP*XP)cv(p)(f) = X P((P*f)

Hence, thanks to (2.64) and (3.27), X p((P*f) = dtf (cp(p) exp(tv))e-o = X VV(p)(f)

Therefore, one sees that cp*XP = XP.

Relation (2.70) now implies that cp*w(Xv) = v.

Using (3.27) and (2.71), for any a E G we obtain Ra *cp*w = cp*Ra *w = Ad(a-1) o

W.

Thus, the g-valued 1-form cp*w defines some connection on P.

Recall that the set {wa} of 1-forms Wa=SaW

completely determines the connection form w. The connection form co*w generates the set {(cp*w)a} of 1-forms, defined as ((P*W)a = s«co*w.

(3.38)

3.2 Zero curvature representation of Toda-type systems

155

Proposition 3.5 The following relation: (V*w). = Ad(cp 1) o wa + V *O,

with 6 being the Maurer-Cartan form of G, is valid.

Proof The statement of the propositions follows from (3.38) and (2.72).

Proposition 3.6 Let Il be the curvature form associated with the connection form w. Then the curvature form corresponding to the connection form cp*w, is cp*S2.

Proof From relation (2.74), with account of (2.50) and (2.51), one obtains cp*52 = dcp*w +

2

ko*w, cp*w]

Hence, cp*SZ is the curvature form of the connection determined by the connection form cp*w. The curvature form 1 is completely determined by the set {Sl } of the 2-forms

Sta = soft The corresponding forms determining the curvature form cp*SZ are defined by s

(3.39)

Proposition 3.7 The relation Ad((pa') o Q« is valid.

Proof The relation in question follows directly from (3.39) and (2.75).

3.2.2 Zero curvature condition Also in this chapter we consider flat connections on a trivial prin-

cipal fibre bundle. Let M be a manifold and let G be a Lie group. Consider the trivial principal G-bundle M x G -* M.

156

Differential geometry of Toda-type systems

Any such bundle has a bundle atlas consisting of only one chart (M x G, idMxG). Hence, as follows from the discussion given in section 2.8.2, we have in the case under consideration a bijective correspondence between connection forms and g-valued 1-forms on M. Bearing this correspondence in mind, we call a g-valued 1-form on M a connection form, or simply a connection. The curvature 2-form of a connection w is determined by the 2-form Q on M, related to w by the formula

Q=dw+2[w,w]. From (2.75) we conclude that the connection w is flat if and only if dw + 2 [w, w] = 0.

(3.40)

We call relation (3.40) the zero curvature condition. For any smooth mapping cp : M -* G, define a g-valued 1-form "w as (3.41)

`°w - cp*9,

where 0 is the Maurer-Cartan form of G. From (2.57) it follows that `°w satisfies the zero curvature condition. Note that if G is a matrix group, then co is a matrix valued function and one can write ww = (p-'dip.

It appears that if the manifold M is simply connected, then any connection satisfying the zero curvature condition has the form `°w for some smooth mapping cp : M -3 G. Note that `°w = 0 if and only if co is a constant mapping.

Proposition 3.8 For any two mappings cp,

M -4 G, the fol-

lowing relation:

Ad(O-i) o `°w +''w is valid. In particular, one has `°-lw = - Ad(V) o `°w.

Proof For any p c M and v E TP (M) we have wlp w(v)

= (w'P)*O(v) = e(W)*n(v)).

(3.42)

3.2 Zero curvature representation of Toda-type systems

157

It is not difficult to show that (,p')*p = RO(p)*w(p) o w*p + Lw(p)*IG(p)

Now, using (2.54) and (2.55), we arrive at (3.42).

o,P.p.

O

The gauge transformations in the case under consideration are

described by smooth mappings from M to G. Let 0 be such a mapping and let w be a connection form. Proposition 3.5 implies that the gauge transformed connection form w''' is given by the relation w" = Ad(V)-1) o w +,0'0. (3.43) For the case of a matrix group G, w is a matrix valued 1-form, ' is a matrix valued function, and (3.43) takes the form w11 =V-1wV)+V-ldVi. From proposition 3.7 it follows that the zero curvature condition is invariant with respect to the gauge transformations (3.43).

In other words, if a connection w satisfies this condition, then the connection w'' also satisfies it. It is convenient to call the gauge transformations defined by (3.43) G-gauge transformations. In fact, using proposition 3.8, one can easily show that (ww)VG = wOow

for any smooth mappings cp and z/.

Proposition 3.9 The equality `°w =

'P

W,

(3.44)

is valid if and only if co'cp-' is a constant mapping.

Proof Performing the gauge transformation corresponding to the mapping co-', from (3.44) we obtain the equality 0,P_iw=0; hence, cp'cp-1 is a constant mapping.

U

Actually, we will consider the zero curvature condition for the case where M is a complex one-dimensional manifold and G is a complex semisimple Lie group. It is convenient to use the notations

z- and z+, respectively, for a local coordinate z on M and its conjugate z. Write for w the representation w = w_dz- + w+dz+,

158

Differential geometry of Toda-type systems

where wt are some mappings from M to g. In what follows the superscripts minus and plus mean for 1-forms on M the corresponding components in the expansion over the local basis formed by dz- and dz+. In terms of wt the zero curvature condition takes the form

a_w+ -a+w-+[w-,w+]=0.

(3.45)

Here and in what follows we use the notation a_ - a1az-, a+ - a/az+.

Choosing a basis in g and treating the components of the expansion of w± over this basis as fields, we can consider the zero curvature condition as a nonlinear system of partial differential equations for the fields. Suppose also that the manifold M is simply connected. In this case any flat connection can be gauge transformed to zero. In this sense system (3.45).is trivial. On the other hand, the majority of two-dimensional integrable equations can be obtained from system (3.45) by imposing some gauge noninvariant constraints on the connection form w. Note that, in general, for the case of infinitedimensional Lie algebras and Lie groups one needs a generalisation of the scheme, see, for example, Leznov & Saveliev (1992), but in

the present book we restrict ourselves to the finite-dimensional case. Consider one of the methods of imposing the conditions in question, giving, in fact, a differential geometric formulation of the group-algebraic approach for integrating nonlinear systems in the spirit of Leznov & Saveliev (1992).

3.2.3 Grading condition Suppose that the Lie algebra g is a Z-graded Lie algebra. The first condition we impose, in accordance with Leznov & Saveliev (1992), on the connection w is the following. Let bt be the subalgebras of g given by (3.19). _Require that the (1,0)-component of the form

w takes values in b_, and that its (0,1)-component takes values in b+. We call this condition the general grading condition. Any connection w satisfying the general grading condition is certainly of the form `0w for some mapping co : M -* G; however not each mapping ep leads to the connection `°w satisfying this condition.

3.2 Zero curvature representation of Toda-type systems

159

Let us formulate the requirements which should be imposed on the mapping (p to guarantee the validity of the general grading condition. Note first that the connected subgroups B± of G corresponding

to the parabolic subalgebras bt are parabolic subgroups. Hence, the homogeneous spaces F± = G/B:F are flag manifolds. Let 7r+ : G -+ F+ be the canonical projections. Define the mappings (pt

M -* Ft by

(pf=7rt0(p. The mappings 7r f are holomorphic. Hence, for any a E G we have 7r±,,, ° Ja = J,r±(a) 0 7r+*a,.

(3.46)

Further, there are defined the natural left actions LF± of the Lie group G on FF, satisfying the condition 7rf o La = Lat o 7rL

(3.47)

for any a E G.

Theorem 3.1 The connection `°w satisfies the general grading conditions if and only if the mapping o_ is holomorphic and the mapping W+ is antiholomorphic.

Proof Suppose that the 1-form for any v E T (M) we have

takes values in b+. Thus, (3.48)

7r-*e['°w(PM(v))1 = 0,

where the linear operator PP projects the tangent vector v to its (0,1)-component, see (2.18). Using (3.41) we obtain from (3.48) the equality PP (v))] = 0. Taking (2.61) into account, we obtain the relation 7r*e o L,-'(P)*,v(P) 0 P (P) °

*r 0

PP = 0.

(3.49)

From (3.47) it follows that 7r+*ab 0 La*b =

LFt

j

(b)

0 7rf*b

for all a, b E G. Hence, -:e 0 I'P

1(P)*G(P)

=

0 7r_*W(P).

(3.50)

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Differential geometry of Toda-type systems

Further, relation (3.46) implies that lrt*a 0 Pa

=P

±(a)

0 7rt*a

for any a E G. Therefore, 7r-1w(P) o P

) = PP-(P) 0 ff-*w(r)

(3.51)

Using (3.50) and (3.51) we obtain the following equality from (3.49): F

F

M 0 PP = 0. Since, for any a E G the mapping La- is a diffeomorphism, we

o

0

-*P

conclude from this equality that P - (P) o (P-*P o PP = 0. (3.52) The mapping cp-*P is real; therefore, after the complex conjugation of (3.52), we obtain

JP

P"-(P) o'P_*p o PP = 0.

(3.53)

It follows from (3.52) and (3.53) that

F "-(P) O P-*P = (P-*P O `7

,

and so the mapping co- is holomorphic. Suppose now that the mapping co- is holomorphic. Reversing the arguments given above, we conclude that the form w(°,1) takes values in b+. The case of the mapping cp+ can be considered in the same way.

We call a mapping co generating a connection which satisfies the general grading condition a mapping satisfying the general grading condition.

EXAMPLE 3.8 Consider the case of G = SL(2, C) and endow the Lie algebra sl(2, C) with the principal gradation corresponding to

the choice of the Cartan and Chevalley generators described in section 1.3.1. In this case the flag manifolds FF are diffeomorphic to the projective space CIP1 and we will identify them. It is not difficult to show that the projections in can be defined as ir-(a) = (all : a21),

7r+(a) = (a12 : a22),

where

a=\all a12) a21

a22

3.2 Zero curvature representation of Toda-type systems

161

Define the functions co j : M - C, i, j = 1, 2, by (ij = gig 0 cP,

where the mappings gij : SL(2, (C) -4 C, i, j = 1, 2, are given by gia(a) = aid. Note that for any p E M either cp12 (p) or V22 (p) is different from zero. So in some neighbourhood of p at least one of the functions (P12/YP22 and cP22/cP12 is well defined. From theorem 3.26 it follows

now that if the mapping cP satisfies the general grading condition, then these functions are holomorphic. Similarly, in some neighbourhood of p at least one of the functions c011 /cP21 and V21 /cP11 is

well defined, and if cP satisfies the general grading condition, then they are antiholomorphic. Now, again following Leznov & Saveliev (1992), perform a further specification of the grading condition. Define the subspaces ii

of g by

m+ = ® gm, 1<m<1+

m- =-l_<m<-1 ® gm,

(3.54)

where It are some positive integers. Let us require that the (1,0)component of the connection w takes values in the linear space m- ® Cl, and that the (0,1)-component of it takes values in 11®m+. We will call such a requirement the specified grading condition. To reformulate this grading condition as a condition imposed on the corresponding mapping cp, introduce some holomorphic distributions on the manifolds F+ and F_ . Note that the subspace b_ ® m+ is invariant with respect to

the adjoint action of the subgroup b- in g. Let p E F+ and let a be any element of G such that 7r+ (a) = p. Define the subspace .M+p C

by M+P = 7r+*a(b-a (D m+a) = 7f++a(m+a)I

where

b-a = La*e(b-), rit+a = La*e(m+) The subspaces M+p, p E F+, generate a distribution on F+ which will be denoted by A4+. In the same way we can define an analogous distribution M_ on F_ .

162

Differential geometry of Toda-type systems

Theorem 3.2 The connection `°w satisfies the specified grading condition if and only if the mapping (p_ is holomorphic and tangent to the distribution M_ while the mapping (p+ is antiholomorphic and tangent to the distribution M+. Proof Suppose that the connection `°w satisfies the specified grad-

ing condition. Let p E M and v E T (M). Following the proof of theorem 3.1, we see that ww(Pp v) -=-

L`°-1(p)*W(p)IP'P

P) ° P*p 0

PP (v))

From this relation it follows that PA(P) o (p*p o Pp (v) E CI ®m+P. Therefore, 7r+*,,(p) o Pi(p) o (P*P O PP (v) E M+,+(p). Since the mapping i+ is holomorphic, while (p+ is antiholomorphic, we obtain +_V(p) o P(p) ° co*p ° PP = PW+(P) °

+*P

Hence, PF+(P)o(p+*P(x) E M+,o+(p); and the mapping (p+ is tangent

to the distribution M+. O Theorem 3.2 shows that the specified grading condition is directly related to the notion of a superhorisontal mapping, see Burstall & Rawnsley (1990).

It is clear that the specified grading condition is not invariant under the action of an arbitrary G-gauge transformation, but it is invariant under the action of gauge transformations (3.43) generated by the mappings taking values in the subgroup H corresponding to the subalgebra 6 = go. Such transformations form a subgroup of the group of G-gauge transformations. We call a gauge

transformation from this subgroup an H-gauge transformation. To obtain a system of equations having no H-gauge invariance, we should impose further restrictions on the connection form. In fact, following Leznov & Saveliev (1992) we choose another way leading to the system in question. It consists of constructing some H-gauge invariant mappings and rewriting the zero curvature con-

ditions in terms of them. To this end, let us first consider the structure of the holomorphic principal fibre bundles G -- F.

3.2.4 Modified Gauss decomposition Consider first the Gauss decomposition for matrices. Taking into account future applications, we deal with matrices over an alge-

3.2 Zero curvature representation of Toda-type systems

163

bra. In other words, we suppose that the matrix elements of the considered matrices are elements of an associative unital algebra. In this case it is convenient to use the theory of quasideterminants in the form given by Gelfand & Retakh (1991, 1992). Let T and J be two ordered sets, each consisting of m elements.

Consider a matrix a = (ai3), i E Z, j E J, with the matrix elements belonging to some associative unital algebra A. Define the family of m2 elements Ialk,, k E Z, l E J of the algebra A which are called the quasideterminants of a. In the case of m = 1 there is only one quasideterminant defined by lalkk - akk. Denote by a(icj) the matrix which is obtained from the matrix a by eliminating the ith row and jth column. The quasideterminant Ialkl of the matrix a is defined by akila(k;t)Ij-i la-l.

Ialki = aki iok,j541

It is clear that the quasideterminant Ialkl exists only if the quasideterminants I a(k;l) J,j for any i k and j 0 1 are invertible elements of A.

For the matrix a = C

all

ail )

a21

a22

one obtains four quasideterminants:

Ialll = all - alga-22 a21,

Ia112 = a12 - alga-la22, 21

I_1

-1

a121 = a21 - a22a12 all,

Ia122 = a22 - a2lalla12

Put bi3 = I aI i1; it appears that aikbkj = kEJ

bikakj = bijkEZ

Hence, the matrix b =- (bi3) is the inverse of the matrix a. If the algebra A is commutative, then Ialki = (_1)k+l deta det a(k;l)

,

(3.55)

and one arrives at the usual expression for the inverse matrix. We now introduce some more notations. Let /C C I and G C J. Denote by a(K;L) the matrix obtained from a by eliminating the rows with the indices from IC and the columns with the indices from G. The notation a(k;c) will be used for the submatrix of a

Differential geometry of Toda-type systems

164

composed of the matrix elements akl with k E IC and I E G. In other words, a(c;c) = a(Z Consider now the Gauss decomposition of matrices over an associative algebra with unit. Suppose that for the matrix a = (aid), i, j = 1, ... , m, there exist the quasideterminants h-kk = Ia(k,...,m;k,...,m)Ikk

Define m+kl = I a(k,1+1,...,m;l,...,m) (kl h _ll'

1
n-kl = ia(k,...,m;1,k+1,...,m) I kl h-fl,

1 < l< k < m,

1

and put h_kl = 0 for k l; m+kl = 0 for 1 < k, m+kk = 1; n-k1 = 0 for k < 1, n-kk = 1. One can now prove that

a = m+n_h_.

(3.56)

This decomposition is called the Gauss decomposition of the matrix a. Such a decomposition, if it exists, is unique. For the case where m = 2 one obtains -1 _ all - al2a22 a21 0 alzazz 1

h_- (

0

m+- (0

'

azz

0

1

n- _

_1

1

azl(all - alzazz azl)

)

1

.

1

Consider the case where the algebra A is commutative. Denote d-k = det a(k,...,m;k,...,m),

1 < k < m,

and put d-(m+1) 1. Now, using (3.55), one obtains d-k

h-kk =

m+kl =

n-kl =

d-(k+l)

1
'

det a(k,1+1,..,,m;l,...,m) d-1

1< k< l< m

det a(k,...,m;l,k+1,...,m)d-(1+1) d-(k+l)d-l

1 < l < k < m.

These are standard expressions for the Gauss decomposition of matrices.

