W-symmetry

^ — 2^<*'»

q and A

i

a r e

funcCi°ns

of x, t with values in 0 ; a is an element of 0 satisfying conditions a ) , b) formulated above. Our purpose is to carry over the main results of the first section to this case. As in Sec. 1, the key feature is the reduction of the operator i-J^+^-la, to canonical form ( s e e Proposition 1 . 2 ) .

#C-(R.«)

(4.1)

However, now if no r e p r e s e n t a t i o n of

0

has been

chosen we cannot conjugate L with a formal s e r i e s in A"1 simply because the operation of m u l t i p l i c a t i o n i s not defined in 0 . c a t i o n of the operator e a " u .

The analogue of conjugation in our s i t u a t i o n i s

This operator a c t s as f o l l o w s :

then *•«•($)— * + [«, ♦l + ^f I". I". Vl]-(- * • • • e"*e" u .

if

« — £Utk~

appli­

, "16©. »(>€0((*~')),

We note that in the case where O— Mat(*, C), £•*"($) —

I t i s not hard to v e r i f y that the mapping e*tf*:0((3l~l))-»>0((X'1)) i s a L i e - a l g e b r a automor­ phism. From the Campbell-tiausdorff formula [40] i t follows that automorphisms of t h i s type form a group. M We consider the Lie algebra C"(R, O).

0

of formal s e r i e s of the form

We note that the operator L belongs to

where £ / — 2 U ' ^ '•

a

/€^"(R. 0) > i s an automorphism of

^

\clj---\-p,\l-1.

c,GC, pfc

I t i s c l e a r that the mapping e a d u ,

0. 0.

We denote the group of a l l

such

(-1

automorphisms by G.

1999

50

Proposition 4.1. has the font

e adU(L)

There exists a formal series f/=JJ,u,X'', a^C"(R, B) such that £ , — '-i

l^-^-Xa

+ ^H,).-,

A^C-(R,«.

(*.2)

i-c

The automorphism e*dU i 8 defined uniquely up to multiplication on the left by automorphisms of the form e

*, where t/0 — 2 C '*" , • *i6C"(R, $). It is possible to choose the series U in (—1

exactly one way so that its coefficients ui belong to C"(R, t>L) • polynomials in q.*i

Here the u^ are differential w

HS.,I-' Proof.

Equating coefficients of X ~

2 * / ^ " ' » w e find that 1-0

n

o n b o t h s i d e s o f the e q u a l i t y

[ u n + 1 , a ] + h n c a n b e e x p r e s s e d in t e r m s of ui

t

'"•

(1)^3

^<*~i"

u n , ho,...,hn-i.

Since the restriction of ad a to $ x is an isomorphism, any element in 0 can be uniquely represented in the form [X, a) * Y, where X&)1-, Keft. Therefore, knowing ui u,,, ho,..., h n - l t it is possible to uniquely determine «»*i€C""(R, Jb1) and k,£C(J(, t}). ui ui Let gi, gjgO be such that the operators £i—g,(L) and £- — g2(I) have the form (4.2). Since G is a group, Sagf' — exp ladJojX.-' I, where t>,6C~(R, 6 ) . We must verify that c,€C"(R, $) •

\ «-'

/

/ -

Equating coefficients of X~ n in the equality exp I ai^v,X''

\ ](Ii) — Ij. we see that if vj,...,

v,eC"(R, * ) , then K ^ , a]6C"(R, * ) , and hence
/_2?(»M.. »i6*

(4.3)

i-o

is an evolution equation for q. For brevity we call this equation Eq. (1.1). 4.2.

The following assertions are proved in analogy to Propositions 1.4, 1.5, 1.7.

Proposition 4.2. If we set A — 9 ( W ) . , where bit) , then Eq. (1.1) has the form 3q/3t Pq(n) + f(q, q' q ^ n _ 1 ^ ) , where P is a linear operator on 8 annihilating 3} and such that its restriction to $ A is equal to adb(ada)~ n . If we assume that q has degree of homo­ geneity i + 1, then f is a homogeneous polynomial of degree of homogeneity n + 1. ■ Proposition 4.3. The functions hj(x) defined by formula (4.2) are densities of conser­ vation laws for Eq. (1.1). Here AJ—^J, ; if i > 0, then up to t c m l derivatives the linear part of hj. is equal to zero, while the quadratic part has the form ■= (ada)"'

Proposition 4.4.

