Maximally non-abelian Toda systems

B ELSEVIER

Nuclear Physics B 494 [PM] (1997) 657-686

Maximally non-abelian Toda systems * A.V. Razumov a,l, M.V. Saveliev b,2,3 a Institute for High Energy Physics, 142284, Protvmo, Moscow region, Russia b Laboratoire de Physique Thgorique de l'Ecole Normale Supgrieure, 24 rue Lhomond, 75231 Paris, France 4

Received l0 December 1996; revised 24 February 1997; accepted 5 March 1997

Abstract

A detailed consideration of the maximally non-abelian Toda systems based on the classical semisimple Lie groups is given. The explicit expressions for the general solution of the corresponding equations are obtained. (~) 1997 Elsevier Science B.V. PACS: 11.20Kk; 11.20.Lm; 02.30.Jr; 02.20.Hj Keywords: Non-abelian Toda systems; Lie algebra; Lie group; Gradation; Quasideterminant D e d i c a t e d to the 60th birthday o f Yuri L M a n i n to w h o m we offer our greatest respect and admiration

1. I n t r o d u c t i o n

In the last two decades the Toda systems permanently attract great attention o f physicists and mathematicians due to their integrability and deep links with a number o f problems in the theory o f differential equations, differential and algebraic geometry, Lie algebras, Lie groups and their representations, etc.; and relevance to many problems of modern theoretical and mathematical physics. In particular, they arise in a natural way in various approaches o f particle physics (e.g., field theory models and supergravity including black holes and p-branes business) and statistical mechanics (e.g., * Partially supported by the Russian Foundation for Basic Research under grant no. 95-01-00125a, and by the INTAS grant no. 93-633. J E-mail: [email protected] 2 On leave of absence from the Institute for High Energy Physics, 142284, Protvino, Moscow region, Russia. 3 E-mail: [email protected]; [email protected] 4 Unit6 Propre du Centre National de la Recherche Scientifique, associ6e ~ I'F~coleNormale SupErieureet h l'Universit6 de Paris-Sud. 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-3213(97)001 83-1

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correlations in the inhomogeneous XY model at infinite temperature). Whereas there are many papers devoted to the classical and quantum behaviour of abelian Toda systems, non-abelian Toda systems remain in many aspects unexplored. This is mainly caused by the absence of non-trivial examples, for which one can write the general solution for the equations under consideration in a rather explicit form. However, there exists a class of non-abelian Toda systems having a very simple structure. We call these systems maximally non-abelian Toda systems for the following reason. A Toda system is related to some Lie group G whose Lie algebra g is equipped with a Z-gradation, and hence a subgroup/q is defined corresponding to the subalgebra formed by the zero grade elements. The Toda fields parametrise a mapping from ]R2 or C to/4. If the subgroup/q is abelian, we deal with an abelian Toda system, otherwise we have a non-abelian Toda system. In the case of the trivial Z-gradation the subgroup coincides with G and we arrive at the Wess-Zumino-Novikov-Witten equations. The maximally non-abelian Toda systems correspond to the case when the subgroup/7/does not coincide with G and is not a proper subgroup of any subgroup of G generated by the zero grade subspace for some non-trivial Z-gradation of g. In the present paper we give a detailed consideration of the maximally non-abelian Toda systems associated with the classical semi-simple finite-dimensional Lie groups.

2. Toda systems and their integration 2.1. Z-gradations The starting point for the construction [ 1-4] of Toda equations is a complex Lie group whose Lie algebra is endowed with a Z-gradation. In the first two sections we give some necessary information about Z-gradations of the complex semi-simple Lie algebras; for more details see, for example, Refs. [5,3]. Recall that a Lie algebra g is said to be endowed with a Z-gradation if a representation of g is given as a direct sum

~=~

l~a,

aEZ

where [0~, 0b] C ga+b for all a, b E Z. Let G be a complex Lie group, and g be its Lie algebra. For a given Z-gradation of ~t introduce the following subalgebras of 0:

a<0

a>0

Here and henceforth we use tildes to have notations different from those usually used in the case of the so-called principal gradation of complex semi-simple Lie algebras. Denote by /~ and ,K/+ the connected Lie subgroups of G corresponding to the subalgebras ~ and fi±, respectively. Suppose that/~ and N+ are closed subgroups of G and, moreover,

A. V Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

/ t n/~/'+ = {e}, N_ n/'tAr+ = {e},

659

~/_ n A/+ = { e } , A/_/-1 n A/+ = {e},

where e is the unit element of G. This is true, in particular, for the finite-dimensional reductive Lie groups, see, for example, Ref. [6]. The set/V_/-~/V+ is an open subset of G. Suppose that G =/V_/~A/+, where the bar means the topological closure. This is again true for finite-dimensional reductive Lie groups. Thus, in the case under consideration for any element a which belongs to the dense set N_/~/V+ one can write the following unique decomposition: a = n _ h n ~ l,

(2.1)

where n_ E /[/-, h E /~ and n+ E N'+. The decomposition (2.1) is called the Gauss decomposition. Note that the Gauss decomposition (2.1) is the principal tool used in the group-algebraic integration procedure for Toda equations. There is a simple classification of possible Z-gradations for complex semisimple Lie algebras. In this case for any Z-gradation of such an algebra 0, there exists a unique element q E 0 which has the following property. An element x E 0 belongs to the subspace 0,, if and only if [ q , x ] = a x . This can he written as o. = {x ~ o l [q,x] = a x } . The element q is called the grading operator. It is clearly semi-simple, and exp(27r~"L-Tad q) = idg. On the other hand, any semi-simple element of 0 which possesses this property defines a Z-gradation of 0. Since the grading operator q is semi-simple, one can always point out a Cartan subalgebra of 0 which contains q [7]. Let [) be such a Cartan subalgebra, and A be the root system of 0 with respect to [). For any element x of the root subspace 0" corresponding to the root a E zl, one has [q,x] = ( a , q ) x , where (ce, q) means the action of the element ce E 0* on the element q E 0. Hence, for any root a E A the number (a,q) is an integer. Furthermore, if we choose a base /7 = {al . . . . . cer} of A corresponding to the Weyl chamber whose closure contains q, then the integers si =-- (ai, q) are non-negative. The grading subspace 0~, a =/= 0, is the direct sum of the root subspaces g,, corresponding to the roots a = ~-~irl niai with ~ i r l nisi = a. The r subspace 00, besides the root subspaces corresponding to the roots ce = ~-~i=l niai with r ~-~i=1 nisi = O, includes the Cartan subalgebra I). It can be shown that, up to a possible reordering related to the freedom in the renumbering of the elements o f / 7 , the numbers si do not depend either on the choice of a Cartan subalgebra containing q, or on the choice of a base possessing the above described property. In other words, a Z-gradation of a semi-simple Lie algebra 0 is described by non-negative integer labels associated with the vertices of the corresponding Dynkin diagram. Two Z-gradations are connected by an inner automorphism of g if and

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only if they have the same set of labels. Two Z-gradations are connected by an 'external' automorphism of g if and only if the corresponding sets of labels are connected by an automorphism of the Dynkin diagram. Let now I~ be a Cartan subalgebra of a semi-simple Lie algebra g, A be the root system of 9 with respect to I~, a n d / / = {al . . . . . a r } be a base of A. Denote by hi and x+i, i = 1 . . . . . r, the corresponding Cartan and Chevalley generators of 9- For any set of r non-negative numbers si the element F

q = ~(k-l)ijsjhi,

(2.2)

i,j=l

where k = ( k i j ) is the Cartan matrix of g, is the grading operator of some Z-gradation of g. Here the numbers si are the corresponding labels of the Dynkin diagram. The principal gradation of g arises when one chooses all the number si equal to 1. Thus, there is a bijective correspondence between the sets of non-negative integer labels of the Dynkin diagram of a semi-simple Lie algebra and the classes of its conjugated Z-gradations. If all the labels of the Dynkin diagram of a semi-simple Lie algebra 9 are different from zero, then the subgroup 90 coincides with some Cartan subalgebra of 0. In this case the subgroup/4 is abelian and we deal with the so-called abelian Toda equations. In all other cases the subgroup/4 is non-abelian and the corresponding Toda equations are called non-abelian. In particular, if all the labels are equal to zero, then there is only one grading subspace 90 = g- In this case the corresponding Toda equations coincide with the Wess-Zumino-Novikov-Witten equations. In the present paper we mainly consider the case when only one of the labels is different from zero; the corresponding equations are called here the maximally non-abelian Toda equations. Sometimes it is more convenient to consider, instead of a semi-simple Lie algebra, some reductive Lie algebra which contains this semi-simple Lie algebra. Here we use Z-gradations defined by the following procedure. Recall that a reductive Lie algebra 9 can be represented as the direct product of the center Z ( 9 ) of g and a semi-simple subalgebra gt of 1~. Choose some Z-gradation of 91 and denote the corresponding grading operator by q. It is clear that q is the grading operator of some Z-gradation of the whole Lie algebra 9. Moreover, the subspaces 9a, a 4: 0, coincide with the subspaces 9~, and the subspace g0 is the direct sum of the subspace 9~ and the center Z ( 9 ) . 2.2. s[(2, C) subalgebras