There is another form of the Gauss decomposition. It is constructed as follows. Let us suppose that for the matrix a = (air), i, j = 1, ... , m, there exist the quasideterminants h+kk = la(1,...,k;1,...,k)Ikk

3.2 Zero curvature representation of Toda-type systems

165

Define

1<1
Tn- kl = l a(1,...,1-l,k;l,...,1) I k(h+111 n+kl = la(1,...,k;1,...,k-1,1)Ikth+11,

and put h+kl = 0 for k # 1; m_kl = 0 for k < 1, m-kk = 1; n+kl = 0 for 1 < k, n+kk = 1. In this case one has

a = m_n+h+. For the case where m = 2 one obtains all 0 h

m-=I

I

+

a22 - a21a111a12

0

n+ _

1

C

(3.57)

1

a21a111

0

- a21a111a12)-1

a12(a22

0

1

Consider again the case where the algebra A is commutative. Here we denote

1 < k < m., and put d+o - 1. Again using (3.55), one obtains dl_, - deta(1,...,k;1,...,k),

h+kk

=

d+k

1
d+(k-1)

det

m_k,=

1,k;1,...,1) ,

1<1
d+1

n+k1=

det a(1,...,k;l,...,k-1,1)d+(1-1) d+(k-1)d+l

1
One can use (3.56) and (3.57) to rewrite the Gauss decomposition in various different forms. For example, defining k- - m_, h - h+ and k+ = h+ln+h+, one obtains from (3.57) the equality

a = k-hk+. In particular, for the case where m = 2 one obtains 0 all 0

-1 a22 - a21all a12

a2lall 1

1

01

1)'

For more information about quasideterminants we refer the reader to the papers by Gelfand & Retakh (1991, 1992). We now proceed to the case of Lie groups. Let G be a complex connected semisimple Lie group. Decomposition (3.11) of the

166

Differential geometry of Toda-type systems

corresponding Lie algebra g leads to the following decomposition of G. Let Nt and H_be the subgroups of G corresponding to the subalgebras n+ and Cl. It can be shown that

G=N+N_H=N_N+H=N_HN+, where the bar means the closure. Thus, any element a belonging to some dense subset of G can be presented in one of the forms

a=m+n-h-, a=m-n+h+, a=k_hk+, where m+, n±, k± E N± and h±, h E H. The representation of an element of G in any such a form is called the Gauss decomposition. It is clear that if an element of G possesses the Gauss decomposition of one or another form, then such a decomposition is unique. EXAMPLE 3.9 Consider the case of the Lie group SL(2, C). This group consists of all complex 2x2 matrices a = C all a21

a12 ) a22

satisfying the condition a11a22 - a21a12 = 1.

The subgroups N_ and N+ are formed by the matrices of the form n-= 1 O l n+ _ ( 1 b )

\

a 1'

0

1J

respectively. The subgroup H consists of the diagonal matrices

_

h

c

0

0

1/c

with c # 0. The first form of the Gauss decomposition exists for all elements of SL(2, C) with a22 0. A concrete expression here is

a=m+n_h_=(01

a12/a22

0

1

)

1

a21 s 22

1)(

1/a22 0

An element of SL(2, (C) can be represented with the help of the second form of the Gauss decomposition if all # 0. Here we have

a= m_n+h+ =

1

a21/all

0)(1 a12a11

all

0

0

1

1

0

1/all

3.2 Zero curvature representation of Toda-type systems

167

The third form of the Gauss decomposition exists simultaneously with the second form. This decomposition is given by 1 1 0 all 0 a12/all a = k_hk+ = 1/all ( 0 1 a21/all 1) 0 As we can see, the Gauss decomposition is not a global one and not each element of a complex semisimple Lie group possesses it. Below we need a decomposition applicable for an arbitrary element. To construct such a decomposition we start by proving three useful lemmas, the second of which belongs to Harish-Chandra (1953).

Lemma 3.1 Let b be an arbitrary element of G. The set N+flbB_ consists of at most one element of G.

Proof Suppose that n1, n2 E N; rl bB_; then there exist the elements bl, b2 E B_ such that

nl = bbl, n2 = bbl. These relations imply the equality nl 1n2 = bi 1b2. Note that N+fB_ = {e}. Hence, we have ni ln2 = bi'b2 = e.

Lemma 3.2 Let G be a Lie group; and let H, K be such Lie subgroups of G that

g=h®t.

(3.58)

The mapping X : (h, k) E H x K H hk E G is regular at any point

(h,k)EHxK. Proof

Actually, due to (3.58) we should prove only that ker X*(h,k) _ {0} for any point (h, k) E H x K. Suppose that for some u E T(h,k) (H x K) we have X*(h,k) (u) = 0.

(3.59)

The discussion given in example 2.16 shows that there are unique tangent vectors v E Th (H) and w E Tk (/K) such that u = tk h(v) + th*k(w),

Differential geometry of Toda-type systems

168

where the mappings tH : H -4 H x K and th : K -* H x K are given by (3.60) th (k) = (h, k). On the other hand, there are unique tangent vectors v' E Te (H) _ and w' E Te (K) = t such that (3.61) w = Lk v = Lh e(v'), Using relations (3.60) and (3.61), we write equality (3.59) in the form

th (h) = (h, k),

e(w').

X*(h,k)(tk*h(Lh**e(v')) + th**k(Lk*e(W'))) = 0.

(3.62)

It is not difficult to verify that X o tk o Lh = Lhk o Lk t o Rk o tH,

Xoth oLk =LhkotK, where tH and tK are the inclusion mappings of H and K into G. These relations show that equality (3.62) is valid if and only if Lh *e (Ad(k-' )v' + w') = 0. Since for any a E G the mapping La is a diffeomorphism, we have Ad(k-')v' + w' = 0, that is, equivalent to v' + Ad(k)w' = 0. This is possible if and only if v' = 0 and w' = 0. Therefore, u = 0, and ker X*(h,k) = {0}.

Lemma 3.3 The set ir+(N+) is an open set. Proof From lemma 3.2 it follows that the mapping X : (n, b) E N+ x B_ nb E G is regular for any (n, b) E N+ x B_. Then the inverse mapping theorem implies that for any (n, b) E N+ x B_ there is an open set U(n,b) such that (n, b) E U(n,b) and the set X(U(n,b)) is open. Hence, the set N+B_ is open. On the other hand, the mapping 7+ is an open mapping; thus 7r+ (9+) is an open set. Using the assertions of lemmas 3.1 and 3.3, we can define a local

section s+ of the fibre bundle G -* F+, assuming that s+ (p) = N+ n (7r+)-1(p),

p E 7r+(N+)

Since N+ is a submanifold of a complex manifold, the section s+ is holomorphic.

3.2 Zero curvature representation of Toda-type systems

169

The subspaces n+a - La*e(n+), a E G, generate an involutive holomorphic distribution N+ on G. The image N+ of the section s+ is an integral manifold of this distribution. We can also say that the mapping s+ is tangent to the distribution N+. Note that for any a E G, the set aN+ is a maximal integral manifold of the distribution A'+, and any maximal integral manifold of N+ has such a form. The following proposition is now almost evident.

Proposition 3.10 There exists an open covering {U+a}aEA of the manifold F+ and a family of local holomorphic sections s+a :

U+a -4 G, a E A, of the fibre bundle G -* F+ such that for any a E A the section s+a is tangent to the distribution N+. Proof As the first element of the required covering _and the cor-

responding section we can take the set U+ = 7r+(N+) and the section G+ defined above. Let p 0 U+, and a E (ir+)-1(p). The set aN+ possesses properties similar to the properties of the set N+. Namely, if aN+ fl bB_ # 0, then the set aN+ fl bB_ contains just one point, and the set it+(aN+) is open. Therefore, we can define a local holomorphic section s+ : U+ -> G, where U. = 7r+(aN+). Here the set U+ contains the point p. Repeating this procedure, we obtain a family of local holomorphic sections of the fibre bundle G -* F+ with the required properties. D Actually, we shall consider families of local sections constructed

with the help of the procedure used in the proof of proposition 3.29. In this case, for any a E A we have s+«(U+a) = a+aN+ = N+a

for some a+a E G. It is clear that a similar family of local sections

can also be constructed for the fibre bundle G -* F_. If the Lie algebra g is endowed with an involutive antilinear automorphism consistent with the Z-gradation, such a family of sections of the fibre bundle G -* F_ can be constructed on the basis of the given family of sections of the fibre bundle G -* F+. The corresponding method for this is considered in section 3.3.4. As was discussed in section 2.8.2, any family of holomorphic local sections of a holomorphic principal fibre bundle covering the base space allows one to define a bundle atlas. The corresponding

170

Differential geometry of Toda-type systems

procedure in our case looks as follows.

For any fixed a E A, consider a holomorphic mapping m+a (7r+)-1(U+a) -* G defined as a E (7r+)-1(U+a). m+«(a) = s+a(7r+(a)), This mapping allows one to introduce another holomorphic mapping b_a defined on (7r+)-1(U+a) by

b-a(a) = m+a(a)a. Thus, for any a E (7r+)-1(U+a) we can write a = m+a(a)b_a(a). Since s+a is a section of the fibre bundle G -+ F+, i.e.,

(3.63)

7r+ o s+a = idu+a ,

we have 7r+(m+a(a))

= 7r+(a),

and from (3.63) it can readily be seen that the mapping b_a takes values in the subgroup B_. Note here that the mappings m+,, and b_a have the following properties: m+a(ab) = m+a(a),

b_a(ab) = b_a(a)b

(3.64)

for any b E B_ . It is clear that the mapping O+a : (7f+)-1(U+a) + U+a X B_, defined as 'zP+a (a) = (ir+(a), b-a (a)),

provides a bundle chart on the fibre bundle G -+ F+. Considering all possible values of the index a, we obtain an atlas of this fibre bundle. Let a be an element of G such that 7r+(a) E U+a f1 U. In this case, using (3.64), we can write b-.(a) = b_a(s+Q(p)b_ (a)) = b_a(s+a(p))b_,Q(a),

where p = 7r+(a). Hence, we have b-.(a) = b_.0(7r+(a))b_ (a),

(3.65)

where b-,,o = b_a o s+a.

It is clear that the mappings b_aa, a, 0 E A, are the transition functions of the atlas we have defined. These transition functions are obviously holomorphic.

3.2 Zero curvature representation of Toda-type systems

171

Again let 7r+ (a) E U+,, (1 U. In this case we have a = m+a(a)b_a = m+,Q(a)b-a(a). This relation, taking account of (3.65), gives

m+a(a) =

m+a(a)b-a,o(7r+(a)).

3.66)

Proposition 3.11 The groups Bt have the holomorphic decomposition

B+ = N+H.

Proof Since b+ = n+ T 6, then, as was done in the proof of lemma 3.3, one can show that iv-+i,- is an open subset of B+. The subspace n+ is an ideal of b+. Therefore, N+ is a normal subgroup of B. Hence N+H is a subgroup of B+. The Lie algebras of N+H and B+ coincide. Thus, these Lie groups also coincide. The case of B_ can be considered in a similar way.

The last proposition implies that we can uniquely represent the mapping b_a in the form (3.67) b-a = n_ah_a, where n_a and h-a are holomorphic mappings from (1r+)-1(U+a)

to the subgroups N_ and H respectively. Analogously, for the transition functions b_aa we have b_aa = n_aah_,O,

where n_ap and h_aa are holomorphic mappings from U+a fl U+a

to N- and H respectively. From (3.64) we obtain the following relations: M-4-a(an) = m+a(a),

m+a(ah) = m_a(a),

n_a(an) = n_a(a)h_a(a)nh-1(a),

(3.68) (3.69)

n-a(ah) = n-a(a), (3.70) h-,(an) = h-a(a), h-a(ah) = h-a(a)h, (3.71) which are valid for any n E N_ and h E H. Using (3.65), for any a E G such that p = 7r+ (a) E U+a n U+Q, we also obtain

n-,(a) = n-cQ(p)h-aa(p)n_ (a)h-'a(p), h-.(a) = h-aQ(p)h_ (a).

(3.72)

(3.73)

172

Differential geometry of Toda-type systems

Proposition 3.12 Any element a E (7r+)-1(U+a)

can

be uniquely represented in the form (3.74) a = m+n_h_, where m+ E N+a, n_ E N_, h_ E H. The elements m+, n_, and

h_ holomorphically depend on a.

Proof We come to representation (3.74) putting h- = h-a (a). n- = n-« (a), The uniqueness of decomposition (3.74) follows directly from the m+ = m+a (a),

fact that N+fN_ =N+lH=N-flH={e}. We now have a proposition similar to the previous one.

Proposition 3.13 Any element a E

N_aB+ can

be uniquely represented in the form

a = m_n+h+, (3.75) where m_ E N_a, n+ E N+, h+ E H. The elements m_, n+, and h+ holomorphically depend on a.

We call decompositions (3.74) and (3.75) the modified Gauss decompositions.

3.2.5 Toda-type systems Now we will use the modified Gauss decomposition to define the required H-gauge invariant mappings and to derive the equations that they satisfy.

Proposition 3.14 Let cp : M -+ G be an arbitrary mapping and let p E M. (i) There exists an open neighbourhood V+ of the point p such that the mapping cp, being restricted to V+, has the unique decomposition O = µ+v-71

(3.76)

for some a E A, while where the mapping p+ takes values in the mappings v_ and rl_ take values in N_ and H respectively.

3.2 Zero curvature representation of Toda-type systems

173

(ii) There exists an open neighbourhood V_ of the point p such that the mapping co, being restricted to V_, has the unique decomposition cP = P-v+71+,

(3.77)

where the mapping µ_ takes values in N__a for some a E A while the mappings v+ and q+ take values in N+ and H respectively.

Proof The proof is based on the modified Gauss decomposition (3.74). It is clear that we can find a E A such that cP(p) E (ir+)-1(U+a) Define a mapping p+ by

+=m+aocp=s+ao7r+ocP=s+aocP+.

(3.78)

The domain of the mapping p+ is the open set V+ = cP+1(U+a). We now introduce the mappings

v-=n_,oW,

7l_=h_a0

(3.79)

with the same domain and arrive at the required decomposition (3.76).

The second part of the proposition can be proved in a similar way.

Corollary 3.1 For any p E M, there exists an open set V such that p E V, and the mapping co, being restricted to V, possesses decompositions (3.76) and (3.77) simultaneously.

Proposition 3.15 If the mapping cp : M -> G satisfies the specified grading condition, then the mapping µ_ is holomorphic and the holomorphic 1-form l`-w takes values in m_, while the mapping p+ is antiholomorphic and the antiholomorphic 1-form µ+w takes values in mom.

Proof Since, by definition, P- = s-a 0 cP(3.80) and the mappings s-a and cp_ are holomorphic, then lt_ is holomorphic.

From (3.80) we see that the mapping y_ is tangent to the distribution N_; therefore, Pµ (r) o p_,r(v) E n_µ_(r) for any v E TP (M). On the other hand, from (3.80) we obtain

1r-O/-=cp_.

174

Differential geometry of Toda-type systems

Recall that the mapping_ cp_ is tangent to the distribution M_; hence, Pµ (r)°N'-*v(v) E b+µ-(r)®m-a-(r). Thus, P' op-.,(v) E m_µ-(P). Writing the equality µ-w(v) =

o

Pµ (r) o N

*r(v),

we conclude that the 1-form µ- w takes values in m_ .

Let xt be some fixed nonzero elements of gt,t . For the case of 1+ = l_ = 1, when we consider a Z-gradation of g associated with an integral embedding of s((2, (C) into g, we can, in particular, take as x+ the corresponding elements defined by this embedding. Let

Cat be the orbits of the elements xt generated by the restriction of the adjoint action of the group G to the subgroup H. Note that the orbits 0± have the following property. If the element x

belongs to 0+ (0-), then, for any nonzero c E C, the element cx also belongs to 0+ (0_). This statement follows from the fact that the grading operator q generates similarity transformations of the subspaces g±,±. Denoting by H+ the isotropy subgroups of the elements xt, we identify the orbits 0± with the homogeneous spaces ii/Ht. More precisely, we establish such identification putting into correspondence to a coset hH±, h E H, the element x(hH±) E Cat given by

x(hH±) = Ad(h)x±. Denote by m+ the subset of m+ which consists of the elements with the l+th grading component belonging to 0+. Similarly, m' consists of the elements of m_, which l_th grading component

belongs to 0_. The corresponding subsets of M± will be denoted M+p. We call a mapping cp : M -* G an admissible mapping, if it satisfies the specified grading condition, and, moreover,

cp_,r(a_r) E .M' P and cp+,r(a+r) E M+r for any p E M. Due

to the properties of the orbits 0t discussed above, this definition does not depend on the choice of the local coordinate z = z. If the mapping cp is admissible, then we can write A±w =

A dz± _

A±mdz

(3.81)

m=1

where p± are the mappings arising from the local decompositions

(3.76), (3.77); and A±,,,, 1 < m < l± - 1, are the mappings tak-

3.2 Zero curvature representation of Toda-type systems

175

ing values in gtm, while the mapping A±1± takes values in O. The factor is introduced in this relation for the future convenience. The mappings A±m are defined in the open set V from corollary 3.1. It follows from proposition 3.15, that the mappings are holomorphic, and the mapping A+,,, are antiholomorphic. Let y± be local lifts of the mappings Ati f to the group H. These

mappings are defined in some open set W C V, and satisfy the relations

Atjt = Ad(y±)xt.

(3.82)

Note that in the case when the groups H+ are nontrivial, the mappings y± are defined ambiguously, but in any case they can be chosen in such a way that the mapping y_ would be holomorphic, and the mapping y+ would be antiholomorphic. In what follows we use in our consideration such a choice.

Theorem 3.3 Let cp : M --4 G be an admissible mapping. There exists a local H-gauge transformation that transforms a connection `°w to the connection w of the form

w=

(x- + E v-m

dz-

m-1

l+-1 V+m + x+)dz+,

+ \/_-l Ad('Y-1)

(3.83)

m=1

where y is a mapping taking values in H and vtm are mappings taking values in gym.

Proof Using representation (3.77) and proposition 3.8 we can write ww = µ-ll+n+w

= Ad(rl+lv+l)(,`-w) +Ad(r1+1)(°+w) +I+w. (3.84)

In the same way, representation (3.76) gives ww = µ+"-'' w = Ad(rl-'v.1)(µ+w)+Ad(r1=1)( w)+n-w. (3.85)

The form µ-w is holomorphic, and the form µ+w is antiholomorphic, therefore,

Differential geometry of Toda-type systems

176

Taking these relations into account, we arrive at the following consequences of (3.84) and (3.85): (`°w)± = Ad(,qf1)(v:Iw)± + (I+w)t. Consider now the mapping K

(3.86) (3.87)

+1/1_.