{ab —

We consider the equations

£-I
51

4.3. In this and the following subsection we shall discuss the Hamiltonian formalism for Eqs. (1.1). As the manifold M on which a Hamiltonian structure is introduced we take C"(R/Z, 9)* while for the class of functionals F we take the set of all functionals from M to C of the form (1.11). We assume that on 0 there is given a nondegenerate, symmetric, invariant bilinear form ( f ) . We recall the invariance of a form means that for any u, v, -w€6 the equality (u, adv(w)) - — (adv(u), w) holds. For any functions g, AfAf is obvious that (g, h') - —(h, g')-

we set (g, A ) — j (g{x), *,z

h(x))dx

•

It

We define the Poisson brackets {* , •}i and {•, ' ] 2 on H b y formulas (1.13) and (1. 14) , In exactly the same way as in part 1.4, it can be verified that any linear combination of these brackets is a Poisson bracket. m

nt

Let « — 2 * i ^ * *'€$•

We

define the functional Hu:M + C by the formula Hu(q)=~2(*.•

1—0

*i) »

i-0

where h£ are the same as in Proposition 4.1. By Proposition 4.3, the functionals Hy are conservation laws for Eq. (1.1). Proposition 4.5. 1) Equation (1.1), where A is defined by formula (4.3), is the Hamil­ tonian equation corresponding to the Hamiltonian Hu and the second Hamiltonian structure. 2) This same equation is the Hamiltonian equation corresponding to the Hamiltonian H^u and the first Hamiltonian structure. 1.9.

The proof of this assertion differs in only minor details from the proof of Proposition ■ COROLLARY.

For any u, u€4 ((*''))•

Wm» w})i =-{//„.

H~)2—0M

4.4. In [37-39] Reiman and Semenov-Tyan-Shanskii constructed an imbedding of the mani­ fold M in the dual space to a remarkable infinite-dimensional Lie algebra 0 such that the second Hamiltonian structure on M is induced by the Kirillov structure on 0*. In this sub­ section we present the construction of Reiman and Semenov-Tyan-Shanskii. It will be used in part 6.5 in constructing a group-theoretic interpretation of the second Hamiltonian structure of Gel'fand—Dikii. We first make a remark of general character. Let X be a Hamiltonian manifold, and let 3 be a subalgebra of the Lie algebra of functionals on X. Then the mapping i; X-*%* as­ signing to each point xdX the functional /,: 8-»-C given by the formula £ x (f) ■ f(x) is a Hamiltonian mapping (see the end of Sec. 3) if the Hamiltonian structure of A. A. Kirillov is considered on 9* . If, moreover, functions in 8 separate points of X, then i is an imbed­ ding. In this case X may be considered a submanifqld of 8* , and the Hamiltonian structure on X is induced by the Kirillov structure on 9*. Of course, an imbedding i of the type de­ scribed above can be useful only if the algebra 8 is not too large. As X we now take the manifold Af—C"(R/Z, 0), equipped with the second Hamiltonian struc­ ture. For any u£M, c6C we define the functional 9.,*:Af-»-C by the formula T«.«(?)"(?. «)+
(4.4)

Thus, the set 0_(9 at |u6iM, c£C) is a Lie algebra. Applying Che construction described above to the pair (Af. 0) , we obtain the desired imbedding l:M -► 0*. It is clear that i(Af)cz (/60*l'(fo.i)—1). This inclusion is not an equality: if /g0', {(To,)—1 , then I has the form /(?,.,) —(<7. u) + c, where q is a generalized function R/Z-*0. Following [38], we understand by 0* below the set of "smooth" linear functionals on 0 (i.e., functionals of the form 9m.*-*(q, u) ♦ ac, where agC, fl=C-(R/Z, 0 ) ) . With this interpretation of $• we have i(H) • We shall discuss the structure of 0 as a Lie algebra. It follows from (4.4) that the one-dimensional subspace in 0\ consisting of functionals of the form f0
2001