It is often useful to consider Z-gradations of a Lie algebra 9 associated with embeddings of the Lie algebra ~I(2,C) into g. Recall that the Lie algebra s[(2,C) is a complex simple Lie algebra formed by all traceless 2 × 2 matrices. This Lie algebra is of rank 1, and the Cartan and Chevalley generators satisfy the following commutation relations: [ x + , x _ ] = h,

(2.3)

A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

[h,x_] = - 2 x _ ,

[h,x+] =2x+.

661

(2.4)

By an embedding of ~l(2, C) into 0 we mean a non-trivial homomorphism from 5l(2, C) into 0. The images of the elements h and x± under the homomorphism, defining the embedding under consideration, are denoted usually by the same letters. The image of the whole Lie algebra 5[(2, C) is called an 5l(2, C) subalgebra of 0. For a given embedding of ~/(2, C) into 0, the adjoint representation of 0 defines the representation of the Lie algebra s/(2, C) in 0. From the properties of the finite-dimensional representations of 5l(2, C) it follows the element h of 0 must be semi-simple, and the elements x± must be niipotent. Moreover, it is clear that exp(2~v/-L-Tad h) = ida. Therefore, the element h can be used as the grading operator defining some Z-gradation of 0. It can be shown that in the case when 0 is semi-simple, the labels of the corresponding Dynkin diagram can be equal only to 0, 1, and 2. In particular, if the labels are equal only to 0 or 2, we deal with the so-called integral embedding. In this case it is natural to consider the Z-gradation defined by the grading operator q = hi2.

2.3. Toda equations Let M be either a real two-dimensional manifold, or a complex one-dimensional manifold. Choose some local coordinates z - and z + on M. In the complex case we assume that z + = z - Denote the partial derivatives over z + and z - by 0+ and 0_, respectively. Consider a Z-graded complex semi-simple Lie algebra 0. Let l be a positive integer, such that the grading subspaces 0a for - l < a < 0 and 0 < a < l are trivial, and c_ and c+ be some fixed elements of the subspaces 0-t and 0+t, respectively. Restrict ourselves to the case when G is a matrix Lie group. In this case the Toda equations are the matrix partial differential equations of the form ~+(y-la_y) = [c_,y-lc+y],

(2.5)

where y is a mapping from M to H. Let h+ be some elements o f / 4 , and the mapping y satisfies the Toda equations. It is easy to get convinced that the mapping

yt = h~lyh_ satisfies the Toda equations (2.5) with the elements c± replaced by the elements /

c± = h~_lc±h±.

(2.6)

In this sense, the Toda equations determined by the elements c± and c~_ which are connected by the above relation, are equivalent. Denote by H± the subgroups of H defined by

H i = {h E H I hc~ h-I = c+}.

(2.7)

The Toda equations are invariant with respect to the transformations

.y = ¢~-l,~z_,

(2.8)

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A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

where s~+ are arbitrary mappings from M to the subgroups/t+, satisfying the relations a:F(+ = 0. 2.4. Integration scheme

To obtain the general solution of Toda equations one can use the following procedure [ 1-4]. Choose some mappings y+ from M t o / 4 such that a:Fy± = 0.

(2.9)

Integrate the equations /.t~lO±/.t-4- = y ± x + y ~ J ,

aT/z-4-= 0.

(2.10)

The solutions of the above equations are fixed by the conditions /x+(p) = a+,

(2.11)

where p is some fixed point of M, and a+ are some elements of G. The mappings /z+ satisfying Eqs. (2.10) and conditions (2.11) are unique and take values in the sets a+~/+. The Gauss decomposition (2.1) induces the corresponding decomposition of mappings from M to G. In particular, one obtains lx+l tx_ = v_rlv + l,

(2.12)

where the mapping r/takes values in/~, and the mappings v+ take values in P~±. It can be shown that the mapping y = y+~rly-

(2.13)

satisfies the Toda equations, and any solution to this equation can be obtained by the described procedure. Note that almost all solutions of the Toda equations can be obtained using the mappings/z+ submitted to relation (2.11) with a+ 6 N+, or, in other words, using the mappings/z+ taking values in the subgroups N± [3,4].

3. Complex general linear group 3.1. Equations

We begin the consideration of maximally non-abel±an Toda systems with the case of the Lie group G L ( r + 1,C). Recall that the corresponding Lie algebra g[(r + 1,C) is reductive and can be represented as the direct product of the simple Lie algebra 5[(r + 1, C) and a one-dimensional Lie algebra isomorphic to g[( 1, C) and composed by the ( r + 1) × ( r + 1 ) complex matrices which are multiplies of the unit matrix. The Lie algebra s l ( r + 1,C) is of type Ar. Let d be a fixed integer such that 1 ~< d ~< r. Consider the Z-gradation of s[(r + 1,C) arising when we choose the labels

A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

663

of the corresponding Dynkin diagram equal to zero except the label Sd which is chosen equal to 1. Construct the grading operator associated with this gradation using relation (2.2) and the well known explicit expression for the inverse of the Cartan matrix, see e.g. Refs. [8,3]. From relation (2.2) it follows that the grading operator in the case under consideration has the form

q-

1 r+ 1

[

±

(r+l-d)~"~ihi+d i=1

(r+l-i)hi

i=d

]

.

It is convenient to take as a Cartan subalgebra for ~l(r + 1, C) the subalgebra consisting of diagonal (r + 1 ) x (r + 1) matrices with zero trace. Here the standard choice of the Cartan generators is hi = ei,i - e i + l , i + l ,

where the matrices ei,j are defined by (3.1)

( ei,.j)kl ---- (~ik~.jl •

With such a choice of Cartan generators we obtain

q = r+----l

(r+l

-d)~~ei,i-d i=1

i

Thus, the grading operator has the following block matrix form:

q=

ml -b-m2 0

--ml

,

(3.2)

m l -t--m2 l m2

where ml = d and m2 = r + 1 - d, so that ml + m2 = r + 1. Here and henceforth lm denotes the unit m x m matrix. We will use this grading operator to define a Z-gradation of the Lie algebra g l ( r + I , C ) . It is not difficult to show that in this case we have three grading subspaces, go and g+l. To describe these subspaces, it is convenient to consider (r + 1) x (r + 1) matrices as 2 x 2 block matrices x = (xq), where xq are mi x mJ matrices. Then the subspace go = ~ consists of all block-diagonal matrices, and the subspaces g - i = fi- and g+l = n+ are formed by all block strictly lower and upper triangular matrices, respectively. In other words, one can say that the subspace ga consists of the matrices x = ( x q ) where only the blocks xij with j - i = a are different from zero. It is easy to describe the corresponding subgroups of G L ( r + 1, C). The s u b g r o u p / ) is formed by all block-diagonal non-degenerate matrices, and the subgroups N_ and N+ consist respectively of all block upper and lower triangular matrices with unit matrices on the diagonal.

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Proceed now to the consideration of the corresponding Toda equations. The general form of the elements c+ is c_=

(0 0) C_ 0

'

c+=

0

'

(3.3)

Parametrise the mapping y as

0)

'Y=

82

where the mappings/31 and/~/2 take values in the groups O L ( m l , C ) and G L ( m 2 , C ) , respectively. Using now the relation

y-'c+y = ( 0 0

]311C+]~J2) 0 '

we can write the Toda equations in the form &+( f l ? l S_1~tl ) = -t~?1C+~2C_,

(3.4)

a+(]~J21tg--]~2) = C_/~IIc+/~J2 .