Proposition 3.8 provides the relation "w = µ-w - Ad(r,-1)(µ+w). Using (3.81), we obtain from this relation the equalities

A. (law)- = Thanks to decompositions (3.76) and (3.77), we conclude that the mapping is can also be represented as ("w)+ = - v / - - l Ad(r,-1)A+,

K = v_r7y+1.

(3.88)

rl = 77-71+

(3.89)

where

Representation (3.88) leads to the equality kw (3.90) = Ad(v+rl-1)("-w -'1-1w - Ad(rl)(v+w)), which results in the formula Ad(rJv+1)A- = (v-w)- - (''-'w)- - Ad(rl)(v+w)-. (3.91) Taking the n--component of (3.91), we obtain the relation (3.92) (v-w)- = vi(Ad(i7v+1)A_)5_. Therefore, the mapping (v-w)_ takes values in m' and we can

write

V.

Ad((rlY-) 1)(v w) _

(3.93)

m.=1

It is not difficult to show that v_1_ = x_. Using (3.93), we obtain from (3.86) the following equality: `°w)_ _ V- I Ad(r1+17-)

V_m +

w)_.

(3.94)

M=1

Similarly, it follows from (3.90) that

-Ad(rI-1v=1)A+ = Ad(rl-1)(v-w)+ + ("w)+ - (v+w)+, and the n+-component of this relation is (v+w)+

(3.95)

3.2 Zero curvature representation of Toda-type systems

177

Hence, the mapping (v+w)+ takes values in m+ and we can write 1+

Ad(`Y+1i)(v+w)-

(3.96)

v+m. m=1

Here v+i+ = x+ and the analogue of (3.94) is 1+

v+m +

(`°w)+ = V-1 Ad(r1-1'Y+)

(3.97)

m=1

We now perform the gauge transformation defined by the mapping q+1-y_. Taking (3.94) into account and (3.97), we conclude

that this gauge transformation brings the connection `'w to the connection of the form (3.83) with the mapping -y given by ly =

(3.98)

y+1711Y-

This is to be proved.

Thus we see that, up to a local H-gauge transformation, we consider the connections of form (3.83). In what follows we deal with the matrix Lie groups only. In this case we have

Vr-

w- _

E v-m +'Y-1a-'y,

y-'v+m

w+ _

y. (3.99)

m=1

M=1

Let us write explicitly the equations for 'y and v±m which follow from the zero curvature condition (3.45). We restrict ourselves to the case of l- = 1+ = 1; the generalisation to the case of l+ is straightforward. Using (3.99), we obtain a_w+ - a+w- _

a-('Y-1v+m'Y) M=1

- , a+2-m - a+(i 1a 7).

(3.100)

m=1

The same relations give t

I

[v-m,

[w-, w+]

`Y-1v+n'Y]

M'n=1

+

[-Y-la-Y, 7-lV+m'YlM=1

Taking into account the identity

we obtain the relation ['Y-1a-'Y,'Y-1v+m'y]

= -a-('Y-1v+m-Y) + y-,(a-v+m)`Y.

178

Differential geometry of Toda-type systems

Therefore, we have l

[W-, w+] =

- m,n=1 [v-m,

'Y-1

v+n'Y]

a-(-y-1v+m-y) +

'

1

m=1

>

v+m)1'. (3.101)

m=1

Finally, relations (3.100) and (3.101) result in the following system of equations: 1-m

['Y_1V+n7,V-(m+n)],

V-101+V-m =

(3.102)

n=1 I

a+(-Y-la--Y) = >2 ['-IV+m7, V_,n],

(3.103)

m=1

!-m i a_V+m

=

YV-n'Y-1, V+(m+n)],

[

(3.104)

n=1

where v±I = xt. This form of writing the equations is equivalent to that given in Gervais & Saveliev (1995). For the case where 1 = 1 one obtains the following equation: [y-1x+y,

(-r-1a-y)

(3.105) x-] = The equations for parameters of the group H which follow from (3.105), are called the Toda equations. In the case of a principal gradation, the subgroup H coincides with some Cartan subgroup H of G and is, by this reason, an abelian subgroup. The corresponding equations in this case are called the abelian Toda equations. In the case of a Z-gradation associated with an arbitrary integral embedding of sI(2, C) into g, the subgroup H is not neca+

essarily abelian and we deal either with the abelian Toda equations or with some of their nonabelian versions.

EXAMPLE 3.10 Let us derive the concrete form of the abelian Toda equations. In the case under consideration, we can locally parametrise the mapping y by the set of complex functions fi as r

y = exp E f ihi

(3.106)

i-1

where r is the rank of and the elements hi E 1l are the Cartan generators. Choose as the elements xt E 9±i the elements

3.2 Zero curvature representation of Toda-type systems

179

describing the corresponding principal embedding of 51(2, C) in g. The concrete form of such elements is given by (3.24). Using (1.18), we obtain the relation y x+'Y = E(2ki)'/2 exp[-(kf )i]x+Z, i=1

where

r

kijfj.

(kf)i = j=1

From this relation one immediately obtains r

2ki exp[-(kf )i]hi

['Y-1x+'Y, x-) = i=1

On the other hand, it is clear that r

a+

(y-1a-y)

a

=

(a+a-fi)hi

i=1

Thus, in the case under consideration, equations (3.105) can be reduced to the system

a+a-fi = 2ki exp[-(kf)i].

(3.107)

Introducing the functions vi = (kf )i - ln(2ki), we rewrite equations (3.107) in the form r

kij exp(-vj),

a+a-vi =

j=1

which is standard for the abelian Toda equations.

3.2.6 Gauge invariance and dependence on lifts Consider now the behaviour of the mappings -y and vt, under the H-gauge transformation. Let cp' = cp2', where the mapping,0 takes values in H. It is clear that to define the mappings -y' and v'+m. corresponding to the mapping gyp', we can use the same modified Gauss decompositions that we have used for the construction of the mapping -y. From relations (3.78) and (3.68) we have p' = p+.

Using the same lift from H/H+ to H as in transition from the mapping p+ to the mapping y+, we obtain y+ = y+. On the other

Differential geometry of Toda-type systems

180

hand, relations (3.79) and (3.71) give rl' In a similar way we have ry' = -y_, and q' = rl+V. Thus, it follows from (3.98) that

ry' = 'y. Further, equality (3.70) gives v' = v_; therefore, from

,

(3.93) it follows that v' = v_,,,,. Similarly, we obtain v',,,, = v+r,,,. Just in this sense the mappings -y and v+,, are H-gauge invariant. One more question that we are going to consider in this section

is the dependence of the mappings y and v±, on the choice of modified Gauss decompositions and local lifts from H/H+ to H. Suppose that we have two local decompositions of the mapping cp:

P=/+v-rl

0 =P+v-r1-,

which are obtained with the help of the modified Gauss decompositions corresponding to the indices a and,3 respectively. Using (3.66) and (3.67), we obtain + = m+Q ° 1p = A+v-a,orl-aQ,

(3.108)

where v-aQ = n-a,Q o w+, 71-a0 = h-,,o ° + It is obvious that the mappings v_ap and 77_ap are antiholomorphic

and take values in N_ and H respectively. From (3.108) it follows

that µ+w

= Ad(rla)o)(Ad(vaa)('+W) + '-1w) +

?

w.

Taking (3.81) and (3.82) into account, one obtains Ad(y+)x+ = Ad(rl_aa'Y+)x+ It follows from this relation that (3.109)

"Y+

where the mapping + takes values in H+. It is clear that the mapping + is antiholomorphic. Further, (3.73) allows one to write 77_ =

'1-a,3gl .

Combining (3.109) and (3.110), we obtain 1-1

1

,

-1

'Y+ ?L _ + 7+ rl and, in a similar way, 1-1 77+

,

-1

'Y- _ q+

-Y--,

(3.110)

3.3 Construction of solutions and reality condition

181

where _ is a holomorphic mapping taking values in H_. Finally, we obtain the relation ry' = +1-y _. (3.111) For the mappings v±,, one obtains (3.112) vfm = Stlvtmt;±. Resuming our discussion, we can say that any admissible map-

ping cp leads to a set bi, (V±m)i}iEZ of local solutions of equations (3.102)-(3.104). These solutions, in the overlaps of their domains, are connected by{{ the relations 7i = S+ j7j6-ij, (vfm)i = c± j(v±m)jc±ij,

where the mappings _ij are holomorphic and take values in H_, while the mappings C-+ij are antiholomorphic and take values in H+. Note here that (3.111) and (3.112) describe symmetry transformations of equations (3.102)-(3.104). For the case l = 1 and the principal gradation, the subgroups Ht are discrete.

Exercises Find explicit expressions for the quasideterminants of a 3x3 matrix over an associative unital algebra. 3.7 Let a = (aij ), i, j = 1, 2, 3, be a matrix over an associative algebra with unit. Prove the following Silvester identities: 3.6

Ia133 = 1a(1,3;1,3) 133 -

1

Ia(1,3;1,2) 1321a(1,2;1,2)122 Ia(1,2;1,3) 123,

IaI33 = Ia(2,3;2,3) 133 - Ia(2,3;1,2) 131 Ia(1,2;1,2)I111 Ia(1,2;2,3) 113

Construct explicit expressions for the matrices entering the Gauss decomposition of a 3x3 matrix over an associative unital algebra. 3.9 For the Lie groups SL(m,C), SO(m,C) and Sp(2n, (C) find the subgroups H± for the case where 1 = 1 and the principal gradation. 3.8

3.3 Construction of solutions and reality condition 3.3.1 General solution of Toda-type systems In the preceding section we have shown that any admissible mapping cp : M -+ G allows one to construct a set of local solutions

182

Differential geometry of Toda-type systems

of equations (3.102)-(3.104). At first glance, the problem of constructing admissible mappings is rather complicated. On the other hand, to construct a solution we do not need to know the mapping cp itself. It is sufficient to deal with the mappings y± entering the Gauss decompositions (3.76) and (3.77); see Leznov & Saveliev (1992). Indeed, once we know the mappings p± corresponding to some admissible mapping cp, then using (3.76), (3.77) and (3.89), we obtain (3.113) µ+lµ_ = v_rly+'.

This relation implies that we can find the mappings rl and of considering the corresponding Gauss decomposition of the mapping tc+lµ_. The mappings yt are determined by relation (3.82). Knowing the mappings 71, v± and yt, we construct the mappings y, and v+m using (3.98), (3.94) and (3.97) respectively. Thus, to construct solutions of equations (3.102)-(3.104) we should specify the mappings pt : M -* G. It appears that not each pair of such mappings gives a solution. Actually we must use only the mappings µ+ arising in the generalised Gauss decompositions of some admissible mapping cp : M -* G. To clarify the situation arising let us return to consideration of the mappings cpf. We call a pair of mappings cpt : M -* F± consistent if there exists a mapping cp : M -+ G such that go±=7rfocp.

If the mappings cp f are consistent, then the corresponding mapping cp is defined up to an H-gauge transformation. To show this, let us prove the following simple lemma.

Lemma 3.4 Let a and a' be two arbitrary elements of G. The equalities

7± (a) = 7r± (a)

(3.114)

are valid if and only if a' = ah for some element h E H.

Proof Suppose that the equalities (3.114) are valid; then a' = ab+ a' = ab_, for some elements b+ E B+. Since b- fl B+ = H, therefore b- _ b+ - h E H. The inverse statement is obvious. D

3.3 Construction of solutions and reality condition

183

Using this lemma, it is easy to show the validity of the following proposition.

Proposition 3.16 Let cp and
are valid if and only if cp' = cp0, where the mapping 0 takes values

in H.

Now formulate a useful criterion which allows one to check the consistency of the mappings cpt. Let us start with one more lemma.

Lemma 3.5 Let p_ and p+ be two arbitrary point of the flag manifolds F_ and F+ respectively. Consider arbitrary elements m= E G satisfying the relation it+(m+) = p±. An element a E G such that (3.115) 7± (a) = p± exists if and only if the element m+lm_ has the following Gauss

decomposition:

m+ lm_ = n_hn+1,

(3.116)

where hEfl and n+E11±. Proof Suppose that the Gauss decomposition (3.116) exists and define

a - m+n_h = m_n+. It is clear that 7r± (a) = 7r± (m±) = p±. On the other hand, suppose that the element a E G, satisfying relations (3.115), exists. Then

a = m+n_h_ = m_n+h+ for some m+, n+ E N+, m_, n_ E N_ and h± E H. Therefore, we obtain the Gauss decomposition (3.116) with h = h_h+'.

Now it is easy to prove the following proposition.

Proposition 3.17 Let cp_ and cp+ be two mappings from M to F_ and F+, respectively, and let p+ : M -* G be arbitrary mappings

184

Differential geometry of Toda-type systems

from M to G, satisfying the condition

f ° µf = cPf. The mappings V± are consistent if and only if the mapping µ+'phas the following Gauss decomposition:

µ+'µ_ = v_iv+',

(3.117)

where the mappings v_ and v+ take values in N_ and N+, respectively; and the mapping q takes values in H.

Recall now that the mappings p± arising from the Gauss decompositions of an admissible mapping cP satisfy the following conditions. First of all, the mapping µ_ is holomorphic, and the mapping µ+ is antiholomorphic. Further, the mapping µ='a_µ_ takes values in m' , and the mapping µ+' a+µ+ takes values in m+. We call a pair of the mappings pt : M -* G satisfying these conditions and having the Gauss decomposition (3.117), regular.

Proposition 3.18 Any regular pair of mappings p± : M -* G arises from the corresponding Gauss decompositions of some admissible mapping cp : M -* G. Proof Let /-z± be a regular pair of mappings. Using (3.117), define

the mapping y : M -+ G by 1P = µ+v-77 = µ-v+.

It is not difficult to show that the mapping co is admissible. More-

over, it is clear that the mappings µt are the mappings arising when we perform the corresponding Gauss decompositions of the mapping cp.

Thus, we obtain the general solution of equations (3.102)(3.104) using all regular pairs of mappings µt.

Proposition 3.19 Let µ' and p± be two regular pairs of mappings. If

it °µt=i+°lif,

(3.118)

then the corresponding solutions of equations (3.102)-(3.104) are

connected by a symmetry transformation of the form (3.111), (3.112).

3.3 Construction of solutions and reality condition

185

Proof The proof of the proposition follows the lines of section

0

3.2.6.

The mappings µt entering a regular pair are, by definition, tangent to the distributions N±. Since these distributions are involutive, the mappings µt take values in maximal integral manifolds of Nt. As we have already noted in section 3.2.4, the maximal integral manifolds of the distributions M± have the form atNN for some elements a+ E G. Suppose that the mappings µ+ have the following Gauss decomposition:

µt = µ}v,77

(3.119)

where the mappings µ} take values in Nt, the mappings v'- take values in N+1 and the mappings q' take values in H. If the pair µ± is regular, then the pair µ' is also regular. Furthermore, the mappings µt and p± satisfy relation (3.118). Therefore, the corresponding solutions of equations (3.102)-(3.104) are connected by a symmetry transformation of the form (3.111), (3.112). Actually, such symmetry transformations are connected with the ambiguity arising in constructing the corresponding lifts from H/H± to H. Thus, almost any solution of equations (3.102)-(3.104) can be constructed from the mappings M± taking values in N±. Such a construction fails only when the mappings µt do not possess the Gauss decomposition (3.119). In fact, we can obtain almost all solutions by choosing the mappings p± taking values in a±NN for some fixed elements a± E G. In constructing a regular pair of mappings, one should satisfy a number of requirements. It is not difficult to satisfy the requirement of holomorphicity or antiholomorphicity. A more difficult problem is to construct mappings µf for which takes values in m' and µ+1 a+µ+ takes values in m+. To solve this prob9rn, lem one can take a set of arbitrary mappings )t+m, : M m = 1, ... , l - 1, and -y± : M -* H, where and ry_ are holomorphic, while A+rn and ry+ are antiholomorphic mappings. Then the mappings /-t± can be determined by the integration of the relations l-1

µflafµt =

A+m + Ytxf Yfl

186

Differential geometry of Toda-type systems

EXAMPLE 3.11 Consider the abelian Toda system corresponding to the Lie group SL(2, Q. The Lie algebra of SL(2, C) is sf(2, (C),

and we can take the identity mapping as a principal embedding of sl(2, C) into s[(2, C). In this, the generators h and x± are of the standard form given in example 1.8. The corresponding Toda system consists of just one equation: a+a- f = e-2f (3.120) This is the famous Liouville equation. Recall that the function f parametrises the mapping y taking values in the subgroup H. Actually, it is a local parametrisation. Indeed, in the case under .

consideration the subgroup H consists of all 2 x 2 diagonal matrices of SL(2, C). Therefore, we can globally parametrise the mapping ry as

'Y = (

0 0 /3

/3-

1

where /3 is a function taking values in C'< . The functions f and ,(3 are connected by the relation (3.121)

)(3 = ef,

and it is clear that the function f provides only a local parametrisation of the mapping y. In terms of the function,3 the Liouville equation is written as ,9+(,8-',9-)3)

(3.122)

=)3-2.