52

algebras C"(R/Z. 6) and C. Thus, ft is a nontrivial central extension of C-(R/Z, 0) by C. It is known that if the algebra 0 is simple, then such an extension is unique up to iso­ morphism. Therefore, in the case where 6 is a simple Lie algebra the manifold H has the following abstract description: let ft be a nontrivial central extension of C-(R/Z, 6) by C; then M is the hyperplane in ft* consisting of functionals /(*£* taking the value 1 on the element 16Cc<J > and the Hamiltonian structure on H is induced by the Kirillov structure on

ft*. In conclusion, we note that the imbedding jW-*-6" described above makes it possible to interpret the Hamiltonian Hu (see Proposition 4.5) in terms of a so-called Adler scheme (see [37-39]). 5.

SOME FACTS CONCERNING SEMISIMPLE LIE ALGEBRAS AND KATS-MOODY ALGEBRAS

As always in this work, in the present section we consider Lie algebras only over C. 5.1. Definition. A Lie algebra is called simple if it is finite-dimensional, nonAbelian, and contains no nontrivial ideals. A semisimple Lie algebra is the direct product of a finite number of simple algebras. We shall recall the structure and classification of semisimple Lie algebras. Definition. A system of Ueyl generators of a Lie algebra © is a system of generators Xi, Yi, Hi, 1 < i < r of it such that a) Xi * 0, Yi * 0, Hi * 0 for all i; b) for any i, j the relations

l"..",]-°. l*i. Yi\ — *ijffi.

\Ht,r&

N,/„

(5.1) (5

2)

^

hold where (Njj) is a nondegenerate matrix such that Nii • 2 for any i. Remark.

If Nil * 0, then, multiplying Xi and Hi by 2/Nii, it can be arranged that Nii -

2. Proposition 5.1. 1) In order that in a finite-dimensional Lie algebra 0 there exist a system of Weyl generators it is necessary and sufficient that 6 be semisimple. 2) Suppose that the Lie algebra © is semisimple and {Xi, Yi, Hi) and {X}, ?f% £,},<€{1 r}. 7€{1 r) are Weyl generators of it. Then a) f ■ r; b) there exist an inner automorphism
(.dAr,)1-"". X,-(»4rl)'-''"Yj-0.

(5.5)

2) Equalities (5.1)-(5.5) form a complete system of relations between the elements Xj, Yj.Hi. 2002

53

A proof is given, for example, in [4], Chap. 8, Sec. 4, Proposition 4. COROLLARY.

A semisimple Lie algebra is uniquely determined by its Cartan matrix.