(3.5)

Recall that the Toda equations defined by the elements c+ and c~: connected by relation (2.6) are equivalent. For the case under consideration this statement is equivalent to saying that Eqs. (3.4), (3.5) are determined by fixing the ranks of the matrices C_ and C+. Let us try to associate the Z-gradation of I~I(r + 1,C), which is considered in this section with some embedding of the Lie algebra s [ ( 2 , C ) into gl(r + 1,C). It is clear that the Cartan generator h of the corresponding s[(2, C) subalgebra should coincide with 2q. The Chevalley generators x_ and x+ should belong to the subspaces ft-i and g+l, respectively. Therefore, we can write for x_ and x+ the block matrix representation x_=

(0 0) X_ 0

'

x+=

0

"

(3.6)

To satisfy relation (2.3) we should have X+X_

-

2m2 Iml ml + m2

,

2ml X_X+ = --I,, ml + m2

2.

From these equalities it follows that the rank r ( X + X _ ) is equal to rnl and the rank r ( X _ X + ) is equal to m2. On the other hand, r ( X + X _ ) <~ m i n ( r ( X + ) , r ( X _ ) ) <<.min(ml,m2).

Hence, one has rnl ~< min(ml,m2) and m2 ~< min(ml,m2). This is possible only if ml = m2 = m, and in this case r ( X + ) = m and r ( X _ ) = rn. It can be easily shown that the equality rnl = m2 is a sufficient condition for the existence of an s[(2, C) subalgebra in question.

A.V. Razumov, M.V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

665

If the condition ml = m2 is satisfied, then without any loss of generality, we can choose =

(,00) -lm

x_ =

'

lm

x+ =

'

.

(3.7)

Now, with c+ = x+, one arrives at the Toda equations of the form o~+(/3110-/31) = --/311/32,

(3.8)

c9+(/321a_/32) =/371/32,

(3.9)

where the mappings /31 and /32 take values in the Lie group G L ( m , C ) . In this case the subgroups/-1+ defined by (2.7) are isomorphic to the Lie group GL(m, C) and are composed of the block matrices of the form

Recall that these subgroups determine the symmetry transformations (2.8) of the Toda equations. 3.2. General solution In accordance with the general scheme described in Section 2, to obtain the general solution for Eqs. (3.4), (3.5) one should start with the mappings y+ taking values in the subgroup/1 and satisfying relations (2.9). Write these mappings in the block matrix form

~'±

=

0

/3+2

'

where the mappings /3+1 take values in the group G L ( m I , C ) , and the mappings /3+2 take values in the group GL(m2, C). Now we have to integrate Eqs. (2.10). Since almost all solutions can be obtained using the mappings /z+ taking values in the subgroups N+, we choose f o r / z + the parametrisation of the form = /Z_

(,m0) k, ~ - 2 1

Ira2

('0') /"L+12

'

l~+ =

lm2

'

with the mapping/x-21 taking values in the space of m2 × ml matrices and the mapping #+12 taking values in the space of ml x m2 matrices. This representation allows one to reduce Eqs. (2.10) to the equations a_#_2~ = / 3 - 2 C - / 3 - I, a+/z+12 =/3+1c+/3+~,

a+/z_2~ = o, a_/z+12 = o.

The general solution to these equations is

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A.V Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686 Z-P

/ z - 2 1 ( z - ) =m-21 + / d Y - 1 3 - 2 ( y - ) C - 1 3 - 1 ( Y - ) , 0

(3.12)

Z t

/~+12(Z +) =m+12 + [ o

dy+13+I(y+)C+13+~(Y+),

(3.13)

where m-21 and m+12 are arbitrary m2 x ml and ml × m2 matrices, respectively. Consider now the Gauss decomposition (2.12). From (3.11 ) one gets ~_t~_ =

Ira,

/-t+12/x-21 [.L 21

Im

--

2

Parametrising the mapping 7/ as

~ =

0 rl22

'

and using (A.4), (A.5) we find that "011 -- Imj --/./,+12//,-21,

'1722 = Im 2 -'l'-]Z--21 ( lnh --/./,+12/./,-21 )-11/'+12.

Note that the mapping/,+l/x_ has the Gauss decomposition (2.12) only at those points of M for which deft/,,, - I z + 1 2 ( z + ) / z _ z l ( z - ) ) 4= 0. Now, using (2.13), we arrive at the following expression for the general solution of Eqs. (3.4), (3.5): 131 = 13+~ (Ira, - 1z+12/./,-21 )13-1,

(3.14)

132 = 13-1+2 [lm2 -}-/Z-21 (Iml -- /-t,+12/A--21)--l/d,+12] 13--2,

(3.15)

where the mappings/x-21 and/x+12 are given by (3.12) and (3.13). Proceed now to the case when the considered Z-gradation is associated with the M(2, C) subalgebra defined by (3.7). Recall that with the choice c+ = x+ the Toda equations have form (3.8), (3.9). Introduce the notation 13 = 131, and write expression (3.14) as 1 3 ( Z - , Z +) ----~ ' + I ( Z + ) ~ ' _ I ( Z - ) @ ~ ' + 2 ( Z + ) ~ ' - 2 ( Z - ) ,

where

#+l = 13d,

~'-1 m ~ - 1 ,

(+2 = --13+~/d'+12,

g'-2 = t*-2113-1.

From (3.8) one obtains

132 = -a+a_13 + a+1313-1a_13.

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667

Consider the family of quasideterminants

Ai =

0+3 3 .

0+0-~--3 3 .

i--I

•.. ••• -.

...

a+ 3

O3+ •i...-- 13 ii C~i--13 .

(3.16)

0+ '-'0 _ 3' - '

The definition of a quasideterminant which is used in our work is given by (A.16). A more general definition of quasideterminants and an investigation of their properties can be found in [9,10]. It follows from (A.17) that fll = A1,

32 = - A 2 .

It is another form of writing the general solution to Eqs. (3.8), (3.9). 3.3. Generalisation

In this section we consider the Toda equations also based on the Lie group G L ( r + 1, C) but with more general Z-gradations of gl(r + 1, C). Strictly speaking, the systems under consideration are not maximally non-abelian Toda systems. Nevertheless, it is possible to find for their general solution some interesting explicit expression. Moreover, maximally non-abelian Toda systems based on classical Lie groups different from G L ( r + I , C ) can be considered as reductions of the systems which are investigated here. Let dl and d2 be two positive integers such that 1 ~< dl < d2 <~ r. Consider the Z-gradation o f 0[(r + 1, C) corresponding to the case when we choose all the labels of the Dynkin diagram equal to zero, except the labels Sd, and Sd2, which are chosen equal to 1. The corresponding grading operator can be constructed as follows. Denoting the grading operator given by relation (3.2) by qd it is clear that the grading operator q in question is q = qa~ + qa2. The explicit form of q is m2 + 2m3 Iml ml + m2 + m3 q =

0

0 --ml + m3

0 Ira2

0

ml + m2 + m3 0

0

--2ml -- m2 Im~

ml + m 2 + m 3

J ,

(3.17)

where ml =dl, m2 = d2 - dl and ms = r + 1 - d2, so that mt + m2 + m3 = r + 1. In this example we consider ( r + 1 ) × ( r + 1 ) matrices as 3 × 3 block matrices x = ( x i j ) with xij being mi × m j matrices. With respect to the Z-gradation of l~[(r + 1, C) defined by the grading operator q given by (3.17), there are five non-trivial grading subspaces ~t+2, g+J and g0. Here the subspace g~ is formed by the block matrices x = (x~i) where only the blocks xij with j - i = a are different from zero. The subalgebra ~ consists of all block-diagonal matrices, and the subspaces fi_ and fi+ are formed by all block strictly lower and upper triangular matrices, respectively• The

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A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

subgroup/7/is formed by all block-diagonal non-degenerate matrices, and the subgroups N_ and N+ consist respectively of all block upper and lower triangular matrices with unit matrices on the diagonal. In the case under consideration the general form of the elements c± E g±l is

c_=

(00 ) C-1 0 0 C-2

,

c+10)

c+=

0 0

C+2 0

•

Parametrising the mapping y as

0 0)

~'=

32 0 0 33

,

we arrive at the following Toda equations: 0+(3710_31) = -311C+132C_1,

(3.18)

0+(321~_32) = -321C+233C_2-~-C_1311C+132,

(3.19)

a+ ( 3 3 ~a_ 33 ) = C - 2 3 ~ ~C+233,

(3.20)

where the mappings Hi, i = 1,2, 3, take values in the Lie groups GL(mi, C). We now look for the conditions which guarantee the existence of an ~/(2, C) subalgebra of gl(r + 1, C) giving the Z-gradation under consideration. Actually we are interested only in an integral embedding of ~l(2, C). Therefore, the Chevalley generators x_ and x+ should belong to the subspaces g-1 and g+] respectively, and the Cartan generator h should coincide with 2q. In this case relations (2.4) are satisfied. Write for x_ and x+ the block matrix representation

X_ =

(o0 1

0

,

X+ =

X_2

0

X+2

0

0

.