Note that the subgroups H± in the case under consideration are formed by the matrices ±12i and the symmetry transformations (3.111) look as /3' = f,3. We now obtain the general solution of equation (3.122). Parametrise the mappings -y± as

_ -Y±

0

1

_(/3t

0

This parametrisation leads to the following expression for the mapping µ_:

- - ( a-,, a-21

a-12 a-22

1

) ( 2/I-

0

1/

where a- =_ (a-Z3) is an arbitrary element of SL(2, Q and Z

G-(z ) =

f_

dy Q-2(y ),

(3.123)

3.3 Construction of solutions and reality condition

187

where c is a fixed complex number. Similarly, for µ+ we obtain µ+ _

1 0,

a+11

a+12

a+21

a+22/ \0

(3.124)

1+/

where a+ - (a+2j) is one more arbitrary element of SL(2, C) and 7p+(z+) _

f dy+/3+(y+),

with c+ being another fixed complex number. Thus, we obtain the following expression: /' `/'+ /' /' 1PC all - a21 `/'+ + a12 b-/ 4'- a12 - a21 Y'+ 1 - a22

--

all + a22 W-

a22

where a = (ati3) = a+1a_. Finally, using the Gauss decomposition of form (3.113) and relation (3.98), we arrive at ,,//'' N = i3+/hQ-1(all - a214'+ + a124'- - a224'+'V-) This is the general solution of equation (3.122). Let us demonstrate the fact that to obtain almost all solutions of (3.122) it is enough to use the mappings µt taking values in the //''

subgroups N. Let the mappings µ- and µ+ be of form (3.123) and (3.124) respectively. Find the mappings 4 entering decomposition (3.119). The mapping µ' has the form 0

1

µf =

a-21 + a_22'a-11 + a_120-

It is easy to verify that this mapping satisfies the equation

Y x-'Y

tz

with the mapping y' parametrised by the function 13' = ±Q-(a-11 + a_12Y-) For the mapping µ' we obtain //''

1

+. =

a+llY'/',+ + a+12 a+21 y'+ + a+22

0

1

and the corresponding function ,6+ is given by a+21 W+ + a+22

A direct calculation shows that the mappings µt and ,Qt give the solution which differs from the solution obtained with the use of

188

Differential geometry of Toda-type systems

the mappings j and 3C3± by a possible symmetry transformation of type (3.111), which, in our case, is simply a change of the sign of the function 0. As we noted above, almost all solutions of Toda-type systems can be obtained with the use of the mappings p± taking values in the subsets a fN+ with fixed elements at E G. Actually, the solutions depend only on the element a = a+la_, and we obtain them using the equality aµ'

P+114-

where the mappings µ} take values in N. Choosing different elements a, we obtain different representation of solutions. In the case under consideration we can write the mappings l+ as 0 m121

1

1

M+12

0

112

0

1I-\

'b-

_

I '/+ 0

1

m-211+

?G-

1J'

1

m+12 + `b+

0

1

'

where m_21 and m+12 are arbitrary complex numbers parametris-

ing general elements of the subgroups N_ and N+. Introducing the notations

- _ -i(m-21 + -),

(+ = -(m+12 + O+),

we obtain the following relation: )3 = )Q+1,3-(all -

a12(_ + a22(+(-) Consider two different choices for the element a:

a=

10 i

a21 S+ +

I,

a= 11

0

I.

These choices give the following formulas:

Q = 3+1Q-(1 + (+ 13 =

which describe almost all the solutions of equation (3.122). Return now to the original Liouville equation. Taking (3.121)

and the relations a_(- = /3=2 and a+(+ = ,(3+ into account, we can formally write the solutions of the Liouville equation (3.120) as of

= (a+(+a-(-)-12(1 + (+(-),

3.3 Construction of solutions and reality condition

189

or as

(a+(+O-(-)-i12((- - +) These are usual the forms for writing the solution of the Liouville equation. of

As we can see, the procedure of the construction of the general solution for the Toda-type systems is based mainly on Gauss decomposition. In some cases it is more convenient not to use the explicit form of Gauss decomposition but, rather, to appeal to the algebraic structure of the construction. The relevant object here is a representation of the corresponding Lie algebra, or, in other words, a module over this Lie algebra.

3.3.2 Modules over semisimple Lie algebras In this section g is a complex semisimple Lie algebra of rank r, Ij is some fixed Cartan subalgebra of g, 0 is a root system of g with respect to C j, and II = {a,. .. , a,.} is a base of A. The corresponding Cartan and Chevalley generators are denoted by hi and xfi respectively. Let V be a g-module and let A E Cl*. Introduce the following notation:

VA- {vEV Ihv=(A,h)vforallhEt3}. If VA

{O}, we say that VA is a weight space and A is a weight of V.

Elements of VA are called elements of weight A, and the dimension

of V' is called the multiplicity of A. It is easy to demonstrate that

for anyAEC)* and aEAwe have gaVA C VA+a.

Since eigenvectors which correspond to different eigenvalues are linearly independent, then, for any two different weights A and p of V, we have V, n Vm = {0}. Therefore, we can consider the following direct sum:

V'VA.

It is clear that V' is a submodule of V. If V' = V, the g-module is called diagonalisable with respect to 1). It can be shown that any finite-dimensional g-module is diagonalisable.

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Differential geometry of Toda-type systems

Let V be a g-module. A nonzero element v E V is called a singular vector of weight A if v E VA and n+v = 0. It is clear that

n+v = 0 if and only if x+iv = 0 for all i = 1, ... , r. Let v E V be a singular vector. The subspace W - U(g)v is a submodule of V. A g-module V is called an extremal module of highest weight A E Cl' if va is a singular vector of weight A and U(g)v,\ = V. Here A is called the highest weight and the vector VA the highest weight

vector of the module V. For any weight p of an extremal module V of highest weight A we have p -< A. This fact explains why the weight A is called the highest weight. It is clear that any extremal module can be thought of as the result of the construction described above.

Let V be an extremal g-module of highest weight A. From representation (3.6) it follows that U(g) = U(n_)U(i)U(n+). This relation implies that U(g)v), = U(n_)vA. Having enumerated the positive roots, we can write A+ = {(31, ... (33} and A- = { - (31 i ... , -133 }. For any Qa E A+ choose some nonzero vector x_aa from g_p From the Poincare-Birkhoff-Witt theorem it follows that the monomials (x_a, )k1 ... (x_p, )'- form a basis of U(n_). Therefore, the elements (x_Q,)k1 ... (x_Q,)k°vA span V, and the weights of V have the form A

-

i-i

mini,

where mi are nonnegative integers. Recall that n_ is generated by the Chevalley generators x_i, i = 1, ... , r, where r is the rank of g. By this reason, it is clear that the vectors of the form xkii

also span M(A). A basis obtained by selecting from these vectors linearly independent ones is called a Verma basis. It is clear that the module V is diagonalisable, the multiplicity of A is equal to 1 and the multiplicities of all weights are finite. Any extremal g-module is indecomposable. If an extremal 9-

module V of highest weight A is irreducible, then any singular vector of V is proportional to the highest weight vector va. It appears that for any A E Cl' there exists an irreducible extremal g-module of highest weight A. Such a module is unique up to isomorphism. To demonstrate the existence of the module in

question, note first that U(g) has the natural structure of a gmodule, induced by the multiplication of the elements of U(g) by

3.3 Construction of solutions and reality condition

191

the elements of g from the left. Consider the left ideal J(A) of U(g) generated by n+ and by the elements h - (A, h), h E 1) and denote

M(A) - U(g)/J(A). The g-module M(A) is an extremal module of highest weight A which is called the Verma module of highest weight A. Here the image of 1 E U(g) is the highest weight vector. It can be shown that any extremal module of highest weight A is isomorphic to a quotient module of M(A).

Any singular vector of M(A) that is not proportional to va generates a proper submodule of M(A). Any proper submodule of M(A) can be represented as a direct sum of such submodules. The direct sum of proper submodules of M(A) is a proper submodule of M(A). Hence M(A) has a unique maximal proper submodule M'(A). The quotient module L(A) - M(A)/M'(A) is an irreducible extremal g-module of highest weight A. It can be shown that any irreducible extremal g-module of highest weight A is isomorphic to L(A).

Since any element of I * can be a weight of some 9-module, we shall often call elements of Cl* weights. A weight A is said to be integral if (A, hi) is an integer for i = 1, ... , r. The set of all integral weights is called the weight lattice and is denoted by A. Note that A c A. An integral weight A is called dominant (regular dominant) if (A, hi) > 0 ((A, hi) > 0) for i = 1, ... , r. The sets of dominant and regular dominant weights are denoted by A+ and A++ respectively.

The module L(A) is finite-dimensional if and only if A is a dominant weight. It can be shown that any finite-dimensional 0module has a singular vector and that the submodule generated by this vector is irreducible. Hence, any finite-dimensional irreducible g-module is extremal and, therefore, it is isomorphic to some g-module L(A) with A being a dominant weight. The dominant weights ei, i = 1, ... , r, defined by (ci, hj) _

jij

I

are called the fundamental weights. The corresponding g-modules (representations of g) are called the fundamental g-modules (fun-

damental representations of g). Here we use the notation Li L(ei).

Recall that any representation of a Lie group generates a representation of the corresponding Lie algebra. If a Lie group G is

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Differential geometry of Toda-type systems

simply connected, then any representation of its Lie algebra g can be `integrated' up to the corresponding representation of G.

Let a be a conjugation of g and let x E g xt = -o,(x) E g be the corresponding hermitian involution of g. The mapping Q defines a real form ga of the Lie algebra g by

gQ={xEgI o,(x)=x}. An element x E g is said to be o -hermitian if xt = x, and it is said to be o-antihermitian if xt = -x. The subalgebra ga is formed by all o--antihermitian elements of g.

Suppose that a can be extended to an antiholomorphic automorphism E of the group G. It is always possible when the group G is simply connected. In that case we have E2 = idG. Define the mapping a E G H at E G, where at = E(a-1) = It is obvious that this mapping is an antiholomorphic antiautomorphism of G satisfying the condition (at)t = a for any a E G. An element a E G is called E-hermitian if at = a, and it is called E-unitary if at = a-1. In the case where the mapping E is determined by the mapping o, defined by (1.29), we simply say `hermitian' and `unitary'. The real Lie group Go corresponding to the real form ga is formed by all E-unitary elements (E(a))-1

of G.

Suppose also that the mapping o, has the property m E Z. Q(g...) = g-.,n, In this case one has

(H)t = H,

(3.125)

(NN)t = N:F .

A representation p : G -+ GL(V) of the group G in the linear space V over the field C is called E-unitary if the space V is equipped with a nondegenerate hermitian form

such that

p(a)t = p(at),

where f on the left-hand side means the hermitian conjugation It can be shown that the with respect to the hermitian form representation p is E-unitary if and only if the restriction of p to the real Lie group Ga is unitary.

3.3 Construction of solutions and reality condition

193

3.3.3 From representations to solutions Below we will use the Dirac notation for complex vector spaces, see Dirac (1958). According to this notation, the elements of a complex vector space V are denoted by the symbol I ). To distinguish different elements of V, the symbol I ) is supplied with labels, for example, Iv), Iu). The elements of the dual space V* are denoted by the symbol (I, also supplied with labels. The ac-

tion of an element (al E V* on an element Iv) E V is denoted by (alv). If a nondegenerate hermitian form ( , ) is defined on V, then one constructs the mapping from V to V* which associates with an element Iv) of V the element (vi of V*, such that (VIn)

(Iv), Iu))

for all Iu) E V. For more details we refer the reader to the book by Dirac (1958).

Let us now consider an arbitrary E-unitary representation p of the group G in a linear space V. Denote by V+ the subspace of V formed by all elements Iv) E V such that p(a)Iv) =

IV)

for all a E N+. For any mapping cp : M -* G and any vectors Iu), Iv) E V we denote by (ulWIv) the mapping from M to C, defined as

(ulcpIv)(p) = (ulp(y(p))Iv).

Theorem 3.4 For any Iu), Iv) E V+ the following relation is valid: (uI-YIv) = (uI(1p+'1+)-1(lL-'Y-)Iv).

(3.126)

Proof Using definition (3.98) of the mapping -y and equality (3.113), we obtain the relation v Yv+ 1 =

(3.127)

where

v=

Y

1 v-'Y+,

v+

= 'Y1 v+'Y-

Then the validity of (3.126) follows from (3.127) and the definition of V+.

O

Thus, we can find some matrix elements of the linear operators corresponding to the mapping -y in a E-unitary representation of

194

Differential geometry of Toda-type systems

the group G by using only the mappings pt, which in their turn are determined only by the mappings (p±. It is natural to suppose here that in using a large enough set of representations, one will be able to recover the mapping -y from the mappings cpt. In more or less the same way one can consider the mappings V±m. Here, however, one deals with matrix elements taken between

the vectors annihilated by the elements of subspaces ® it gtn, see Gervais & Saveliev (1995). Note also that for 1 = 1 form (3.126)

of writing the general solution of a Toda system coincides with that given in Leznov & Saveliev (1992).

3.3.4 Real solutions Let us now consider a special class of solutions which are called real solutions. To this end, we introduce two antiholomorphic mappings Et : Ft -3 F=F defined by

Et(aBT) It is easy to show that the mappings Et are defined correctly. Directly from the definition of these mappings, we obtain the equalities E+o7r+=7r_0E, (3.128) E_o7r_ =7r+oE. Moreover, these mappings are mutually inverse:

E+oE_=idF_,

E_oE+=idF+.

(3.129)

A mapping p : M -+ G is said to satisfy the reality condition if

E+°cp+='p

(3.130)

which can be also written as E_ 0

(3.131)

Proposition 3.20 A mapping cp : M -> G satisfies the reality condition if and only if

E0

=(p

,

(3.132)

where the mapping & takes values in H. Proof Let y satisfy the reality condition. Using (3.128) in (3.130), we obtain 7r_ 0E0 =7r_0(p.

3.3 Construction of solutions and reality condition

195

Hence, E o cp = co' 1', where b takes values in B+. In a similar way,

from (3.131) one sees that % takes values in b- Since B+ n H, we conclude that 0 takes values in H. The inverse statement of the proposition is obvious. A mapping : M -* G is called E-hermitian if for any p E M the mapping 7p(p) is a E-hermitian element of G.

Proposition 3.21 The mapping 0 entering proposition 3.20 is E-hermitian.

Proof Since E' = idG, from (3.132) one obtains

=(E0W)(E0v)), which can be written as

= y(E 0 Comparing (3.133) with (3.132), we have E0

Hence, the mapping

(E o'&)-1 = is E-hermitian.

(3.133)

10.

Proposition 3.22 The reality condition is H-gauge invariant. Proof Let a mapping cp_satisfy the reality condition, and gyp' = cp6, with taking values in H, be a gauge transformed mapping. Using (3.132) we have

Eocp'=(p, o,

"

where 0

Since E(H) = H, the mapping i' satisfies the reality condition. Let s+,, a E A, be a family of local holomorphic sections of the fibre bundle G -+ F+ with the properties described in proposition

3.29. The mapping E allows us to construct the corresponding family of local holomorphic sections of the fibre bundle G - FFor each open set U+a, we define the open set U_a by U-a = E+(U+a).

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Differential geometry of Toda-type systems

Using (3.128) and (3.129), it is easy to show that for any a E A the mapping s_a : U_a -4 G given by S_a - E0s+a 0E_ is alocal holomorphic section of the fibre bundle G -+ F_. Since E(N+) = N_, we also have

E,N+=N_; hence, the section s_, is tangent to the distribution N_. Thus, we obtain a family of holomorphic sections of the fibre bundle G -} F_ with the required properties. Now, to construct the mappings needed to define the mapping y, one uses for any section s+,,, the corresponding section s_a defined with the help of the procedure described above.

Proposition 3.23 If a mapping (P satisfies the reality condition, then the mappings u+ and, µ_ entering proposition 3.14 are connected by the relation AT = E o IL f.

(3.134)

Proof Recall that the mappings pf are given by Pf = Sta 0 IPf From this relation we have 11f =E 0 Sfa0ET_ 0OF = EOSta O(Pt = OF/,±. This chain of equalities provides the assertion of the proposition.

0 Proposition 3.24 If a mapping (P satisfies the reality condition, then the mappings yt satisfying (3.82) canoe chosen in such a way that

Eoy±=y::F

-

(3.135)

Proof First, let us show that the mappings A+ and A_ are connected by the relation From (3.134) we have µ-w = (p+ 0 E*)B.

3.3 Construction of solutions and reality condition

197

Now, using equality (3.4), one shows that µ-w(v) = E*9(A+*(v)) = Q (µ+w(v))

for any v E TP (M), p E M. In particular, there is the following equality: P_w(a-)

= 0, (µ+w(a+))

which leads directly to (3.136). From (3.136) it follows that Ad(-y-)x- _ (o, o Ad('Y+))x+ Suppose now that the elements x± entering (3.82) are chosen in such a way that

Q(x+) _ -x_. Taking relation (3.2) into account, we conclude that one can choose the mappings 'y+ and y_ satisfying (3.135).

O

Theorem 3.5 If a mapping cp satisfies the reality condition, then the mapping y can be chosen E-hermitian; while for the mappings one has (v+m)t = v-.,n.

(3.137)

Proof From proposition 3.20, using (3.76) and (3.77), we obtain ) =IL_v+r1+'. Since decomposition (3.77) is unique, then EOW

=(EOt+)(Eov-)(Eoi

(3.138)

Eon-=q+,O, and hence, one can write the equality which leads to the relation As follows from proposition 3.21, the mapping 0 is E-hermitian, thus the mapping 77 is also E-hermitian. Taking proposition 3.23 and the definition of the mapping -y into account, we conclude that it can be chosen to be E-hermitian. From (3.138) it follows that vt = v-1 Using relations (3.93) and (3.96) now one obtains (3.137).

0

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Differential geometry of Toda-type systems

Thus, we can say that using the mappings cp which satisfy the reality condition, one can construct hermitian, in a sense real, solutions of equations (3.102)-(3.104). Since the solutions are actually

determined by the mappings cpt, then to obtain real solutions of the equations under consideration, we should choose the mappings cp± satisfying relation (3.130).