Instead of writing out the Cartan matrix, it is customary to present che corresponding Dynkin scheme — a graph with vertices ci,...,c r such that 1) the number of segments connect­ ing c£ and cj, i * j is equal to NijNji; 2) these segments are equipped with an arrow pointing to c^ if and only if Nij < Nji. From Proposition 5.2 and the equality Nii - 2 it follows that the Cartan matrix can be uniquely recovered from the Dynkin scheme. Proposition 5.4. t) The Dynkin schemes of simple Lie algebras are those and only those graphs presented in Table 1. 2) Let 9,,..., 9» be simple Lie algebras. Then the Dynkin schemes of the algebra 9 i X . . X 9 * is the disjoint union of the Dynkin schemes of the alge­ bras 9j,.,.,0k. Assertion 1) is proved in Sec. 4, Chap. 6 of the book [3], while assertion 2) follows from the definitions (as a system of Weyl generators of the algebra 9,x...X9,. it is pos­ sible to take the union of the systems of Weyl generators of the algebras 9t, . . . , 6 k ) . Remark. When we speak, say, of a Lie algebra of type E« we have in mind the simple Lie algebra whose Dynkin scheme has type Eg. The index 6 designates the number of vertices of the Dynkin scheme, and Ijejjce the rank of the algebra is equal to 6. Examples of simple Lie algebras are the algebras *l(n) for a ^ 2 , o(2n+l) for n ^ l , *»(2n) for n > 1, and o(2n) for n > 3 which are usually called the classical Lie algebras (see Ap­ pendix t). Their Dynkin schemes are An-i, Bn, C n , and Dn, respectively. Since Ai • Bi - Ci, B 2 - C 2 , A 3 - D 3 , it follows that sl(2) ~o(3) -*»(2), o<5)<=-*»(4), *I(4)-o(6) . We note further that o(4) is a semis imple Lie algebra isomorphic to *I(2)X*I(2) (see Appendix 1 ) . We recall the structure of the group of automorphisms of a semi simple Lie algebra 9. We denote by Aut°9 the set of inner automorphisms of 9. Aut°9 is a normal subgroup of Aut9. Let T be the Dynkin scheme of 9, T^Aut I\ and let T(C()=»=CO{O . Then there exists exactly one automorphism 9t:9-»-9 such that 9 t (Xt) *=Xcil), 9T(K,) — Ka(j>. f t ^ - Z / o i / j (the existence of 9 t follows from Proposition 5.3). We call mT the automorphism 9 induced by T . It follows from assertion 2) of Proposition 5.1 that any automorphism of 9 can uniquely be represented in the form f<#t , where /eAut°9, teAut T . Therefore, Aut9/Aut° 9 = A u t T . If ipeAut9 and ip_/<>9T, where /€Aut°9, tgAut T , then we call T the automorphism of r determined by the auto­ morphism . In conclusion we note that with each system of Weyl generators of a semisimple Lie alge­ bra 9 there are connected three subalgebras in ©; which will play an important role in Sec. 6: the Cartan subalgebra $, generated by all elements Hi, the Borel subalgebra $, generated by all elements Hi and Y£, and also the subalgebra n, generated by all elements Y^. It is easy to see that a) $ is commutative; b) the elements H£ form a basis in $ ; c) $ — $ © $ . If for the classical Lie algebras we choose the Weyl generators as done in Appendix 1, then 9 is the set of diagonal matrices in 9, £ is the set of upper triangular matrices in 9, and n is the set of matrices in •§ with zeros on the main diagonal. 5.2. algebra.

Let V be a Lie algebra; then on <[C,C"'] there is a natural structure of a Lie If
C"1! | / (C« " )—
is a Lie subalgebra in

*JC. C"1]-

Definition. A Kats-Moody algebra is a Lie algebra of the form L(%, simple Lie algebra and 9 : %-+% is an automorphism of finite order.


I is a

The next result follows from results of the work [23]. Proposition 5.5. 1) If !(«,,
2003

54

*•»« n r t ! : ^ . '•*••>' s i s . cZi* ,_

."• „.,?.][ „ W * J | { J J, ,n ?

°" *

..Te.

»!t.n»)

C„,n*1 «n.n>3

Cl 't

Cn-1 'n

Ci c.

Cm c„

t< c» tin 4h-i

■ ■j ■ ■

H

•■j

— -

fsn

^

. —

17

^

*S-l.

^ K i »7 <*>*

**"»' en,

*n.»><

5

*,'(, t,""fc7e.

"»■"*' SI «.

*i"



-

has type Ej ' The Rats-Moody algebras of types B^(l) cj l >- A n k ) D n k ) (k - 1, 2) are called classical. Their explicit realizations are presented in Appendix 2. Proposition 5.5 shows that all Kats-Moody algebras are exhausted by algebras of the form /-(«,
An automorphism C of a simple Lie algebra

31 is called a Coxeter auto*

morphism if 1) the algebra * c — {jet* \Cx — x) i s Abelian; 2) among all automorphisms % such that the algebra «» is Abelian C has least order. It follows from the results of [23) that for any automorphism T of the Dynkin scheme g in the corresponding coset K T of the group A u t * by the subgroup Aut°« there exists a Coxeter automorphism C, and C is unique up to conjugation with inner automorphisms. The order h of the automorphism C is called the Coxeter number of the algebra i(*, C). The Coxeter numbers of Kats-Hoody algebras are presented in Table 3 (see part 5.6) taken from (57). From the results of [2 3] it is possible to extract the following means of constructing C if h is known. In « we consider a canonical system of generators {Xj, Y j , H j ) , 1 < j < r. Suppose that T takes the j-th vertex of the Dynkin scheme into a vertex with index o(j). Then the action of C on the generators can be given by the formulas Cr.YJ — e*"'**'.,,, CfJM — e-*"»Y ,,< K C(Hj) - Ho(j). " '"" " °1"' Let C be a Coxeter automorphism of a simple Lie algebra h is the Coxeter number. set 0'=-Up.