To satisfy relation (2.3) we should have m2 + 2m3 Im,, ml + m2 + m3 - m l q- m3 X+2 X_ 2 - X_ 1X+ 1 = 2 1,,2, X+lX-t = 2

(3.21) (3.22)

m l + m 2 -[- m 3

X-zX+2 = 2

2rnl + m2 m] q- m2 -b m3

1,, 3.

(3.23)

From (3.21) it follows that ml ~< m2 and that the ranks r ( X + l ) are equal to ml. On the other hand, relation (3.23) implies m3 ~< m2 and r(X±2) = m3. Multiplying (3.22) from the left by X+I and taking into account (3.21), one obtains X+IX+2X-2

2

- m l + m2 + 3m3X+l. ml + m2 + m3

A. V. Razumov, M. V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

669

This relation gives that ml ~< m3. Similarly, multiplying (3.22) from the right by X+2 and taking into account (3.23), we get the relation X_ I X+l X+2 =

23ml q- m2 -- m3 X+2, ml + m 2 + m 2

which implies that m3 ~< ml. Thus, one can satisfy relations ( 3 . 2 1 ) - ( 3 . 2 3 ) only if ml = m3. It is easy to see that the conditions ml = m3 and ml ~< m2 are sufficient conditions for the existence of an s[(2, C) subalgebra leading to the Z-gradation under consideration. In the simplest and most symmetric case arising when ml = m2 = m3 = m, we can take, without any loss of generality, as the sought-for 5[(2, C) subalgebra the subalgebra generated by the elements

h=

x_ =

0 0 0

0 0

(3.24) -21m

0 V/21,,

0 0

0 0

0

V/21m

0

,

x+ =

(i'm 0) 0

x/~l.,

0

•

( 3.25 )

0

With the choice c+ = x + / v / 2 one arrives at the Toda equations of the form

Cg+(/~ll~_~l ) = --/~11~2,

(3.26)

a+(/3~-~a_/32) = -/3=-, /33 +/3~-~/32, a+(/3~-'a_/33) . = / ~ 2 -1 ]33,

(3.27) (3.28)

where the mappings /~i, i = 1,2, 3, take values in the Lie group G L ( m , C). Return now to the case of arbitrary m~, m2 and m3. To obtain the general solution of Eqs. ( 3 . 1 8 ) - ( 3 . 2 0 ) we start with the mappings y + which are parametrised as

~/± ----

/~4-2 0

/~-t-3

Writing for the mappings IX+ the following representation:

0 0) IX- = ~ IX-21 \ Ix-31

Ira2 //,-32

Eqs. (2.10) take the form

0 Ira3

('ml+12 ,

Ix+ =

0 0

lm2 0

IX+13 IX+23 I/'/l 3

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A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

O-/z-21 = fl-2C-lfl--l,

O+/z-21 = O,

0-/.z-32 = fl-3C-2fl_-1,

0+/z-32 = O,

tg_/t,t_31 = fl,_32/[~_2C_11~_-I,

0+//,_31 -- O,

O+/z+12 = fl+lC+lfl+ 1 ,

O-/z+12 = O,

0+/-/,+23 = B+2C+2B~ 1 , O+/x+13 =

0-/t/,+23 = O, -1

tz+12fl+2C+2fl+3,

O-/x+13 = 0.

The general solution to these equations is z-

f dy[-fl-2(y[-)C-1 fl-I (Yl),

/Z-21(Z-) =m-21 q-

o z-

/z-32(Z-) =m-32

-I- /dy[B-3(y[)C

-~ ( Y-l ) -2fl-2

o

/-6_31 ( Z - ) = m - 3 1

-Jr-

f dy

j

m-32+

o

x fl-2 (Y2)C-lfl-I

,i

dy[fl-3(yl)C-2fl-~(y[-)

0

)

(Yf),

Z+

/z+,2(z +)

=m+,2+ f dy+fl+,(y~)C+,fl+1(y+), o Z+

/x+23 ( z + ) = re+z3

+ / dy+fl+2 (y~-) C+2fl+~ (y~-), 0

/J,+13(Z +) =m+13 q-

f dyy m+,2 + 0

dy+fl+l(y+)C+lfl+g(y~) 0

X/~+2 (Y+) C+2J~+~ (Y~"), where m-21, m-32, m-31, m+12, m+23 and m+13 are arbitrary constant matrices. The next step of the integration procedure is to obtain the mapping 7/ from the Gauss decomposition (2.12). Using the relation

f lm,

--P~+12 --(/-/,+13

12'+12]~+23) "~ --/2"+23 / Ira3 -

-

we get for the blocks determining the mapping/x+l/x_ the following expressions:

A. V Razumov, M. V. Saveliev/ Nuclear Physics B 494 [PM] (1997) 657-686

(p+l#_)ll

671

= 1,,, - #/,+12#./,_21 - (#./-+13 - #+12#+23)#2.-31,

( # + 1 t l . - ) 12 = - # + 1 2 - (#+13 - # + 1 2 # + 2 3 ) # - 3 2 , ( # + 1 # _ ) 1 3 = - - ( # + 1 3 -- #+12]Z+23), (#/'+1#/'--)21 = #2'--21 -- #'/'+23#-£--31, ( # + 1 # - - ) 2 3 = --#+23, (#+1#--)32=#--32,

( # + 1 # - - ) 2 2 = lm2 -- # + 3 3 # - 3 2 ,

(#+1#--)31 = #-31, ( # + 1 # - - ) 3 3 =Ira3.

Now, using relations ( A . 1 0 ) - ( A . 1 2 ) , one can write down the expressions for the mappings 71~, 7122 and/']33 entering the parametrisation of the mapping 71,

7/=

7122 0

:)

•

7133

The corresponding expressions are rather cumbersome, so we give here only the one for 7111, 7111 = Ira, -- #+12#--21 -- (#+13 -- # + 1 2 # + 2 3 ) # - 3 1 .

Finally, relation (2.13) allows us to write the general solution of the Toda equations ( 3 . 1 8 ) - ( 3 . 2 0 ) in an explicit form. In particular, the expression for the mapping/31 is /31 = i ~ , ~

(Inq

-- #+12#--21 -- (#+13 -- #+12#+23)#--31 )/3--1.

(3.29)

Consider now the case rnl = m2 = m3 = m. Recall that in this case the Z-gradation under consideration can be associated with the s l ( 2 , C ) subalgebra generated by the elements h and x± defined by (3.24) and (3.25). Choosing again c+ -- x±/x/2, we arrive at the Toda equations ( 3 . 2 6 ) - ( 3 . 2 8 ) . Denote/3 =/31, and write/3 in the following form: / 3 ( Z - - , Z +) = ( + I ( Z + ) ( _ I ( Z - ) 4- ( + 2 ( Z + ) f - - 2 ( Z - ) 4- ( + 3 ( Z + ) ( - - 3 ( Z - )

which follows from (3.29). Here

(+l =/3~,

(-I =/3-1,

(+2 = --/3+~#+12,

(-2 = #-21/3-1,

(+3 = --/3+~ (#+13 -- #+12#+23),

~"-3 = # - 3 1 / 3 - 1 -

U s i n g Eqs. (3.26) and (3.27), one obtains /31 = A1,

/32 = --A2,

/33 = A3,

where the quasideterminants A i a r e defined by (3.16). It is interesting to go further and consider the case when r 4- 1 = p m, where p and m are positive integers. Suppose that Sm= s2,, . . . . . &p-l)m = 1, and all remaining

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A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

labels of the Dynkin diagram are equal to zero. The corresponding grading operator can be written as a p x p block matrix q = (qij) with

qij

1

-~(P + 1 -

=

2i)t~ijl

m.