Suppose now that the involutive antilinear automorphism or is defined by (1.29), so that the corresponding real form u of g is compact. In this case the grading operator is o-hermitian, and (3.125) is valid. Consider the realification OR of the Lie algebra g. Let J be the linear operator in OR corresponding to multiplication by in g. It is clear that we have the expansion

OR =u®Ja®n+R, where a is a maximal abelian subalgebra of u. This expansion is called the Iwasawa decomposition of OR. Note that hR = a ® Ja. There is the corresponding decomposition of the Lie group G considered as a real Lie group, see Helgason (1978). It has the form GR = UN+RA*,

(3.139)

where A* is the real connected Lie group corresponding to the subalgebra Ja. Theorem 3.6 If mappings cpt : M -* Ft satisfy relation (3.130), then they are consistent.

Proof It is enough to show that if two points p+ E F+ and p_ E F_ are connected by the relation p- = E+(p+), then there exists an element a E G, such that p+ = 7r+(a),

p- = it-(a).

This fact can be proved using the Iwasawa decomposition (3.139).

Let a' be any element of G, such that 7r+ (a) = p+. This element can be written as a' = un+a*,

3.4 Toda fields and generalised Pliicker relations

199

where u E U, n+ E N+ and a* E A*, and the subgroups U, N+ and A* are defined above. We have N+A C B+ C B+, hence i+(u) = p+. Using (3.128) one now obtains (r-+ ° 7r+)(u) = (7r_ ° E)(u) = ir-(u) = E+ (P+)

Thus, the element u can be taken as the element a that we are looking for.

0

The generalisation of the results proved in this section and in the previous one, to the case of the semi-integral embeddings of s[(2, Q into 9 is straightforward, and can be performed following completely similar reasons. Recall that in that case we deal with Z/2-gradations of g.

Exercises 3.10 Construct explicitly Verma bases of the first fundamental representation for the classical series of simple Lie algebras .sl(r, (C), .so (2r + 1, C), io(2r, Q, sp(r, C), and for the exceptional Lie algebra G2-

3.11 Let Ii) be the highest weight vector of the ith fundamental representation of a complex semisimple Lie algebra g normalised by (iji) = 1. Using the defining relations (1.16)(1.18), find an explicit expression for the matrix elements (ilx+a, x+jmx-im x_il ji) in terms of the corresponding Cartan matrix.

3.4 Toda fields and generalised Pliicker relations In this section we give a derivation of the generalised Plucker relations.

3.4.1 Riemannian and Kdhler manifolds A smooth tensor field g of type (2) on a real manifold M is called a Riemannian metric on M if for any p E M one has (RM1) g,. (v, u) = gp(u, v) for all v, u E TT(M); (RM2) gp(v, v) > 0 for all nonzero v E Tp(M).

200

Differential geometry of Toda-type systems

In other words, g is a Riemannian metric on M if for any p E M the bilinear form gp on TT(M) is symmetric and positive definite. It can be shown that any real manifold possesses a Riemannian metric. Here it is important that we consider only the manifolds which are second countable topological spaces. A manifold endowed with a Riemannian metric is called a Riemannian manifold.

Let M and N be two manifolds and let cp : M -* N be an immersion. Suppose that N is endowed with a Riemannian metric

gN. Define a tensor field g' on M by (v) '

gp (v, u) = gw(P)

*P

(u))

for any p E M and v, u E Tp(M). The tensor field gm is a Riemannian metric on M, which is called the Riemannian metric induced

from gN by V. In particular, any submanifold of a Riemannian manifold can be provided with the Riemannian metric induced by the inclusion mapping. As it is for an arbitrary tensor field, a Riemannian metric g on

a manifold M can be extended to a complex tensor field on M. Denote this tensor field also by g. Here, for any p E M one has (CRM1) gp(v, u) = gp(u, v) for all v, u E T (M); (CRM2) gp(v, v) > 0 for all nonzero v E TP(M); (CRM3) gp(v, u) = gp(v, ii) for all v, u E T7 (M).

On the other hand, any complex tensor field g on M having the properties (CRM1)-(CRM3) is the natural extension of a Riemannian metric on M. A Riemannian metric on a complex manifold M is, by definition, a complex tensor field of type (°) satisfying (CRM1)-(CRM3).

Let g be a Riemannian metric on a complex manifold M and let (U, z1, ... , zm) be a complex chart on M. From (CRMI) we obtain the following equalities: gij = gji, 9ij = gj-1, while (CRM3) implies g13,

gij = 9j=,

9;- = g1j .

A Riemannian metric g on a complex manifold M is called hermitian if g(JMX JMY) = g(X, Y) for all X,Y E 3Co(M). Let g be a hermitian metric on a complex manifold M. It can easily be shown that g(X, Y) = 0 if both

3.4 Toda fields and generalised Pliicker relations

201

vector fields X and Y are either of type (1,0) or of type (0,1). Therefore, for any complex chart (U, z1, ... , zm) on M one obtains the equalities 9i3 = 0,

g2, = 0,

which imply the following local representation of g:

9I u = gijdzi 0 d2 + g2;dzi 0 dz3.

(3.140)

Let g be a hermitian metric on a complex manifold M. The 2-form 4 defined by -T (X, Y) = 9(X, JMY), is called the fundamental form associated with g. Using a complex

chart (U, zl, ... , z-), one has 'I'I u = -2 gi,dzi A dz'. (3.141) A hermitian metric g on M is called a Kahler metric if the fundamental form associated with g is closed, i.e., d4(b =0.

A complex manifold endowed with a Kahler metric is said to be a Kahler manifold. The fundamental form 1 associated with a hermitian metric g can be locally represented as

-D = -2fJaaK, where K is a real valued function. The function K entering the last relation is called a Kahler potential of the hermitian metric g. Note also that the expression for the Ricci curvature tensor R in the case under consideration has the form d(10)d(°'1) In g, (3.142) R=2 where g is the determinant of the matrix (gig), see Kobayashi & Nomizu (1963).

EXAMPLE 3.12 Let V be a complex linear space and let P(V) be the projective space of V. Recall that P(V) is the set of onedimensional subspaces of V. Denote by pr the canonical projection from the set V" - V - {0} onto P(V). Suppose that V is endowed

with a positive definite hermitian scalar product ( , ), and define a real valued function F on V" by F(v) - In (v, v).

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Differential geometry of Toda-type systems

There exists a unique 2-form 4 VS on P(V) satisfying the relation

pr* IFS = -2id(1'0)d(°'1)F.

(3.143)

This form determines a Kahler metric on P(V), called the FubiniStudy metric.

3.4.2 Verma modules and flag manifolds Let G be a complex connected semisimple Lie group of rank r and

let g be a Lie algebra of G. Fix a Cartan subalgebra f of g and a base II of the corresponding system of roots. As above, denote by hi and xfi the Cartan and Chevalley generators of g associated with the base H. Let L(A) be the Verma module of the highest weight A and let va be the corresponding highest weight vector. By definition, one has

hva=(A,h)va,

hEC7,

(3.144)

xva = 0, x E n+. (3.145) Since L(A) is a finite-dimensional g-module, then A, - (A, hi), i = 1'... , r, are nonnegative integers. The conditions (3.144)(3.145) are equivalent to hiva = AivA, x+iva = 0,

i = 1,...,r, i = 1,...,r.

(3.146) (3.147)

Using the relations [x+j, x-i)

= jihi

and (3.147), one obtains x+j(x_iva) = SjiAivA.

(3.148)

If for some i one has Ai = 0, then (3.148) implies x+j (x_iva) = 0. Hence, either x_iva = 0 or x_iva is a singular vector. Suppose that

the vector x-iva is a singular vector. Any singular vector in L(A) should be proportional to va. The vector x_iva is of weight A - 7ri and, for this reason, it cannot be proportional to vA. Therefore,

x_iva = 0. On the other hand, if for some i one has Ai # 0, it follows from (3.148) that

x+i(x-iva) = AivA 0 0, and, therefore, x_iva # 0.

3.4 Toda fields and generalised Plucker relations

203

Thus, if Ai = 0, then x_iva = 0; and if Ai = 0, then x_iva 0 0. Let p.. be the representation of G which is obtained by `integration' of the representation of g in L(A). Consider the projective space P(L(A)) and denote by pr, the canonical projection from L(A)" - L(A) - {0} onto P(L(A)). The representation pa, as any representation, defines the left action of G in P(L(A)) satisfying the condition a- pr,\ (v) = pr,\ (pa(a)v)

(3.149)

for any a E G and v E L(A). Denote by pa the point of P(L(A)) corresponding to the vector va. As follows from (3.144) and (3.145), the Lie algebra gpa of the isotropy subgroup Gpa contains the Borel subalgebra b+. Hence, gp, is a parabolic subalgebra of g. Furthermore, the above discussion implies that this Lie algebra

contains the vectors x_i for all i such that Ai = 0 and does not contain the vectors x_i for all i such that Ai 0. Taking into account the general structure of parabolic subalgebras described in subsection 3.1.2, we conclude that Spa = P+il,...,ik'

where the positive integers i 1 , . . . , ik make up the set of all i E {1, ... , r} for which A, = 0. Since a parabolic subgroup of a connected Lie group is connected, then one concludes that Gpa = P +i1,...,ik

and the orbit of p,\ is a submanifold of P(L(A)) diffeomorphic to the flag manifold Define the embedding t_i...... ik of the flag manifold F_i...... ik into the projective space P(L(A)) by c-ii ..... ik (aP_ ,...,ik) = a, pa.

(3.150)

With this embedding, the Fubini-Study metric on ]P(L(A)) induces a Kahler metric on F_i,,...,ik; hence, the flag manifold F_i,..... ik is a Kahler manifold.

3.4.3 Generalised Pucker relations Let
204

Differential geometry of Toda-type systems

any parabolic subgroup P+i1,...,ik there is the natural projection 7r-i...... ik from F_ onto the flag manifold F_2,..... ik, defined by (3.151) (aB+) = aP+i1,...,ik. Using this projection we can define a holomorphic curve cp_i.....,ik in as 7r-21,...,ik o (0--

Thus, any holomorphic curve (p_ in F_ generates a family of holomorphic curves (p-il,...,ik labelled by the parabolic subgroups containing the Borel subgroup B+. Consider the flag manifold F_i1 ...,ik as the orbit in the projectivised representation space P(L(A)). The mapping (p_i1i,,.,ik is not, in general, an embedding. Nevertheless, the Kahler metric on F_i1i...,ik specifies a symmetric tensor field of type (°) on M. This

tensor field is called a pseudo-metric on M; we will denote it by ga. The fundamental form 4)a associated with this pseudo-metric is given by (3.152)

4)a = ((p* ° *i1....ik ° L*

where IFS is the fundamental form associated with the FubiniStudy metric on P(L(A)). Proposition 3.25 Any holomorphic local lift ip_ of the mapping (p_ to G leads to the following local representation of the fundamental form IA: 4)a = -2v/_-_1d(10)d(°'i) ln((p_vA, (p-va).

Proof By definition, the mapping (p_ satisfies the relation (p_ = 7r_ 0 (p_.

(3.153)

Define the mapping TA : G -+ L(A)" by TA(a) - pA(a)vA,

a E G,

and prove the following equality: pra °TA =

o

o 7r_.

3.154)

Indeed, using (3.149), for any a E G we obtain

(pra °T,\)(a) = pra(pa(a)vA) = 9'pa On the other hand, it follows from (3.151) and (3.150) that 0 7r_i1,...,ik 0 it-)(a) = t-i1,...,ik (aP+il,...,ik) = a - pa.

3.4 Toda fields and generalised Pliicker relations

205

Hence, equality (3.154) is true. Relations (3.153) and (3.154) result in t-ii,...,ik o 7r_il,...,ik o P- = pra oTa o 'y_ -

Taking (3.152) and (3.143) into account, we obtain 4)a = = *oTa*opr*)4DFS

(3.155)

The assertion of the proposition is the direct consequence of (3.155).

O

Proposition 3.25 shows that the function KA

ln(cp_va, ip'_va)

is a Kahler potential of the pseudo-metric gA on M, having a as its fundamental form. Choosing different lifts cp-, we obtain different Kahler potentials.

Let us suppose now that cp_.p(a_r) E M' r for any p E M. Here the subset M'J, of Tw_ tr) (F_) is defined in section 3.2.5. The

most convenient, f o r our purposes, choice of the lift cp_ can be con-

structed as follows. Suppose that the hermitian bilinear form ( , ) in Va is chosen in such a way that the corresponding representation of the group G is o-unitary with or being the Chevalley conjugation. Denote by E the antiholomorphic automorphism of G corresponding to o, and define the mapping E_ : F_ -> F+ = G/B_ by

E-(aB+) = E(a)BNow introduce the mapping cp+ : M -* F+ = G/B_ given by The mapping cp+ is antiholomorphic and, as follows from theorem 3.5, there exists the mapping cp such that

cp±=7rf0(p. The mapping cp, by construction, satisfies the reality condition. Construct for cp a local decomposition of form (3.77). With the help of (3.82), determine the mapping ry_ corresponding to the mapping p_. Finally choose the mapping iP_ as It is clear that cp_ is a holomorphic local lift of cp_ to G. Thus, tak-

ing the o -unitarity of the considered representation of the group

206

Differential geometry of Toda-type systems

G into account, we obtain the following expression for the Kahler potential K,,:

K,\ = ln(p_ _vA,µ_ _vA) = ln(va, (p_ _)t(p_ _)va) Further, propositions 3.23 and 3.24 imply that the above expression can be rewritten as KA = ln(vA, (/-p+'Y+)-1(p_ y_)va),

where µ_ is the mapping arising in the decomposition of cp of form (3.76) and y_ is the mapping determined from (3.82). Using theorem 3.4 we now conclude that KA = In (v,\, -yv,\),

where y is the mapping satisfying the abelian Toda equations (3.105). Using parametrisation (3.106), we obtain r

KA=

fiAi

Here the Toda fields fi satisfy equations (3.107).

For the fundamental representations Li, i = 1, ... , r, denote the corresponding pseudo-metrics by gi and the associated fundamental forms by In this case the Kahler potential Ki of the pseudo-metric gi coincides with the Toda field fi. Theorem 3.7 Under the conditions described above, the Ricci curvature tensors Ri of the pseudo-metrics gi on M are connected with the corresponding fundamental forms -Di by the relations r

Ri = Ekij4Dj.

(3.156)

j=1

Proof Since the Toda field fi is the Kahler potential of gi then 4>i = -2v"-'_1d(1,°)d(°'l)fi.

(3.157)

Comparing (3.140) and (3.141) and using equations (3.107), for the pseudo-metrics gi we find the expression gi = a_ a+ fi (dz- 0 dz+ + dz+ ® dz- )

= 2ki exp[-(k f)i](dz 0 dz+ + dz+ 0 dz-).

3..4 Toda fields and generalised Plucker relations

207

Taking (3.142) into account and using again equations (3.107), we have

r

Ri _ -2

E ki,d(''0) d(°'').f,; j=I

and, taking account of (3.157), we arrive at (3.156).

O

Relations (3.156) are called the generalised infinitesimal Plucker relations. The validity of these relations was first conjectured in Givental' (1989), and proved in Positsel'skii (1991). Our proof is based on the special choice of the Kahler potentials of the pseudometrics, and in this sense is more close to the proof of the standard Plucker formulas, see Griffiths & Harris (1978). Note also, that the relation between the abelian Toda fields and the Kahler potentials in question has been established for the Ar series in Gervais & Matsuo (1993), and for the other classical series (Br, Cr, Dr) in Gervais & Saveliev (1996), using explicit calculations in a local coordinate parametrisation of the flag manifolds. The discussion above follows the lines of Razumov & Saveliev (1994) and

is valid for an arbitrary simple Lie algebra g. The corresponding generalised global Plucker formulas are also valid (F. E. Burstall, personal communication).

4

Toda-type systems and their explicit solutions

4.1 General remarks Roughly speaking, the Toda-type systems (3.102)-(3.104), in the form given in section 3.2.5, represent systems of elliptic partial differential equations. In physical applications one more often deals with systems of hyperbolic partial differential equations. For this reason we start the present chapter with a brief review of the procedure for obtaining the Toda-type systems and constructing their general solutions for the case of hyperbolic systems. We begin by introducing the notion of a chiral manifold, see Gervais & Matsuo (1993). Let M be a two-dimensional manifold. Suppose that there exists an atlas {(Ua; za , za)}aEA of M, such

that aza/aza = o

Oz-/aza = o,

for all a,,3 E A, and UUEA = M. In this case we say that the atlas {(UU; za , za)}aEA endows M with the structure of a chiral manifold. Here, any chart (U; z-, z+) is called a chiral chart on M. A smooth function f on M is called (anti) chiral if of/az+ = 0 (Of /Oz- = 0) for any chiral chart on M. A mapping y from a chiral manifold M to a manifold N is called (anti) chiral if, with respect to any chiral

chart on M and any chart on N, it is described by (anti)chiral functions.

Now let M be a simply connected chiral manifold, G a complex matrix semisimple Lie group, and g the Lie algebra of G. A connection on the trivial fibre bundle M x G -* M is described by a g-valued 1-form w. The form w corresponding to a flat connection satisfies the zero curvature condition (3.40). For any such 208

4.1 General remarks

209

connection we can point out a mapping cp : M -* G, such that w = cp-ldcp.