The corresponding Z-gradation of the Lie algebra [tl(r+ 1, C) can be associated with the ~[(2,C) subalgebra generated by the elements h = (hij), x_ = (x-ij) and x+ = (x+ij), where hij =

(p + 1 - 2i)6ijlm,

X_(i = ~

-- i)~i,j+llm,

X+ij = ~

-- i)t~i+l,iI m.

Defining the elements c_ and c+ by the relations C--(i = t~i,j+llm,

C+ij : ~i+l,jlm,

one arrives at the following Toda equations:

O+(fl?l O_fll ) = - f l ~ l fl2,

(3.30)

Cg+(i/~10_ti) = - - i ] - I i i + l + i/--_llti,

1 < i < p,

O+( t p l O_tp ) = t / l l t p .

(3.31) (3.32)

Denoting i = ill, then one can show that from Eqs. (3.30)-(3.32) it follows that

t i = (--1)i-1Ai,

(3.33)

where the quasideterminants Ai are defined by (3.16). On the other hand, analysing the structure of the Gauss decomposition of block matrices, we conclude that t can be represented as

P (+i(Z + ) ~ - i ( Z - ),

t ( Z - , Z + ) "= ~

(3.34)

i=1 where the mappings ~'-t-i take values in the space of m x m matrices. Using arbitrary mappings (±i, we obtain the general solution to system (3.30)-(3.32). A proof of relation (3.33) is given in Appendix B. Slightly modifying the consideration given in Ref. [ 10], one can state the following. Consider the one-dimensional infinite system of ordinary differential equations d /

_ldfll'~

tP, -s-)

-dt -

fl~l

-~

__BZ lii+l

+ f l i --1 _lfli

,

i >

1,

where fli, i = 1,2 . . . . . are G L ( m , C)-valued functions of the real variable t. Denoting fl = flI, one can write the general solution to this system as fli = (-- 1 ) i - l / ' i ,

(3.35)

A.V. Razumov, M.V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

673

where El, i = 1,2 . . . . . are the quasideterminants defined by d fl #

dt

I'i =

d i - l /3

--

d /3

d2/3

dt

dt 2

d i - l /3

di /3

dti-1

dt i

"""

dt i- J de~3

"'"

dt i

d2i- 2/3 """

dt 2i-2

ii

Hence, relation (3.33) can be considered as a two-dimensional generalisation of (3.35). Actually, relation (3.33) for arbitrary/3 gives the general solution to the infinite system of equations having the form (3.30) and (3.31) without the condition i < p. Here, if /3 has form (3.34), then one gets Ap+l = 0 and arrives at the general solution of the finite system (3.30)-(3.32). Note that the expression for/31 in the form/31 = / 3 with /3 given by (3.34) was also obtained in Ref. [11] by some other method.

4. C o m p l e x orthogonal group

The complex orthogonal group O(n, C) is the Lie subgroup of the Lie group GL(n, C) formed by matrices a E GL(n, C) satisfying the condition (4.1)

[natl. =a -l ,

where [. is the antidiagonal unit n x n matrix, and a t is the transpose of a. The corresponding Lie algebra o ( n , C) is the subalgebra of ~[(n, C), which consists of the matrices x satisfying the condition (4.2)

[nx'[. = -x.

For an ml x mz matrix a we will denote by

ar

the matrix defined by the relation

a v = Im2atlml.

Using this notation, we can rewrite conditions (4.1) and (4.2) as a r = a - l and x r = - x . The Lie algebra o ( n , C) is simple. For n = 2r + 1 it is of type Br, while for n = 2r it is of type D r . We discuss these two cases separately. Consider the Z-gradation of o ( 2 r + I , C ) arising when we choose sa = 1 for some fixed d such that 1 ~< d ~< r, and put all other labels of the Dynkin diagram be equal to zero. Using relation (2.2), one gets r-I

q= ~

1 ihi + "~rhr,

d = r,

i=1 r--l

q=Zihi+ i=1

1

-~(r - l)hr,

d=r-1,

A.V. Razumov, M. V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

674 d

r--I

q=Y~

ihi+d

~

i=1

1 hi+'~ dhr,

l~d
i=d+l

It is convenient to choose the following Caftan generators of o(2r + 1, C): hi =ei,i - ei+l,i+l + e2r+l--i,2r+l-i -- e2r+2-i,2r+2-i,

1

<. i

< r,

hr = 2 ( er, r - er+2,r+2),

where the matrices ei,j are defined by (3.1). Using these expressions one obtains d

d

q =Z

ei,i-

Z

i=1

e2r+2-i,2r+2-i.

i=1

Denoting ml = d and m2 = 2(r - d) + 1, we write q in the block matrix form lint 0 0

q=

0 0 0

0 ) 0 , -Iml

(4.3)

where zero on the diagonal stands for the m2 x m2 block of zeros. It is easy to verify that in the case of the Lie algebra 0(2r, C) r--2

q = ~1 Z i h i +

~1( r - 2 ) h r - I + lrh 4

d=r,

r,

i=1

~-2

1

q=~1 Z i h i +

1

rhr_l+~(r-2)hr,

d=r-1,

i=1 d

r-2

q=Zihi+dZhi+~d(hr_l+hr), i=1

l~
i=d+l

Choosing as the Cartan generators of o(2r, C) the elements 1 <~ i < r,

hi = ei,i - ei+l,i+l nt- e2r-i,2r--i -- e 2 r + l - i , 2 r + l - i , hr = er--l,r--1 -1- er, r -- er+l,r+l -- er+2,r+2 ,

one easily obtains

r

1

q= ~

1± ei,i -- ~

i=1

1

r--I

q = ~ ~_~

1

ei,i

1

-- ~er,~ + ~ e r + l , r + l -

i=1 d

1

r--I

~_~ e 2 r + l - i , 2 r + l - i , i=1

d

q = ~_~ ei,i -- ~_~ e 2 r + l - i , 2 r + l - i , i=1

d = r,

ez~+l-i,2r+l-i, i=1

i=1

1 <~ i < r -

1.

d=r

-

1,

A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

675

Note that the grading operators corresponding to the cases d = r and d = r - 1 are connected by the automorphism o- of o(2r, C) defined by the relation o-(x) = a x a -I, where a is the matrix corresponding to the permutation of the indices r and r + 1. There is a corresponding automorphism of the Lie group O(2r, C), which is defined by the same formula. Thus, the cases d = r and d = r - 1 actually lead to the same Toda equations, and we can exclude one of them, for example d = r - 1, from the consideration. For the case d = r the grading operator has the following block form: 1(,,, 0 ) q = 9_ 0 -Ira '

(4.4)

where we denoted m = r. In the case 1 ~< d < r - 2 , denoting ml = d and m2 = 2 ( r - d ) , one sees that the grading operator q has form (4.3). Resuming the consideration, we can say that for any representation of the positive integer n in the form n = 2ml + m2, where ml and m2 are positive integers such that m2 4= 2, there is a Z-gradation of o(n, C) corresponding to a maximally non-abelian Toda system. This gradation is generated by the grading operator (4.3). In the case n = 2m, there is one more Z-gradation defined by the grading operator (4.4). Probably, the absence of the case m2 = 2 requires a special explanation. Actually the operator q given by (4.3) with m2 = 2 defines some Z-gradation of the corresponding complex orthogonal algebra, but this gradation corresponds to the case when two labels of the Dynkin diagram, s r - I and st, are equal to 1. Therefore, the corresponding Toda system is not maximally abelian. Nevertheless, it is convenient to consider the case m2 = 2 together with those which do correspond to maximally non-abelian systems. The Z-gradation defined by the grading operator (4.4) can be associated with an SL(2, C) subalgebra of the Lie algebra o ( 2 m , C) if and only if the integer m is even. In this case the corresponding element h coincides with 2q, and the elements x± have form (3.6), where the matrices X± satisfy the conditions X+X_ =lm,

X r, = - X ± .