(4.1)

On the other hand, any mapping cp : M -* G generates, via this relation, a flat connection which is denoted `°w. Let w be a flat connection on M x G -* M. Choose a chiral chart (U; z-, z+) on M and write w = w_dz- + w+dz+. In terms of the g-valued functions wt, the zero curvature condition (3.40) takes the form (3.45), which can be considered as a system

of nonlinear partial differential equations. To obtain a nontrivial system we should impose some restrictions on the form w. Suppose that the Lie algebra g is endowed with a Z-gradation. The first condition we impose on the form w is the requirement that w_ takes values in b_, and w+ takes values in b+, where bt are the parabolic subalgebras of g defined by (3.19). We call this requirement the general grading condition. A mapping cp : M -+ G is said to satisfy the general grading condition if the corresponding connection "w satisfies the general grading condition. Now fix two positive integers l f and define the subspace mj of g by (3.54). We say that the connection w satisfies the specified grading condition if w_ takes values in m_ (D 6 and w+ takes values in [ ®m+. Here the subalgebra h is defined by (3.20). A mapping cp is said to satisfy the specified grading condition if the corresponding connection Ww satisfies this condition. Denote by N+ the connected Lie subgroups of G corresponding to the nilpotent subalgebras of defined by (3.18). Further, let H be a connected Lie subgroup of G corresponding to the subalgebra 1. It can be shown that any mapping cp : M -+ G has the following local modified Gauss decompositions: cp = µ+v-71- = µ-v+rl+,

(4.2)

where the mappings v± take values in the-Lie groups N±, the mappings q± take values in the Lie group H, and the mappings µt take values in subsets a±N± for some elements a+ E G. The latter condition is equivalent to the requirement that takes values in n_, while µ+la+µ+ takes values in n+. A mapping cp : M -* G satisfies the specified grading condition if and only if for any local modified Gauss decompositions (4.2)

210

Toda-type systems and their explicit solutions

the mapping p_ is chiral and p'8_Et_ takes values in m_; while the mapping p+ is antichiral and p+' 8+µ+ takes values in fn-+. In this case one can write tf A±m, m=1

where the mappings A 1 take values in gtm. Here the mappings A-,,,, are chiral, and the mappings A+m are antichiral. Now choose some fixed elements xt belonging to the subspaces

gttf. The adjoint representation of G generates the action of the subgroup H in the grading subspaces gtm. Denote the orbits of the elements xt by Cit. Consider a mapping cp satisfying the specified grading condition, and suppose that the mappings Af entering representation (4.3) take values in 0±. We say in such a case that the mapping c' is admissible. It is clear that there are local mappings -yt : M -* H, such that

A}tf = Ytxt7fl Now one can prove that there exists a local H-gauge transformation which brings the connection `°w to the connection w with l_-1

V_m + X_ +'Y-'8-7Y,

w- _

(4.4)

m=1

t+-1

w+ _

'Y-1

V+m + x+

Y

(4.5)

M=1

Here the mapping ry : M -+ H is given by 'Y = 7+1r1-r1+17- = Y+17'Y-

and the mappings v+m, : M

(4.6)

g±,, are defined as

l_-1 V_m

1Y1?7-1(v18-v-)7l7-'

(4.7)

M=1

1+-1

v+m = Y+ 1 (v+ 1a+v+) J

1

'Y+

(4.8)

m=1

We restrict ourselves by the choice l_ = l+ - 1; asymmetric systems corresponding l_

1+ can be considered in the same way.

4.1 General remarks

211

In this case the zero curvature condition for the connection with the components given by (4.4) and (4.5) leads to the equations a+v-m = [X-,7-1V+(l-m)7] +

[v-(m+n),7-lv+nY],

(4.9)

n=1

l-1 [v-m.,7-1V+m7],

a+(7-1a-7) = [x-,7-1x+7] + E

(4.10)

M=1

1-m-1 a-v+m =

[x+,7v-(l-m)7-1] + >

[v+(m+n),7v-n7-1]. (4.11)

n=1

Denote by HH the isotropy subgroups of the elements x+. System (4.9)-(4.11) possesses the symmetry transformations of the form

7' _ +17c-,

vfm = c± V±rc±,

(4.12)

where - : M -+ H- is a chiral mapping, and + : M -+ H+ is an antichiral one. Call a system of partial differential equations of form (4.9)-(4.11) a Toda-type system. In this chapter we illustrate the general system (4.9)-(4.11) by considering concrete examples for l = 1, 2. It will also be shown how the general formulas from the preceding chapter for the solution of system (4.9)-(4.11) work for the equations under consideration. Recall that to find the general solution to equations (4.9)-(4.11) we choose some mappings 7t : M -+ H and Atm : M -4 gfm, m = 1, ... ,1 - 1. Here the mappings y_ and A-m are chiral, while the mappings 7+ and A+m are antichiral. Then we define the mappings A± by 1-1

At = 7±x±7±1 +

A±m M=1

The next step is to find the mappings p+. They are obtained by the integration of the equations

atµt = P±A±.

(4.13)

One can easily check that the general solution to these equations

212

Toda-type systems and their explicit solutions

is represented by the series of nested integrals t fyl yn-1 f zt dy± f f ... dy2 P± (z ) = a± dyn J c±

f

cf

x At (yam) ... A± (y±)

,

(4.14)

where a± are some elements of G, and c± are some real numbers. It is clear that for such a solution we have t) = a±. P± (C Moreover, the mappings pt take values in a+N+. In fact, as was discussed in section 3.3.1, to obtain almost all solutions it suffice to take the elements a± belonging to the subgroups Nt. Further, we use the Gauss decomposition

µ+lµ- = v-rlv+1 (4.15) to obtain the mapping q. Then the mapping 7 is calculated with the help of (4.6). Finally, the mappings v±rn, m = 1, ... , l - 1, are determined using (4.7) and (4.8), where the mappings vt can be found from (4.15).

4.1

Exercises Derive the generalisation of equations (4.9)-(4.11) for the case where l+

l_.

4.2 Abelian Toda systems For the case where 1 = 1 equations (4.9)-(4.11) take the form (4.16) a+(Y-la- Y) = [x-,'y-lx+'y], which are called the Toda equations. Consider the principal embedding of the Lie algebra st(2, cC) into g, and the Z-gradation

defined by the grading operator q = h/2, where h is the Cartan generator of the s((2, C)-subalgebra. Recall that for such a gradation we do not use tildes in notations. The subgroup H is generated here by a Cartan subalgebra [ of g and, therefore, it is an abelian group. The discussion of abelian Toda systems given in section 3.2 is based on a local parametrisation of the mapping

4.2 Abelian Toda systems

213

y : M -4 H. Any such parametrisation leads to a loss of some global information. Therefore, at the beginning of this section we consider the transition to a global parametrisation of the mapping y.

First note that, for the case under consideration, a convenient local parametrisation of the mapping 'y is

fihil

y = exp I -

(4.17)

.

Using this parametrisation, we obtain from (4.16) the equations r

a+a-fi = 2ki exp E kid f; Introducing the notation

Qi = exp(-fi), we arrive at the following system of partial differential equations:

a+(Qz la-Qi) = -2ki 11

k`',

i = 1,... , r.

(4.18)

j=1

Consider now classical Lie groups and investigate, to what extent the introduced parametrisation is a global one.

4.2.1 Lie group SL(r + 1, C) Let us begin with the Lie group SL(r + 1, C). Using the formulas given in section 1.3.1, one sees that the Cartan matrix k in this case has the form 2 -1 0 0 0 0 -1 2 -1 . 0 0 0 0 -1 2 ... 0 0 0 k= . .

0 0 0

0 0 0

0 0

..

0.

2

-1

0

-1

2

-1

-1

2

0

Toda-type systems and their explicit solutions

214

After some calculations one obtains the following expression for the inverse of k: 2 1 r r-1 3

k-1

=

r-2 f r-1 2(r-1) 2(r-2) ...

6

4

2

r-2 2(r-2) 3(r-2) .

9

6

3

-

1

r+1 3

6

9

... 3(r-2) 2(r-2) r-2

2

4

6

... 2(r-2) 2(r-1) r-1

1

2

3

r-1

r-2

r

Using this expression we obtain r

2ki = 2 1:(k-1)ij = i(r - i + 1). j=1

The group H consists of all complex nondegenerate diagonal matrices from SL(r+1, C), and we have the following parametrisation of the mapping -y: Q1

0

... Qi1I2 ... 0

0

0

\0

0

...

0

0

0

0

/r i/ Or 0

01

0'-1

It is clear that the parametrisation of -y via the functions /3z is global. Note that in this case equations (4.16) have the form

a+p

1a-'31) _ -r/31

2,32,

+ a+\Nr

1 < i < r,

-r/3r-1Nr 2

The simplest abelian Toda system, the Liouville equation, and its general solution were considered in example 3.11. Let us consider a more difficult case of the abelian Toda system based on the Lie group SL(3, (C). As was discussed above, a global parametrisation of the mapping -y can be chosen here as 131

'y =

0 0

0

0

/3i 100 0

'8

(4.19) 1

4.2 Abelian Toda systems

215

where i31 and /32 are functions taking values in (C'. The elements

xt are given by 0 0

x+=

0

0 0

0

0

0

x_= V 0

0 0

0/ 0

0

.

The corresponding abelian Toda equations have the form a+(Qi la-01) = -201 2/32, a+(Q2 la-Q2) = -2/3 /3 2.

(4.20)

(4.21)

Parametrise the mappings p±' as follows:

µ+ =

1

/2+12

0

1

/2+23

0

0

1

/1+13

P-

1

0

0

2-21 /l-31

1

0

2-32

1

Then, for the mappings µt1 we obtain the expressions -/1+12 1

-/1+13 + /1+12/2+23 -/2+23

0

1

1

11=1 =

11_21

-P-31 + /-1-32P-21

0

0

1

0

-P-32

1

Hence, one can write 0

a+p+12

a+/2+13 - P+1249+P+23

0

0

(9+ P+23

0

0

0

0 19-µ-21

a-A-31 - /-L-3249-/2-21

0

0

0

0

0-11-32

0

Representing the mappings -yt in the form Yf =

0 0

/331 /t2 0

(4.22)

Toda-type systems and their explicit solutions

216

we obtain 0 'Y+x+ry+-1 =

V2-

0

0

0 V,Q+1 /3+2

0

0

0

0

V2-Q-1Q-2

- _ 'Y-'Y-1 =

0

0

0

0

V2-Q-2

0

0

Using the above relations we see that the matrix equations (4.13) are equivalent to the ystem of equations a+µ+12 =

a+µ+23 = V LQ+i /+2

V"2)32 1)3+i,

a+/1+13 - //+112a+/1+23 = 0,

a-N'-21 =

V42-,3--2

a-P-32 =

/3-2,

V

a41-31 - µ-32a-/1-21 = 0.

The general solution of this system is

dyi '32+1 (Y+),3;2 )f+2 (yi ),

µ+12('Z+) = m+12 + V/2 J /z+

= m+23

+

J

dyi Q+i (yi )Q+2 (yi ), z+

= m+13 + m+12.Vf2 fz+

+2

J+

M-21

f dyi+i (yi ) Q+2 (yi )

y2

dye f dC+yi -'+l (y1 )&'+2 (y1 )O+1(y2 ),Q+2 (y2 ),

+ fz

= m-32 +

dyi -i (yi ),3-2 (yi dy1 -,3-1 (Y1 )Q-2(y1 ),

c

fz

m-31 + m_32 V L J _ dy1 Q-1(y1

+

2f

z

dye

J

y2

dy1 - 0 - 1

),3z

(y1 )

) Q-2 (y1 ) 6-i (y2 )Q-2 (y2 ),

where mfij are arbitrary complex numbers. Now we should find the mapping i entering the Gauss decomposition (4.15) of the mapping p+lµ_. Here we use the follow-

4.2 Abelian Toda systems

217

ing Gauss decomposition of a general element a of the Lie group SL(3, (C):

a = n_hn+1,

where n+ E N± and h E H. As follows from the formulas of section 3.2.4 the explicit form of the nonzero matrix elements of the matrices n_, h and n+ is

n-21 =a21-, n-31 =a31-, n-32 = all

hll = all ,

a11a22 - a21a12

all

h22 =

n+12=

a11a22 - a21a12 ,

all a12 all

a11a32 - a12a31

n+13=

n+23=- a11a23

h33 =

1

a11a22 - a21a12 a12a23 - a13a22 a11a22 - a21a12

- a13a21

a11a22 - a21a12

Actually, here we need only the expressions for the matrix elements of h. The expressions for the matrix elements of nt will be used later for the construction of the general solution of a higher grading generalisation of the system under consideration. Having introduced the notation is = µ+lµ_, we find for the matrix elements of the mapping is the expressions K11 = 1 - P+12/1-21 - P+13/1-31 + /1+12µ+23/1-31, K12 = -I1+12 - /1+13/1-32 + /d+12/1+23/1-32, K13 = -/1+13 + 11+1211+23,

K22 = 1 - /1+23/1-32, K31 = /1-31 i

k21 = P-21 - /1+23/1-31,

K23 = -/1+23,

K32 = P-32,

K33

= 1.

After some calculations we obtain 7711 = 1 - /1+12/1-21 - /1+13/1-31 + /1+12/1+23/1-31

µ+13/1-31 - P+23/1-32 + /1+13/1-32/1-21),

7722 = 77111(1 1

7733 = 7/1111/22

From relations (4.6), (4.19) and (4.22) it follows that 01 = Q+i -17/11,

32 = 0+2P-21117722-

Thus, we finally come to the following expression for the general solution of system (4.20), (4.21): 01 = O+1/3-1 (1 - P+12P-21 - /1+13/1-31 +/1+12/1+23/1-31), (4.23)

02 = 3+20-2(1 - /1+13/1-31 - /1+23/1-32 + p+13/1-32/1-21); (4.24)

Toda-type systems and their explicit solutions

218

which is parametrised by two chiral, ,3_1, /3_2; and two antichiral, /3+1, /Q+2 functions taking values in C'.

4.2.2 Lie group SO(2r + 1, C) The situation is more complicated for the case of the Lie group SO(2r + 1, C). The Cartan matrix has now the form 2

-1

-1

2

0

-1

2

0 0 0

0 0 0

0 0 0

..

0

-1 ...

0

0

0 0

0 0

0 0 0

k= 2

-1

-1

2

-2

0

-1

2

...

0

After some algebra, one arrives at the following expression for the inverse matrix: 2

2

2 .

2

2

2

2

4

4

...

4

4

4

2

4

6 .

6

6

6

2

4

6

2(r - 2)

2(r - 2)

2(r - 2)

2

4

6

2(r - 2)

2(r - 1)

2(r - 1)

1

2

3

r-2

r-1

r

1

k-1 2

This expression allows one to show that

2ki =i(2r-i+1), 1
2k,. =r(r+1)/2.

The group H consists of all diagonal matrices from SO(2r + 1, C), and the mapping -y has the form

y=

a0 0

1

0

0

0 0 (aT)-1

where a is a mapping taking values in the Lie group D(r, (C) of all complex diagonal rxr matrices. Using the explicit expressions for the Cartan generators h, given in table 1.4, we obtain that, in

4.2 Abelian Toda systems

219

terms of the functions ,32, the mapping a can be written as 0

,(3

'02

0

0

a= 3-1 r2Nr-1

0

0

0

0r 11 0r O

0

/

From this expression we conclude that the function 0, enters the parametrisation of the mapping ry only in the form /3, 2. Therefore, to obtain a global parametrisation of -y one must use the function 02 as a basic object. Introducing the functions r

of=,Q2,

1
6r

we obtain the equations (9+(6i 1(9-61) = -2r6i 262,

0+(62 1,9-bi) = -i(2r -,_1Si i +2bi+1, 1)(S11 < i < r, 18_Sr) _ -r(r + 1)6r-1(Sr 1,

which describe the abelian Toda system associated with the Lie group SO(2r + 1, C).

4.2.3 Lie group Sp(r,C) We proceed to the case of the Lie group Sp(r, C). The Cartan matrix in this case is the transpose of the Cartan matrix for SO(2r + 1, C). The inverse matrices are certainly connected by the transposition. After some calculations we obtain

2ki = i(2r - i),

1 < i < r,

2kr = r2.

The group H consists of all diagonal matrices from Sp(r, C); hence, the mapping -y can be represented as ry =

(a)-1 0

\ I

,

(4.25)

where a is a mapping taking values in the Lie group D(r, (C). Using the explicit expressions for the Cartan generators hi given in table

1.4, we obtain that, in terms of the functions Pi, the mapping a

Toda-type systems and their explicit solutions

220

has the form Ql

0

01 1/32 0

0

0

0

...

00

1

a= 0

... ...

0

0

,6r laar-1

0r-10'

0

Therefore, in this case the functions ,Qi provide a global parametrisation of the mapping -y. The corresponding abelian Toda system is

a+(01 la-,al) = -(2r - 1),Q 2/32,

a+(3 la-0j) = -i(2r - i)Qi-l/3a

2/jz+l,

1
4.2.4 Lie group SO(2r,C) As in the last example take the Lie group SO(2r,C). It follows from the formulas of section 1.3.4 that the explicit form of the Cartan matrix for this case is 2 -1 0 0 0 0 -1 2 -1 0 0 0 2 0 -1 0 0 0 k= 2 -1 -1 0 0 0 0 0 0 -1 2 0 0 0 0 -1 0 2 and for the inverse of k one has the expression

k_1

_

4

4

4

4

2

2

4

8

8

4

8

12

...

8

4

4

...

12

6

6

4

8

2

4

12

4(r - 2)

2(r - 2)

2(r - 2)

6

2(r - 2)

r

r- 2

2

4

6

2(r - 2)

r- 2

r

1

4

It is not difficult to see that

2ki = i(2r - i - 1),

1 < i < r - 1,

2kr_1 = 2kr = r(r - 1)/2.

4.2 Abelian Toda systems

221

Here the group H is the same as for the case of the Lie group Sp(r, C), and the mapping ry has the form of (4.25), where the mapping a, in terms of the functions 032, is /31

0

... ...

0 31 1 a2

0

0

0

0

)3r 12I3r-l/3r

0

0

ar 11Nr

a= I

0

O

0

/

From this expression we conclude that a global parametrisation of the mapping -y can be realised with the functions 6

13

xx

1
8r-1

>

-Or-10r,

8r =R2r

and the corresponding Toda system is a+(S11a-81) = -(2r - 2)81 252,

a+(6z la-oi) = -i(2r - i - 1)82-16i 262+1, a+\Sr,la-Sr-1)

_-

r(r - 1) 2

(l r-26

1 < i < r,

_Z1

6r + Sr-26r 1),

a+(Srla-Sr) _ -r(r - 1)6r_28r1.

Exercises 4.2 Due to the symmetry of the root system or the Dynkin diagram for sl(r, C), one can perform the reductions, sometimes called foldings, s1(2r + 1, C) -+ so (2r + 1, C) and 5((2r, C) -4 sp(r, C). Quite naturally this symmetry is manifested in the Cartan matrix entering the corresponding abelian Toda sys-

tem. Obtain the abelian Toda systems associated with the algebras so(2r + 1, (C) and sp(r, C) from those for the complex special linear algebra using the relevant equations for of the Toda fields fi, see (4.17). 4.3

Using the folding so(7, C) -> G2, obtain the abelian Toda equations for the case of the algebra G2.

4.4

Show that the general solution of equations (4.20) and (4.21) can be described by the chiral and antichiral functions

fl, 0 < a < 2, entering the decomposition 61 = E2=0 fa fa

and submitted to the condition det at fb = ±2\. Repre-

222

Toda-type systems and their explicit solutions

sent the solution of this condition as nested integrals of two arbitrary (anti)chiral functions. 4.5 Starting from representation (4.14) with the boundary condition at = e and relation (3.126) with ju) = Iv) = Ii), where Ii) is the highest weight vector of the ith fundamental representation, find the solution for the abelian Toda system (4.18) in the form of finite sums of nested integrals. Use a Verma basis.

4.3 Nonabelian Toda systems 4.3.1 Lie group Sp(r, C) Let us begin our discussion of nonabelian Toda-type stems with an example based on the complex symplectic group Sp(r, C). En-

dow the corresponding Lie algebra sp (r, C) with a Z-gradation associated with the st(2, C)-subalgebra constructed as follows. Define the Cartan generator h as the element with the characteristic 0

0

0

2

0

Using the explicit form of the inverse of the Cartan matrix for the Lie algebra sp(r, C), given in the previous section, and the relation r

h = E (k-1)ijnjhi,

(4.26)

i,j=1

where nj are the labels entering the characteristic of h, we obtain n

h = E ihi. i=1

The explicit form of the element h is _ Ir 0 h

0

-Ir

Introduce a Z-gradation of sp(r, C) choosing as the grading operator the element h/2. We obtain three grading subspaces, g0 and g± 1. The subspace g0 - Cl is formed by 2r x 2r matrices a of the block form

4.3 Nonabelian Toda systems

223

where x is an arbitrary complex rxr matrix. The subspace g+1 = n+ is composed of the matrices a of the block form _ 0 a

0

0

'

with y being an arbitrary complex rxr matrix which satisfies the relation yT = y. Finally, the subspace g_1 = n_ consists of the matrices

O O1

a

z

0 J'

where the complex rxr matrix z satisfies the condition zT = Z. Consider now the corresponding subgroups of Sp(r, Q. The subgroup H is formed by 2r x 2r matrices a of the block form

X

a=

0

0

(XT)-1

where X is an arbitrary complex nondegenerate rxr matrix. The subgroup N+ is composed of the matrices a of the block form a

0

CIr

Y)

Ir

with Y being an arbitrary complex rxr matrix which satisfies the relation YT = Y. Finally, the subspace N_ consists of the matrices CIr

Z Ir

a

°)

where the complex rxr matrix Z satisfies the condition ZT = Z.

One can see that the Chevalley generators of an s((2, C)subalgebra corresponding to the Cartan generator h can be chosen in the form

x+- (0

0

)'

X

(Ir

0

Parametrise the mapping -y as

7-\0 (/)-1) '3

0

where the mapping /3 takes values in the Lie group GL(r, Q. Using this parametrisation we obtain the following matrix equation: -/3-1/3T-1.

Now construct the general solution of this equation.

(4.27)

224

Toda-type systems and their explicit solutions

Write the mappings p± in the form Ir /i+12 P+ = Ir 0

0 Ir

'

where the mappings µ+12 and /1-21 satisfy the relations p+12 = +12 and pT 21 = P-21 Parametrising the mappings 'Y+ as 0

Q+

Y+ =

0

(Q+)-1

,

we find that, in our case, equations (4.13) are equivalent to the equations

041-21 = ('3T)-1)3-1. The general solution of these equations is given by a+/L+12 = )3+)3+,

fz+

/1+12(z+) = m+12 +

dy+,3+(y+)/+(y+),

J z

dy-OT-1(y

) = m_21 + f(y ), where m+12 and m_21 are constant complex rxr matrices satisfying the relations m+12 = m+12 and mT 21 = m-21. For the mapping n p+1µ- we have the representation P-21 (z

/L+12/1-21 /1-21

-/t+12 1

Ir

.

Here it is convenient to consider all the matrices arising in our construction as 2x2 matrices over the associative algebra Mat(r,(C).

The Gauss decomposition for such matrices is given in section 3.2.4. Using the formulas obtained there, we obtain the following representation for the general solution of equation (4.27): N = Q+1(Ir - /1+12/1-21)13-,

where /3_ and j3+ are arbitrary chiral and antichiral mappings taking values in GL(r, Q. Note also that the subgroups H+ in the case under consideration are isomorphic to the Lie group O(r, C), and the symmetry transformations (4.12) look as follows:

where the mappings e+ satisfy the relation T

E+E+ = Iri

the mapping E_ is chiral and the mapping E+ is antichiral.

4.3 Nonabelian Toda systems

225

4.3.2 Lie group SO(2r + 1, C) Consider now some nonabelian Toda systems associated with the complex special orthogonal group SO(2r + 1, Q. Introduce a 7G-

gradation of the Lie algebra o(2r + 1, C) connected with the s((2, C)-subalgebra having as its Cartan generator h the element with the characteristic n r-n

2 00 OO

2

Here n is a positive integer, such that 0 < n < r. Using the explicit form of the inverse of the Cartan matrix for the Lie algebra o(2r + 1, (C), given in section 4.2.2, and relation (4.26), we obtain n

hi(2n-i+1)hi+ n(n2+ ) 1

i-1

r-' 2

hi+hr

.

i=n.+1

One can verify that the corresponding Chevalley generators of the sl(2, C)-subalgebra can be chosen in the form

xt = "-1 i(2n - i+ 1)x±i +

n(n + 1)

2

xtn,...,r

i=1

Here and in what follows we use the notation x+ik,...,23,i2J1

x-ik,...,13,i2J1

-

[x+ik, ... [x+i3 f [x'+i2) x'+il .. 11 [.. . [[x_ti1, x_i2], x_i3] ... , x_ik

It is convenient to write the matrices xt in the following block form:

x+ =

a+

b+

0

0 0

0 0

-b+T

a_

X_

=

,

-a+

b_

0 0

0 0

,

-b? -a-

0

where a± are n x n matrices, b+ is an n x (2(r - n) + 1) matrix, and

b_ is a (2(r - n) + 1)xk matrix. The explicit form of the matrix a+ is 0 I

0

1'2n 0

a+ _

...

.

I

0

0

0

0

.

0

0

..'

0

0

0

...

0

.

(n - 1)(n + 2)

0

Toda-type systems and their explicit solutions

226

while for the matrix b+ we have

10

0

0

0

0

0

0

0

0 1

0

0

0

0

0

b+ _ n(n + 1)

The matrices a_ and b_ are the transposes of a+ and b+, b- = (b+)t. a- = (a+)t, The matrix valued function y can be also written in a block form

a0 ry =

0 0

0

(4.28)

0

j3

0

(aT)-1

where the mapping a takes values in D(n, C), and the mapping /3 takes values in §-O-(2(r - n) + 1, (C) which can be written as /3T // N I' - 12(r-n)+1

Using the above formulas we obtain the relation 0 a-la+a a-'b+/3 7-1x+'Y =

0 0

-(a-'b+0)T -(a-la+a)T

0

0

which allows to write equations (4.16) as

Ma-ia-a)

a-'a+a]

-

[a-, a-'b+,aba+()3-'a-)3) = b-a-1b+)3 - (b-a-1b+(.3)T. To be more concrete, consider the Lie group SO(7, (C) and put n = 2. In this case the SL(2, C) subgroup is generated by the elements

h = 4h1 + 6h2 + 3h3 i

x± = 2x±i + v'-3xt2,3,

and the grading operator is

q=2h1+3h2+Zh3.

(4.29)

The grading subspaces have the form 90 = g-a3 ®h ®9+a3, 9t1 = 9ta, ®gta2 ® 9±(092+093)

(4.30) (4.31)

®Qt(a2+2093)'

0±2 =

9±(091+092) ®g±(a1+092+a3) ®9±(091+092+2093)

9t3 =

gf(al+2092+2093)

,

(4.32) (4.33)

4.3 Nonabelian Toda systems

227

Parametrise the mapping 'y as 1=

of+1+3ef-x-3ef1h1+f2h2+f3h3

Such a parametrisation leads to a mapping of form of (4.28) with

a

- /efi (\

0 e-fl+f2

0

/

' -ef2-2f3f+

e-f2+2f3(1 + f-.f+)2

I

v2f+(1 + f-f+) 1 + 2f-f+ e-f2+2f3f-(1 + f_f+) -e-f2+2f3 f2

-

ef2-2f3 f'+

-YLf-

ef2-2f3

Direct calculations give

y-'a

=a-fl hl+a_f2h2+(a-f3+f-a-f+) h3 +

ef2-2f30_f+ x+3 + e-f2+2f3

(a-f-

- f-' a- f+) x-3

Further, one can see that

-

-4e-2J1+f2h1

3eh-f2(1

+ 2f-f+) (2h2 + h3) + X+3+ + f-f+) x_3. Using the above relations, we obtain the following system of equa[x-,'Y-lx+'Y] =

6ef1-2f3f+

6ef1-2f2+2f3 f-(1

tions:

a+a-fl = -4e-2fl+f2 -6eh-f2(1 0+0-f2 = + 2f-f+),

0+(a-f3+ f-a-f+) = /`"+(ef2-2f3 `8_f+)) = Me-f2+2f3(a-J-

-3ef1-f2(1

+ 2f-f+),

6efh-2f3f+,

- J?a_f+)) =

6ef1-2f2+2f3 f_(1

+ f-f+).

Note that in some physical applications, in particular, in relation to black holes and relativistic string models, a constrained version of these equations arises, namely, ,9+,9-vi = 2ev1 - ev2 - ev3,

a+a-V2 = -ev' + 2e"2 - 2

sinh y2-y3 3 a+ V449-V4, cosh 3,2 4

a+8-v3 = -e'" + 2e'2 +2

a+ tank 2 y2

13

4

sinh y2-V3

cosh

a_v4\1 +a_

3 v2 - v 4

'9+V40-V4,

(ta.h

2 y2

4

y3

a+v4 I = 0;

Toda-type systems and their explicit solutions

228

for more detail see Gervais & Saveliev (1992); Barbashov, Nesterenko & Chervyakov (1982) and Leznov & Saveliev (1992). Here

v1=-2fi+f2, V2 = fl - f2 + 2Arsinh (f_ f+) 1/2,

v3 = fl - f2 - 2Arsinh (f- f+)1/2 a+v4 = -(1 + f_ f+)a+[ln a_v4 = 1 + f- f+ &-[ln

1+2f-f+

(f_e-f2+2f3)] + a+[ln (f+ef2-2f3)]

.f-.f+]/2,

- a-[ln f-f+]/2.

We suggest that the reader perform such a reduction as an exercise.

Exercises 4.6

Prove the compatibility condition 19+(19-V4) = 09-(19+V4), and

obtain the above given constrained system.

4.4 Higher grading systems In general, system (4.9)-(4.11) describes the Toda-type fields coupled to matter fields parametrising the mappings v± .. ; see Gervais & Saveliev (1995). For the case of affine Lie algebras this system was studied in Ferreira et al. (1996). Here we conventionally call the fields parametrising the mapping -y the Toda fields, while the fields entering a parametrisation of the mappings vt are called the matter fields. The reason for this becomes clear from the obser-

vation that, using a relevant specialisation of the Inonii-Wigner contraction, one can bring to zero the back reaction to the Toda fields for some or all matter fields. In particular, we can define the subalgebra to of g as to :_ go (D 0±1 (D 0±21 (D

and take as t1 the direct sum of all the remaining grading subspaces. Performing the Inonu-Wigner contraction now, as is described in section 1.1.11, we arrive at the case where equation (4.10) does not contain the mappings As a result, we obtain the equation which looks similar to the equation describing some standard Toda system, but with a different meaning for the

4.4 Higher grading systems

229

elements x+ which belong here to the subspaces g±1. Note that this type of system has been discussed in Ferreira, Miramontes & Guillen (1995). Evidently, there are many other meaningful possibilities for obtaining contracted systems. In this section we discuss the equations corresponding to the case where l = 2 when a system of type (4.9)-(4.11) is rewritten in the form a+('Y-la-'y) = [x-,-y-lx+'y] + [v_,-y-1v+7], a+v- = [x-,'y-1v+7], a-v+ _ [x+,7v-7-1),

(4.34) (4.35) (4.36)

where we denote vt1 simply by vt. Note that this system looks very similar to a system based on a semi-integral gradation of the Lie algebra g; see Leznov (1985) and Feher et al. (1992); and to the supersymmetric Toda system associated with a superalgebra, see Leites, Saveliev & Serganova (1986), where, however, the mappings vt are parametrised by odd functions. Some special abelian cases of system (4.34)-(4.36) were also considered in Chao & Hou (1994); moreover, in Chao & Hou (1995) these authors have studied an asymmetric, as they called a heterotic Toda system, corresponding to the Lie algebra sr(r+1, (C) endowed with the principal gradation, and a choice when l_ = 1, 1+ = 2.

4.4.1 Lie group SL(r + 1, C) Consider the Lie group SL(r + 1, C) and endow the corresponding Lie algebra sr(r + 1, C) with the principal gradation. Parametrise

the mapping y as in (4.17) and the mappings vt by v± =

gfix±ii=1

The general form of the elements xt E g±2 is r-1

xf =

cix±i,i+i i=1

with some constants ci. One can easily see that the corresponding equations look as follows: r

a+(0i la-0i) _

c?

j=1

kij-ki+1,i

Toda-type systems and their explicit solutions

230

r

_C2 1 r

II

11

j=1

j=1

r

afgzFi = ±(Ci_1

q-iq+i, r

11 3,_

gf(i_1) - Ci

j=1

11 13,-k=+l,i

Nj

gf(i+1)),

j=1

where ci - 0 for i = 0, r; and qfi - 0 for i = 0, r + 1. Introducing the functions r Si

13- k,i

=

11 j=1

we arrive at the equivalent system of equations r

19+(62la-Si)

- j=1 kijsj(cj6j+1 +Cj_1Sj-1 +q-jq+j),

afgT-i = ±(ci-lsi-lq+(i-1) - cisi+lq±(i+l));

compare this with the equations given in Gervais & Saveliev (1995).

For the case where r = 2 there is actually only one possibility of choosing the elements x±, namely,

x+=

0 0 0

0 0 0

1

0 0

,

x_=

0 0 1

0 0 0 0 0 0

This choice gives us the following equations:

-()1N2)-l - J1 202q-lq+l,

a+(Nl

-(,3112)-l

a+q-1 = -0102 2q+2, a-q+l ='3102-2q-2,

- 01'3z-2 q-2q+2,

a+q-2 =

)3 12

'2q+1,

0- q+2 = -N3l 2 /32q-1.

The subgroups Ht are isomorphic to the Lie group GL(1, C) and the mappings fit, entering symmetry transformations (4.12), can be parametrised as follows: et 0

0 _2

0

0

et

where e_ is a chiral function, e+ is an antichiral one, and both of them take values in CX Using such a parametrisation, we see .

4.4 Higher grading systems

231

that in our case symmetry transformations (4.12) are given by the relations i

3

q-1 = -q-1,

)32

= E+E-)32,

i

-3

q2 = E

q-2,

q+2 = E+q+2

q+1 = E+3q+1,

We now proceed to the construction of the general solution of the system under consideration. Using (4.22) we obtain the expressions A+ =

0

S+1

0 0

0 0

0

0

0

(_1

0

0

a+10+2

A+ =

(+2

)3-1

(-2 0 where C-1, S-2 are arbitrary chiral functions, while (+1, S+2 are arbitrary antichiral functions. The system of equations determining the mappings p± looks in our case as follows: 0

r

a+µ+12 = 5+1,

-2

a+/+23 = 5+2,

494413 - 12+12a+µ+23 = Q+1/3+2,

a-µ-32 a-P-31 - P-320-t -21 = 19-P-21 = S-1,

(-2, 1,3-2*

The general solution of this system is fz+

/-1+12(z+) = m+12 + {1+23(Z+)

= m+23 +

J + lz+

dyi S+1(yi ),

dyi'+2(Yi ),

[1+13(Z+) = m+13 + m+12 z+ I+

I J

z

dyi (+2'Y1 )

+

z+

y+ f 2

dyi O+1 (Yi )F'+2 (y2) +c+ f dy2 Jc+ dYi S+1 (Yi )(+2 (y2 ),

p21(z)

= m-21 +

p-32(z ) = m-32 +

dyl S-1 (y1

r

dy l S-2(y1 ),

[1-31(Z-) = m-31 + m-32

dyl (_1 (yl )

Toda-type systems and their explicit solutions

232

_

+ f z dyl

i (yl )Q-2 (y1) + f z dye

f

_

yz

c

dyl (-2(yl)(-1(y2 ),

where m+Z3 are arbitrary complex numbers. It is clear that we have for the functions N1, ,132 expressions (4.23) and (4.24), where the matrix elements of the mappings p+ are given above. To find the expressions for the functions qf1 and qt2 we use the analogues of relations (3.95) and (3.92), which in our case look as follows:

v+ ,9+v+_= (7j-v- A+v-rl)n+, 1

1

V-1.9-V-

_

1

(7lv+'A-1/+71-1)n_.