With a Z-gradation generated by the grading operator of form (4.3) one can find the corresponding SL(2, C) subalgebra of o(n, C) if and only if ml ~< m2. Here h = 2q and the elements x± are

x_=

(o0 0) 0

0

--XT

0

,

x+=

0

-Xr+

0

0

,

where the matrices X± satisfy the conditions X+ X _

=

21.,,

,

r X _7 X+

X_X+

= O.

We now proceed to consider the Toda equations associated to the gradations described above and start with the gradation generated by the grading operator (4.4). The general

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A.V. Razumov, M. V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

form of the elements ca: E ga:l is given by (3.3), where the matrices C i should satisfy the relations C r, = - C + .

(4.5)

The subgroup/~ in the case under consideration is formed by the 2 x 2 block matrices of the form 0

where hll E G L ( m , C ) . Hence, the mapping y has the following block form:

0 )

y =

(/3_1) r

,

(4.6)

where the mapping/3 takes values in the Lie group GL(m, C). The corresponding Toda equations are 0+(/3-10_/3) = -/3-1C+(/3-1)rC_.

(4.7)

It is clear that these equations can be considered as the result of the reduction of Eqs. (3.4), (3.5) to the case /32 = (/311)r, which is possible if relations (4.5) hold. Therefore, the general solution to Eq. (4.7) can be obtained from the general solution to Eqs. (3.4), (3.5) by the method described in [4]. Certainly, we can get the general solution directly, using the general scheme of Section 2.4. The mappings 3/+ in the case under consideration have the form

• where the mappings/3+ take values in the Lie group GL(m, C). Note that the subgroups N_ and N+ in the case under consideration are formed respectively by the block matrices gt_~

( ,m 0)

,

(,o

rt+:

n-21 In,

,

In,

where the matrices n-21 and n+12 obey the equalities nT-21 = - - n - 2 1 ,

nT+12 = --n+12.

Representing the mappings/~+ as

tz_=

\ tz-21 1,,,

'

Iz+ =

0

In, J '

one obtains z

tz-2j

(z -) = m-21 + f dyo

(fl-1)T(y-)C_fl-_l

(y-),

(4.8)

A. V. Razumov, M. V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

677

~+ ]J,+I2(Z + ) = m+12 +

/ dy+/3+(y+)C+/3r+(y+),

(4.9)

0

where the constant matrices m-21 and m+12 satisfy the relations mT_21 =

--m-e1,

mTl2 = --m+12.

The final expression for the general solution to Eqs. (4.7) is (4.10)

/3 =/3~1 (lm-/z+12/x-21 )/3-,

where the mappings/z_2j and/x+12 are given by (4.8) and (4.9). Consider now the Toda equations arising when we choose the Z-gradation of o(n, C) generated by the grading operator q defined by (4.3). In this case the general form of the elements c+ is

c_=

0

,

c+=

-C T

(1o o)

0

0 - C +r

0

0

.

0

The mapping y has the following block form:

Y=

/32 o

0 (/3?l)r

,

(4.11)

where the mappings/31 and 132 take values in the Lie groups GL(ml, C) and O(m2, C), respectively. The corresponding Toda equations are 6~+(/3/1(9__/31)

= -/311C+/32C_,

0+(/3216"t_/32) = - ( C _ / 3 1 1 C + / 3 2 ) T + C - / 3 / I c + / 3 2

.

These equations can be considered as the reduction of Eqs. (3.18)-(3.20) to the case /33 =/3j and /3T = /3-1 2 " Such a reduction is possible if the matrices C+l and C±2 in ( 3. i 8) - (3.20) satisfy the conditions C-1 = -C_r2 = C_,

C+l = -C+r2 = C+.

The general solution in the case under consideration can be obtained either by the reduction of the general solution to Eqs. (3.18)-(3.20), or directly. The corresponding expressions are rather cumbersome and we do not give them here. Note only that the corresponding subgroups N_ and A/+ are composed respectively from the block matrices of the form

o o)(,m

n_ = [ n-21

1.,2

0

\ n-31

n-32

lnq

where

.

n~ =

n+3)

0

I., 2

n+23

0

0

lm~

,

(4.12)

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A. V Razumov, M.V. S a v e l i e v / N u c l e a r Physics B 494 [PM] (1997) 6 5 7 - 6 8 6 T

T

n+23 -- --n+12,

n+l 3 --- --n+13 • n+12rt+23,

nT_31 ------n-21,

nT3, = --n-31 -{-n-32n-21.

The maximally non-abelian Toda system based on the Lie group 0(5, C) and with the choice d = 1 in relation to the physics of black holes was investigated in Ref. [ 12] (see also Ref. [ 13] ). In these papers a local parametrisation of the Lie group/4, which is isomorphic here to GL( 1, C) x 0(3, C), was used. With the approach developed in the present paper, one can write the general solution in terms of corresponding matrices without using any local coordinates. Actually, for the system considered in [ 12,13] it is easier to use the fact that the Lie group 0(5, C) is locally isomorphic to the Lie group Sp(4, C) and to consider the corresponding maximally non-abelian Toda system based on Sp(4, C). This will be done in the next section.

5. Complex symplectic group We define the complex symplectic group Sp(2r, C) as the Lie subgroup of the Lie group GL(2r, C) which consists of the matrices a E GL(2r, C) satisfying the condition La'L

= -a-',

where ]r is the matrix given by Jr:

-L

0

"

The corresponding Lie algebra ~p(r, C) is defined as the subalgebra of the Lie algebra ~l(2r, C) formed by the matrices x which satisfy the condition Lx'L

= x.

The Lie algebra s p ( r , C ) is simple, and it is of type Cr. Therefore, the Cartan matrix of ~p(r, C) is the transpose of the Caftan matrix of 0(n,C), and the same is true for the inverse of the Cartan matrix of sp(r,C). For any fixed integer d such that 1 ~< d ~< r, consider the Z-gradation of s p ( r , C ) arising when we choose all the labels of the corresponding Dynkin diagram equal to zero, except the label sa, which we choose to be equal to 1. Using relation (2.2), we obtain the following expressions for the grading operator: q = ~

ihi,

d = r,

q=

i='

ihi + d i=1

hi,

I <~ d < r.

i=d + 1

Using the following choice of the Cartan generators: h i = e i , i - e i + l , i + l -~- e 2 r - i , 2 r - i h r = e r , r -- e r + l , r + l ,

e2r+l-i,2r+l-i,

1

<~ i < d,

A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

679

one sees that the grading operator for the case d = r has the form (4.4) with m = r, and for the case 1 ~< d < r it has the form (4.3) with ml = d and m2 = 2 ( r - d). The corresponding S L ( 2 , C ) subalgebra always exists for the case d = r, and for the case 1 ~< d < r it exists if and only if mj <~ m2. In the case d = r the general form of the elements c± is given by (3.3), where the matrices C± satisfy the conditions c,T = c±.

(5.l)

The mapping y has here the form (4.6) and the Toda equations coincide with (4.7) where the matrices C± satisfy (5.1). The obtained equations can be considered as the reduction of Eqs. (3.4), (3.5) to the case El = (fl~-l)r = fl which is possible when (5.1) holds. The general solution of the Toda equations is described by relation (4.10) where the mappings /-t-21 and /.t+12 are given by (4.8) and (4.9) with the constant matrices m-21 and m+12 satisfying the relations mT_21 = m-21,

mTl2 = m+12.

The simplest choice of the matrices C± is C± form

=

lr. Here the Toda equations take the

~+(3--103 /~) = __(~T/~)--I The subgroups /.it± determining the symmetry transformations (2.8) are isomorphic to the Lie group O ( r , C ) and are composed of the matrices of the form (3.10), where the matrix hll satisfies the condition hlrl = h~ I. With C+ = [r we arrive at the equations

a+ (3-~ 0_3) = - ( i f 3 ) -I Returning to the discussion given in the end of the previous section, it is clear that choosing a consistent local parametrisation for the Lie groups Sp(4, C) and 0 ( 5 , C), we can obtain from the general solution of the Toda equations for Sp(4, C) the general solution for the corresponding Toda equations based on 0 ( 5 , C). It can be verified that this solution coincides with the solution obtained in [ 13]. Proceed now to the case 1 ~< d < r. In this case the general form of the elements c± is

(o

c_ =

o

C_

0

,

o -6c'__Z-d

c+ =

0

]r_dCt+[d

o

o

"

the mapping y has the form (4.11) where the mappings fll and f12 take values in the Lie groups G L ( d , (2) and S p ( 2 ( r - d), C), respectively. The Toda equations are

0+(/~llo3-/~l)

=

-/~llc+/~2C_,

0+(3;10_32)

= f l ~ l ~ _ a C ~ ( 3 ~ l ) t C _t J -r - d + C - 3 1 1 C + f l 2 .

(5.2) (5.3)

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A. V Razumov, M. V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

These equations are the reduction of Eqs. (3.18)-(3.20) to the case /33 = /31 and Jr-d/3~Jr-d = --/321, which is possible if the matrices C+1 and C+2 in (3.18)-(3.20) satisfy the conditions C-1 = --L-dCt_..2[d = C _ ,

C + l = [dCt+2L-d = C+.

It is quite clear that using the integration procedure described in Section 2.4 one can easily write down the general solution for Eqs. (5.2), (5.3).

6. Concluding remarks The main goal of our study was to obtain explicit expressions for the general solutions for some class of non-abelian Toda systems, namely for the maximally nonabelian Toda systems, which has a very simple structure. Our consideration concerns finite-dimensional complex semi-simple Lie groups, but can be extended to exceptional groups and infinite-dimensional loop groups. In particular, starting with the loop group associated with the complex general linear group it is possible to arrive at the periodic two-dimensional non-abelian Toda chain which was obtained by Capel and Perk in the context of the inhomogeneous XY model [ 15 ], and also by Mikhailov in the framework of the reduction scheme [ 16]. Finite-gap solutions to this equations were found by Krichever [ 14]. Actually one can consider an infinite system of equations of the form (3.31) with i = 0, ± 1,4-2 . . . . This system was introduced by Polyakov at the end of seventies. The finite two-dimensional non-abelian Toda chain and the periodic one can be considered as special cases of such a system corresponding to the boundary conditions/3o I flp+l = 0 and/3o =/3i,+1, respectively. We believe that non-abelian Toda systems are quite relevant for a number of applications in theoretical and mathematical physics, especially in particle and statistical physics, and in the near future their role in the description of non-linear phenomena in these areas will not be less than that for the abelian systems. Finishing the paper, we would like to mention one more reason why non-abelian Toda systems are not yet popular and known enough in mathematics, in particular in algebraic and differential geometry. The problem is that such important notions as the Gauss-Manin flat connection, the Griffiths transversality of variations of the Hodge structures, superhorizontal distributions, local and global Plticker relations, etc., which are perfectly fitted in the scheme with the abelian Toda system, need to be re-understood or extended for the non-abelian case. The same goes for the systems generated by flat connections with values in higher grading subspaces of a complex Z-graded Lie algebra. We hope that progress in this direction will be very fruitful for studies in theoretical physics as well as for mathematics itself. =

A.V. Razumov, M.V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

681

Acknowledgements The authors are grateful to J.-L. Gervais, I.M. Krichever, Yu.I. Manin and J.H.H. Perk for the fruitful discussions. One of the authors (M.V.S.) wishes to acknowledge the warm hospitality and creative scientific atmosphere of the Laboratoire de Physique Throrique de l'l~cole Normale Suprrieure de Paris.

Note added After our paper was submitted for publication an electronic preprint of P. Etingof, I. Gelfand and V. Retakh [17] appeared concerning in particular non-abelian Toda systems. In this interesting paper some of our results are reproduced by some other method.

Appendix A. Gauss decomposition of block matrices Let a = (a~i) be a non-degenerate block matrix formed by m i × mj matrices. By the Gauss decomposition of such a matrix we mean its representation as the product of a lower triangular block matrix with unit matrices on the diagonal, an upper triangular block matrix with unit matrices on the diagonal, and a block-diagonal matrix, taken in some order. In this appendix we construct the Gauss decomposition of 2 x 2 and 3 × 3 matrices. Consider first a non-degenerate 2 x 2 block matrix all a 1 2 ) . a21 a22

a=

(A.I)

Suppose that the matrix a can be represented as

a = n_hn+ 1,

(A.2)

where n_, h and n+ are block matrices of the form

\ n_12 Ira2

n_

0 h22

'

0

Ira2

Note that since the matrix a is non-degenerate, the matrices hjj and h22 must be also non-degenerate. It can be easily shown that -1 n+

(10~ - n + 1 2 ) =

Ira2

Using this relation, one can write equality (A.2) in the form

a,,

el2) =(

a21

a22

hi,

-h,,n+,2

n21hll

h22 - n-21hlln+12

) "

A. E Razumov, M. If. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

682

From this equality it follows that the Gauss decomposition of the matrix a of the form (A.1) exists if and only if the matrix all is non-degenerate. In this case one has n-21 = azl ( a l l ) - l ,

(A.3)

hll = a l l ,

(A.4)

h22 =a22 -- a 2 1 ( a l l ) - l a l z ,

(A.5)

n+12 = - ( a l l ) - l a l 2 .

(A.6)

It is worth noting that the obtained Gauss decomposition is unique. Consider now a non-degenerate 3 × 3 matrix

j

ail a12 a 1 3 ) a = | a 2 1 a22 a23 • / \ a 3 1 a32 a33 Suppose that a can be represented in the form (A.2), where

hlj 0 0 ) h=

n_ =

0 0

h22 0

0 h33

,

(Ira 0 o) n_21 //-31

Ira2

0

n-32

Im3

,

n+ =

(,nl

n+12 Imz

0

n+13 //+23 lm3

.

Now we have

[

lnq

//+1= /

0

\o

--//+12 Imz 0

--n+13 + n+12//+23 "~ --//+23 ) , Ira3

and in the same way as was done for the case of 2 × 2 block matrices one finds that the Gauss decomposition in question exists if and only if the matrices all and a22 a21(all)-la21 are non-degenerate. The explicit expressions determining the matrices n_, h and n+ are n-21 = a 2 1 ( a l l ) -1,

(A.7)

n-31 = a 3 1 ( a l l ) -1,

(A.8)

n-32 = (a3z -- a31 ( a l l ) -la12) (a22 -- a21 ( a l l ) - l a l 2 ) - l ,

(A.9)

hll = a l l ,

(A.10)

h22 = a22 - a21 ( a l l ) -la12,

(A.11)

h33 =a33 - a 3 1 ( a l l ) - l a l 3 + (a32 - a 3 1 ( a l l ) - l a l 2 ) x

(a22 -

a21

n+12 = - ( a l l ) - l a l 2 ,

( a l l ) -1 a12) -1 (a23 -- a21 ( a l l ) -la13),

(A.12) (A.13)

A. V. Razumov, M. V. Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686 n+L3 = - - ( a l l ) - I ( a l 3

683

-- a12(a22 -- a 2 ! ( a l l ) - l a l 2 ) - I

x(a23 -- a 2 1 ( a l l ) - I a l 3 ) ) , n+23 = --(a22 -- a 2 1 ( a l l ) - l a l 2 ) - I (a23 -- a 2 1 ( a l l ) - l a l 3 ) .

(A.14) (A.15)

The Gauss decomposition in this case is again unique. It is interesting to compare the relations obtained in this appendix with those arising in the theory of quasideterminants of matrices over an associative unital ring, in the form proposed by Gelfand and Retakh [9,10]; see also Ref. [3]. Let us give here some relevant definitions. Let 27 and f f be two ordered sets, each consisting of p elements. Consider an invertible matrix a = (aij)iCZ,jE3" with the matrix elements belonging to some associative unital ring R. Define the family of p2 elements lal~, i C 27, j C if, of the ring R, which are called the quasideterminants of a. In the case p = 1 there is only one quasideterminant defined by lalij = aij. For p > 1 the quasideterminant ]al~j of the matrix a can be defined by the relation lal~ I = ( a - j

)ji"

(A.16)

It is clear that the quasideterminant [alij exists only if the matrix element (a -l).i~ is an invertible element of the ring R. A more general definition of a quasideterminant applicable to the case of non-invertible matrices can be found in [10], but for our purposes the above definition is the most convenient. Let KS and £ be subsets of 27 and if, respectively. Denote by a (x:;~) the matrix which is obtained from the matrix a by removing the matrix elements aij with i C KS or j E £. Consider the matrix a (i;j) and suppose that it is invertible. It is not difficult to show that

lal,'j =

ai.j -

~ aik(a(i;j)-l)klali. k~=i,l~j

( A . 17)

This equality is used in Appendix B to prove the validity of relation (3.33). Further, denote by a(pc;L) the submatrix of a composed of the matrix elements aii with i E KS and j E £. Consider the case when Z = ff = {1 . . . . . p}. It can be shown that an invertible matrix a = (aij)iEZ,jEJ has a Gauss decomposition of the form (A.2) if and only if there exist quasideterminants [a(l,...,i,1,...,i)lii, i = 1 . . . . . p. Here the non-trivial matrix elements of the matrix h are determined by these quasideterminants. Actually, one has hii = [a( l,...,i;l,...,i) [ii.