Substituting these equalities into (4.8) and (4.7), one obtains

v+ _

/ 'Y+-1v- 1"+v-'Y+)n+

Now, taking into account the explicit formulas for the Gauss decomposition (4.4) given in section 4.1.2, we come to the expressions q+1 = N+1'N+2(+1

+

P-32 + /2+12P-31 - /-P+12/-1-32P-21

;11

+2

1

- /413/1-31 - P+23/t-32 + P+13/-x-32P-21

q+2 =+1)3+25+2

R2 /?1

/1-21 + /1+23/2-31

h'+1/'+2 1

- /L+12P-21 - /2+13P-31 + P+12/1+23µ-31

q-1 = 210-25

/2+23 - /1+13/-21

2

/-1+13/1-31 - /1+23/1-32 + /L+13/2-32/1-21 -1)32

q-2 =

-2(-2 2

q

+ Q -1 -2

P+12 + /1+13/--32 - 4+12P+23/1-32 1 - /2+12/L-21 - /2+13P-31 + /1+12/1+23P-31

where, as above, the functions p±ij are determined by four chiral functions /3-1,-2 and (-1,-2, and by four antichiral functions ,3+1,+2 and (+1,+2

4.4 Higher grading systems

233

4.4.2 Lie group SO(7, C) Now consider the Lie group SO(7, C), and provide the corresponding Lie algebra o(7, C) with the gradation defined by the grading operator (4.29). The corresponding grading subspaces are given by (4.30)-(4.33). The subgroup H is isomorphic to D(2, C) x SO (3, (C). Introduce for it the following local parametrisation:

7=

eah3eb(x-3+x+3)edh3ea2h2ehlal

where a, b, d, a1 and a2 are complex functions. The mapping -y is here of form (4.28) with CY =

\

eai 0

e-a2+2(a+d) cosh2 b 1 e-a2+2d sink 2b v'2

0

e-ai+a2 e2a sinh 2b

-ea2+2(a-d) sinh2 b

cosh 2b

- 1vr2ea2-2d sinh 2b

-e-a2-2(a-d) sinh2 b -e-2a sinh 2b ea2-2(a+d) cosh2 b After some lengthy but simple calculations, we obtain -y-la_ry = a_al h1 + a_a2 h2 + (cosh 2b a_a + a_d) h3 + ea2-2d (sinh2b a_a + a_b) x+3 sinh2b a-a +,9-b) X-3+ Now we should choose the elements x± E 9±2. Seemingly, the simplest, though rather nontrivial, possibility arises when one takes e-a2+2d(_

x± = x±1,3,2,3

For such a choice of x± we obtain [x-, Y-1x+'Y]

=

2e-2a-al sinh2b x_3. + h2 + h3) + Introduce now an appropriate parametrisation of the mappings v±. Convenient bases in the subspaces g±1 are formed by the elements -4ea'+a2-2(a+d) cosh2 b(h1

e±o = x±1,

etl =

- 1

1

et2 = 1 X±2,3,

2 x±3,2,3

e±3 = X±2-

Using these bases, we parametrise the mappings v± as 3

v± = E 4±i e±i i=o

Toda-type systems and their explicit solutions

234

Define the functions q+i by 3

7-lv+'Y

=

4+ie+i i=O

The explicit forms of the functions q+i are q+o = e

-2a1+a2

q+o,

4+1 = eal -2(a+d) cosh2 b q+1

-1

ea,

-2d sinh 2b q+2 -

eat+2(a-d) sinh2 b

q+3,

sinh2b q+1

4+2 =

+ eai-a2 cosh 2b q+2 +

e2a+a1-a2 sinh 2b q+3,

-ea1_2a2+2(-a+d) sinh2 b q+1

q+3 =

+

1ea, -2a2+2d sinh 2b q+2 + eai -2a2+2(a+d) cosh2 b q+3

We will also need the expressions for the functions q-i defined as 3

'Yv-'Y-1 = E q-ie-i, i=0

Using the parametrised form of the mapping 'y, we come to the representation q-o = e -2a1+a2 q-o, 4-1 = eal-2(a+a) cosh2 b q_1

-

Q_2 =

eai-a2-2a sinh2b

-- e 1

q_2 -

+ eai-a2 cosh 2b q-2 +

q-3 =

-eat+2(a-d) sinh2

+

eal-2a2+2(-a+d) sinh2 b q-3,

ai -2d sinh 2b q_1 ea1_2a2+2d

sinh 2b q_3i

b q_1

2eai-a2+2a sinh2b q-2

+eai-2a2+2(a+d) cosh2 b q_3.

Now one can obtain the following relation: [v-,'Y-lv+'Y]

-

= -q-oq+o

h1

(q-1q+1 + q-2q+2 + q-34+3) h2

-

-

(q-1q+1 +

Zq-2q+2) h3

f(q-2q+1 - q-3q+2) x+3 - f (q-1q+2 - q-24+3) x_3.

With these formulas we come in the case under consideration to the following equations for the Toda fields: a+a_a1 =

a+a_a2

-4e-a1+a2-2(a+d) cosh2 b =-4e-ai+a2-2(a+d) cosh2 b

- q-o4+o,

- q-14+1 - q-2q+2 - q-3q+3,

4.4 Higher grading systems

235

-4e-al+a2-2(a+d) cosh2 b

a+(a_d + cosh 2b a_a) =

a+(ea2-2d(a_b + sinh2b a_a)) = a+(e-a2+2d(a-b - sinh2b a- a)) =

- 2(q-1q+i + q-2q+2), (q-3q+2 - q-2Q+1), 2e-2a-a' sinh2b

+ 1(q-2q+3 - q-1q+2); see Gervais & Saveliev (1995). Now, using the relations [x-,'Y-1v+'Y] _ -2q+1e_o + 2q+oe-1, 1]

= 2q-1e+0 - 2q-oe+1, we obtain the equations for the matter fields [x+, ryv-'Y

a±q+0 = ::F2q±1, a±gT-2 = 0,

a±q+1 = ±2q±o, a±qT3 = 0.

Let us give an example of a contracted form of the above considered system. Here we use the notation of section 1.1.11. To perform the Inonii-Wigner contraction of the algebra so (7, C) with the chosen gradation, define E0=9-2®90®g+2, t1 = 9-3 ® 9-1 ®9+1 ®g+3

After the contraction we arrive at the algebra g' which is not already simple; it is the semi-direct sum of the Lie algebra g((1, cC) x

5(5, C) and the ten-dimensional commutative subalgebra with a basis formed by the elements x±1, x±2, x±3,2, x±3,2,3, x±2,1,3,2,3 Here g[(1, C) is generated by the element 2h2 + h3i while the subalgebra 5(5, (C) is spanned by the elements x±1,2, x±1,3,2, x±1,3,2,3,

x±3. The resulting system has the form

aaa=

-4e-a1+a2-2(a+d) cosh2 b,

a+ a- a2 =

-4e-al+12-2(a+d) cosh2 b,

a+(a_d + cosh 2b a- a) _

-4e-al+a2-2(a+d) cosh2 b,

a+(ea2-2d(a_b + sink 2b a_a)) = 0, a+(e-a2+2d(a_b - sinh2b 9- a)) =

a+q-o =

-2ea,-2d(e-2a cosh2 bq+1

- 1 sink2bq+2 a_q+o =

2e-2a-al sinh2b,

2ea1-a2-2a

e2a sinh2 bq+3),

(eat-2d cosh2 b q-1

- 72 sink 2b q-2- e-a2+2d sinh2 b q-3),

236

Toda-type systems and their explicit solutions

a+4-i = 2e -2a1+a2 9+0, a-4'+i = -2e -2al+a2 4'-0, atgT2 = 0, 9±q:F3 = 0; compare with those in Gervais & Saveliev (1995). In accordance

with the general integration scheme discussed above, this contracted system, as well as the initial one, can be solved in an explicit way.

Exercises 4.7 Using a substitution, analogous to those for the functions va,1 < a < 4, given in the end of section 4.3.2, obtain the constrained version for nonabelian systems considered in the present section.

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Index

action of a Lie group effective, 115 free, 119 left, 117 right, 115

transitive, 117 admissible mapping, 174 algebra, 1 abelian, 2 associative, 2 commutative, 2 complex, 2 graded, 139 quotient, 8 real, 2 unital, 2 almost complex structure, 87 canonical, 88 integrable, 88 antiautomorphism, 4 antihomomorphism, 4 antilinear, 5 atlas, 62 bundle, 122 maximal, 62 automorphism algebra, 4 of a root system, 24 base

of a root system, 27 of a topology, 57 Bianchi identity, 128 bilinear form invariant, 13 Borel subalgebra, 133 opposite, 134 Borel subgroup, 133 Borel-Morozov theorem, 134

bundle, 121 trivial, 122 bundle morphism, 121

Cartan generator, 34 Cartan matrix decomposable, 37 generalised, 36 indecomposable, 37 of affine type, 36 of finite type, 36 of indefinite type, 37

Cartan subalgebra, 31 centraliser, 31 centre of an algebra, 8

chart, 61 admissible, 62 bundle, 122 chiral, 208 complex, 82 Chevalley generator, 34 Chevalley involution, 35 hermitian, 53 closed mapping, 57 closed set, 55 closure, 57 commutator, 3 of vector fields, 70 complex covector field of type (0,1), 90 of type (1,0), 90 complex structure, 21 canonical, 21 Lie, 21 complex vector field of type (0,1), 86 of type (1,0), 86 complexification of a bilinear form, 50 242

Index of a Lie algebra, 22 of a real vector space, 21 component of a topological space, 59 conjugation of an algebra, 5 connection, 125 connection form, 126 consistent pair of mappings, 182 continuous mapping, 57 contraction of a Lie algebra, 19 coordinate functions, 61 coordinates of a point, 61 cotangent space, 74 cotangent vector, 74 covector, 74 covector field, 75 antiholomorphic, 90 holomorphic, 90 cover of a topological space, 60 open, 60 Coxeter number, 29 curvature form, 128 curve, 69

at a point, 69 cyclic vector, 11

defining relations, 34 dense subset, 57 derivation of an algebra, 8 inner, 8 diffeomorphism, 61 differentiable structure, 62 smooth, 63 differential of a function, 75 differential of a mapping, 69

direct product of algebras, 9 of Lie groups, 100 of manifolds, 66 distribution, 98 complex, 99 holomorphic, 99 involutive, 98 smooth, 98 Dynkin diagram, 30 extended, 133

embedding, 96 embedding of sl(2, C), 142 integral, 143 principal, 147

243

semi-integral, 143 endomorphism, 2 module, 11 equivariant mapping, 115 Euclidean metric, 56 Euclidean space, 25 exponential mapping, 103 exponents, 148 exterior derivative, 80 exterior form, 79 exterior product, 79 fibre, 121 fibre bundle, 122 complex, 122 holomorphic, 122 principal, 124 fibre preserving mapping, 121 flag, 138 full, 138 flag manifold, 137 flow of a vector field, 73

Frobenius theorem, 98 Fubini-Study metric, 202 function antichiral, 208 antiholomorphic, 82 chiral, 208 holomorphic, 82 fundamental form, 201 gauge transformation, 151 Gauss decomposition modified, 172 of a matrix, 164

of an element of a Lie group, 166 general linear group complex, 101 real, 100 generalised Plucker relations, 207 G-manifold left, 117 right, 115

gradation of a vector space, 138 grading condition general, 158 specified, 161

grading operator, 140 characteristic of, 140 Grassmann manifold, 119

244

Grassmannian, 119 group of automorphisms of a root system, 24 of an algebra, 4 group of inner automorphisms, 129

height of a root, 27 hermitian form, 20 hermitian metric, 200 highest weight vector, 190 homeomorphic spaces, 57 homeomorphism, 57 homogeneous coordinates, 116 homogeneous element, 139 degree of, 139 homogeneous space, 117 homomorphism algebra, 3 antilinear, 5 module, 11 homotopy equivalent loops, 60

ideal of an algebra, 7 generated by a set, 7 left, 7 right, 7 trivial, 7 two-sided, 7 identity element, 2 immersion, 94 Inonii-Wigner contraction, 19 inner automorphism, 129 integral curve, 72 integral manifold, 98 maximal, 98 inverse mapping theorem, 69 involution of an algebra, 4 hermitian, 5 isomorphism algebra, 3 module, 11 isotropy subgroup, 117 Iwasawa decomposition, 198 Jacobi identity, 3 Jordan decomposition, 31 Kahler manifold, 201 Kahler metric, 201 Kahler potential, 201 Kac-Moody algebra, 38

Index Killing form, 14

length of a root, 26 Levi theorem, 17 Lie algebra, 3 associated, 3 compact, 53 free, 33

nilpotent, 16 reductive, 12 semisimple, 17 simple, 17 simply laced, 29 solvable, 16 Lie algebra of the Lie group, 102 Lie bracket, 71

Lie group, 99 complex, 100 semisimple, 132 Lie subgroup, 107 Liouville equation, 186 loop, 60 constant, 60 manifold almost complex, 87 chiral, 208 complex, 83

real analytic, 63 smooth, 63 topological, 62 mapping antichiral, 208 antiholomorphic, 82 chiral, 208 holomorphic, 82 of class C°°, 61 of class C', 61 of class C', 61 real analytic, 61 regular at a point, 69 smooth, 61 matrix of a bilinear form, 6 matrix of an endomorphism, 4 Maurer-Cartan form, 112 module, 10 cyclic, 11 diagonalisable, 189

dual, 12 extremal, 190

Index finitely generated, 11 fundamental, 191 indecomposable, 12 semisimple, 12 simple, 11

245

regular pair of mappings, 184 regular point of a mapping, 96 regular subgroup, 132 regular value of a mapping, 96 regular vector, 28

representation of a Lie algebra

neighbourhood of a point, 57 nilpotent element, 31 normaliser, 31 open ball, 56 centre of, 56 radius of, 56 open mapping, 57 open set, 55 operator nilpotent, 31 semisimple, 31 orbit, 115 orthogonal algebra complex, 7 real, 6 orthogonal group complex, 109 real, 109 parabolic subalgebra, 135 parabolic subgroup, 135 path, 60 7r-system, 133

Poincare-Birkhoff-Witt theorem, 18 principal gradation, 142 projective space, 116 pseudo-metric, 204 pseudo-orthogonal algebra, 6 pseudo-orthogonal group, 109 pull-back of a differential form, 81 of a tensor field, 78 push-forward of a tensor field, 78 quasideterminant, 163 radical of a Lie algebra, 17 rank of a Lie algebra, 31 rank of a root system, 24 real form of a Lie algebra, 52 normal, 53 realification of a vector space, 19 reality condition, 194 reflection, 23

coadjoint, 12

dual, 12 fundamental, 191 orthogonal, 145 symplectic, 145 representation of a Lie group, 103 complex, 103 real, 103 representation of an algebra, 9 adjoint, 12 completely reducible, 12 complex, 9 irreducible, 11 real, 9 reducible, 11 representation space, 10 Riemannian manifold, 200 Riemannian metric, 199 root, 24 decomposable, 28 indecomposable, 28 long, 29 maximal, 29 minimal, 29 negative, 27 positive, 27 short, 29 simple, 27 root system, 24 dual, 24 irreducible, 29 reduced, 24 Schur lemma, 11 section, 124 semisimple element, 31 Serre relations, 35 sesquilinear form, 20 singular vector, 190 sl(2, C)-subalgebra, 142 semiprincipal, 148 special linear algebra complex, 7 real, 7

246

special linear group complex, 108 real, 108 special orthogonal group complex, 109 real, 109 special pseudo-orthogonal group, 109 special unitary algebra, 20 special unitary group, 110 standard differentiable structure

onC"`,63 on Rm, 63 on a real vector space, 63

standard topology

onC"`,56 on Rm, 56 on a complex vector space, 57 on a real vector space, 56

structure constants, 22 structure group, 123 subalgebra, 5 generated by a set, 7 Levi, 17 regular, 132 subcover, 60 submanifold, 94 complex, 94 embedded, 96 open, 65 submersion, 96 submodule, 10 trivial, 10

subsystem of a root system closed, 132 symmetric, 132

symmetric algebra, 8 symplectic algebra complex, 7 real, 7 symplectic group complex, 109 real, 109 system of simple roots, 27 extended, 133

tangent bundle, 124 tangent space, 67 to a complex manifold, 85 tangent vector, 67 complex, 84

Index horizontal, 126 real, 84 to a curve, 69 vertical, 126 tensor algebra, 2 Tits cone, 28 Toda equations, 178 abelian, 178 topological product, 58 topological space, 55 compact, 60 connected, 59 disconnected, 59 discrete, 55 Hausdorff, 60 path-connected, 60 second countable, 57 simply connected, 60 topologically equivalent spaces, 57 topology, 55 discrete, 55 indiscrete, 55 induced, 58 product, 58 quotient, 59 stronger, 56 trivial, 55 weaker, 56

unit element, 2 unitary algebra, 20 unitary group, 110 universal enveloping algebra, 17 vector field, 70 antiholomorphic, 87 complete, 73 complex, 86 holomorphic, 87 horizontal, 126 left invariant, 101 cp-projectible, 72 real, 86 smooth, 70 vertical, 126 vector fields W-related, 72 vector space graded, 139 Verma basis, 190

Verma module, 191 weight, 189 dominant, 191 fundamental, 191 highest, 190 integral, 191 regular dominant, 191 weight lattice, 191 weight space, 189 Weyl chamber, 28 fundamental, 28 Weyl group, 24 Weyl theorem, 17

zero curvature condition, 156