( A . 18)

Returning to the case o f block matrices, suppose that all blocks of the matrices under consideration are square m x m matrices. In this case we can treat such a block matrix as a matrix over the ring of m x m matrices and use the formulae relevant for matrices over the general associative unital ring.

684

A. V Razumov, M.V. S a v e l i e v / N u c l e a r Physics B 4 9 4 [ P M ] (1997) 6 5 7 - 6 8 6

Appendix B. Proof of relation (3.33) In this appendix we prove relation (3.33) which gives the general solution to Eqs. ( 3 . 3 0 ) - ( 3 . 3 2 ) . Probably our proof is not the shortest one but it is quite direct. First of all note that from (3.31 ) it follows that 1~i+1

= - - ~ + ~9-- i~ i "~- ~ + l~ i l~ z l ~9-- t~ i "~- /~i1~/----11~i"

Therefore, to prove (3.33) it suffices to prove the equality Ai+ l =

c 9 + c 9 - A i - - tg+ A i A Z I c g - A i "4-

(B.1)

A/A]-_IIA/.

Let p = (Prs)r,~=l,2 .... be the infinite block matrix with the matrix elements defined by ,~r--lr~s--ll" 4 Prs = v+ ~_ t-',

and let ~ be a submatrix of p formed by the matrix elements Prs with r, s ~< i. Recalling (1) the definition of Ai one can write di = IPlii. Represent the matrix ~ in the form p=

.._,,

(B.2)

.

Pii ti--I)

where the matrix elements of 1 x (i - l) matrix or and (i - 1) x 1 matrix " -7." are given by

(,,,,)

(,,/,)

From equality (A.17) we obtain the following expression: li--(Hi--()__ | (i--l) Ai = Pii --

or

P

7"

(B.3)

.

Using this expression and representation (B.2) we arrive at the relation / Ii~)ll-- 1 pI ~ _ I =

--

O--1)--1 (i--11 _l 0--1)(i--I)_ l p 7"/I i O" p

ti~h _ 1 ti-t) _ _ 1 --

(B.4)

A? 1

For the quasideterminant Ai+ l one has = Pi+l,i+l

(1)(1) __ (i) -- orP 1 T.

Note that the following relations hold: °3+firs = f l r + l . s ,

a-,Ors =flr, s+l.

Therefore, one can represent ~ and ~2 as

=

7" J i

,

~ _ 1((--(HI--D_ l --/..Ii Or p

Ai+l

19

c9+ or

O+Pii

,

7" =

cg-Pii

.

A.V Razumov, M.V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

685

Using this representation and relation (B.4) one easily obtains

Ai+ 1 = cg+O-Pii

-

-

~+('~11('~1) --I 0-- t'~.')

-- (a+Pii- a+('-')"-" o- p - I"~')) ,aT l ( a - p i i - "~r'"-p"-I-a_ "-')'~ 7").

(B.5)

Differentiating (B.3) with respect to z - we arrive at the relation

(i~11_1 (,~1).

(B.6)

It is easy to convince oneself that

b4=i-I

ti ll__1 (i--1)Nt p

C9_ p )ab =

{ ~a,b+l,

i--I Z C'-pl'--l)acPci,

b=i-1.

c=l This relation implies the equality ((~ ¢'O-')_ "O-'l¢ipt)--lc9 "-pl))a = Ai6i_l,a. Therefore, from (B.6) it follows that li_l)(,_l)_la (i~l) (i ii O_A i = cT_pi i -- O" p __ Ai ( -p -I "-") 7" i-I.

(B.7)

Now differentiating (B.3) with respect to z + and using the relation "-""-I)-l\ tg+ p

p

/ 6a+l,b,

a 4: i-- 1

)ab = I i--1 Z Pic [l'-ll-l'~ ~P )cb,

(B.8)

a=i-l,

c=l we arrive at cg+Ai

=

c?+Pii--

a+"o.l)(ipl)-ll'~ .l)

,i-1)(/-i)_ 1

- - ( O" p

(B.9)

)i_lai .

Using relations (B.7) and (B.9) one obtains

~+a,~:'~, : (~+~ii-- ~+,,,,~ P' ,,i ,,~rJ ~:' (~Pii

~ i,,~,,)p ,~

The differentiation of (B.7) with respect to z + with account of (B.9) and (B.8) gives

A. V Razumov, M. V Saveliev/Nuclear Physics B 494 [PM] (1997) 657-686

686

~ +~-- Ai = (9+¢9_pii _ ~ +.~.ljti~l)_ 169_ .~11 __ (O+Pii_o+'i~)ti-pl)--l~i~rl)) (ti-pl)--lli~-'~)i_l

f"-""-'~-l~o,p

)i-I (\tg-Pii- 'i~"i-ff''-Ic~-"~''')

--J-("~'"pl-l)i_ldi~

t9

7" )i_l--AiAT_llAi .

(B.11)

Now with (B.5), (B.11) and (B.10) we ensure that relation (B.1) holds. Hence, relation (3.33) also holds true.

References [ 1 ] A.N. Leznov and M.V. Saveliev, Group-theoretical methods for integration of non-linear dynamical systems (Birkhauser, Basel, 1992). 12] A.V. Razumov and M.V. Saveliev, Differential geometry of Toda systems, Commun. Anal. Geom. 2 (1994) 461. [ 3 ] A.V. Razumov and M.V. Saveliev, Lie algebras, geometry and Toda-type systems (Cambridge University Press, Cambridge, 1997). 14] A.V. Razumov and M.V. Saveliev, Multidimensional Toda type systems, hep-th/9609031. [5] V.V. Gorbatsevich, A.L. Onishchik and E.B. Vinberg, Lie groups and Lie algebras, 3. Encyclopaedia of math. sciences, Vol. 41 (Amer. Math. Soc., New York, 1994). [6] J.E. Humphreys, Linear algebraic groups (Springer, New York, 1975). [ 7 ] J. Dixmier, Alg/~bres enveloppantes (Gauthier-Villars, Paris, 1974). [8] M. Goto and ED. Grosshans, Semisimple Lie algebras (Marcel Dekker, New York, 1978). [9] I.M. Gelfand and V.S. Retakh, Determinants of matrices over non-commutative rings, Funct. Anal. Appl. 25 (1991) 91. [ 10] I.M. Gelfand and V.S. Retakh, A theory of non-commutative determinants and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992) 1. [ 11 ] A.N. Leznov and E.A. Yusbashjan, The general solution of two-dimensional Toda chain equations with fixed ends, Lett. Math. Phys. 35 (1995) 345. [12] J.-L. Gervais and M.V. Saveliev, Black holes from non-abelian Toda theories, Phys. Lett. B 286 (1992) 271. [ 13 ] A. Bilal, Non-abelian Toda theory: A completely integrable model for strings on a black hole background, Nucl. Phys. B 422 (1994) 258. [ 14] I.M. Krichever, The periodic non-abelian Toda chain and its two-dimensional generalization, Russ. Math. Surv. 36(2) (1981) 82. 115] J.H.H. Perk and H.W. Capel, Transverse correlations in the inhomogeneous one-dimensional XY model at infinite temperature, Physica A 92 (1978) 163. [ 16] A.V. Mikhailov, The reduction problem and the inverse scattering method, Physica 3D ( 1981 ) 73. [17] P. Etingof, I. Gelfand and V. Retakh, Factorization of differential operators, quasideterminants, and non-abelian Toda field equations, q-alg/9701008.