We set G — L(%, Q.

It is clear that <3—®G>,

%, and let u - e 2 "^" 1 , where

We have C - © % p ,where

We

[O', C*|c;C *», so that G is a graded Lie algebra.

According to the definition of the Coxeter automorphism, the algebra number r — dlmCJ"

«, — {X€*|Cjc— a>x).

,

G°—«„

is Abelian. The

we call the rank of the Kats-Moody algebra G.

Proposition 5.6. 1) There exist elements «o,.... e,(fj\ /0, -./,€C-'. A„ M O ' such that ») «« er form a basis in C 1 , fo f r form a basis in G": ho h r generate G°; b) the following relations hold:

l*,.M-0, 2004

(5.6) (5.7)

55

lAj.eJ-*,^, I*i. /,!--•**,/,.

(5.8) (5.9)

where Aii - 2 for all i. 2) The elements e£, fj, h£ are uniquely determined up to enumera­ tion and transformations of the form e^ - c exp (ada)e{, ?i • c - i exp (ada)f£,h£ ■ h^, where A proof is given at the beginning of part 4 of the work [23]. We note that if concrete Sand C are given, then it is not hard to find the explicit form of the elements e^, f{, hi (ei and f{ are eigenvectors of the operators ada. a(-(P). The matrix (Ajj) is called the Cartan matrix of the algebra G. It is proved in part 5 of the work [23] that this matrix possesses the following properties. Proposition 5.7. 1) At,€Z. 2) If i * j, then Aij < 0, AijAji < 4. Aji ■ 0. 4) The space of solutions of the system of equations -t 2i*i,Jfi-0. / - 0

3) Ajj » 0 ■**■

(5.10)

r,

(-0

is one-dimensional. From the results of part 4 of the work [23] and Proposition 13 of the work [21] we ob­ tain the following results. Proposition 5.8. 1) The elements ei, fi, hi, 0 < i < r generate G. 1

1

2) If i * j , then

1

(ad*,) -*''«,-<«!/i) "* '/,-<>. 3) Let ( d o , — , a r ) be a nonzero solution of the system (5.10).

(5.11) Then

2a,*,-0.

(5-12)

4) Equalities (5.6)-(5.9), (5.11), (5.12) form a complete system of relations between the elements ei, fi, hi. We call the system of generators {ei, fi, hi> of the algebra G canonical. Remark 1. Assertion 1) of Proposition 5.8 means that G is generated as a space by mul­ tiple commutators of the elements e;, fj, hj, Using the Jacobi identity and relations (5.6)(5.9), from this it is not hard to deduce that G is generated as a vector space by elements of the form [ei l ,... ,ei n ], [fil,.••,fin] (n - 1, 2,...) and hi. It is clear that for any rt€N the space G n is generated by elements of the form [ei,(... ,ein] and G~ n by elements of the form [fi x ,...,fi n ]. Remark 2.mi Frequently not the algebra G but the algebra G with generators ei, fi, hi and defining relations (5.6)-(5.9), (5.11) is called the Kats—Moody algebra. The algebra G has a one-dimensional center generated by the element ^. a,A( , and its factor by the center is i-o

isomorphic to G. We note that the most interesting applications of Kats—Moody algebras (see [22, 31, 47, 48, 56, 66, 68}) are connected with G rather than G. The concept of the Dynkin scheme for a Kats—Moody algebra is introduced in the same way as for semis imple Lie algebras. It follows from Proposition 5.7 that the Cartan matrix can be uniquely recovered from the Dynkin scheme of a Kats—Moody algebra and thus so can the al­ gebra itself. The Dynkin schemes of Kats—Moody algebras are presented in Table 2 borrowed from [23]. Let G be a Kats—Moody algebra with canonical generators ei, fi, hi, 0 < i < r. Proposition 5.9. Let Scz{0, 1,..., r) be a proper subset. Then the subalgebra gener­ ated by the elements e,t fu A,, i€S, is semisimple, and these elements themselves are Weyl gen­ erators. Proof. Comparing Tables 1 and 2, it is not hard to see that if from the Dynkin scheme of a Kate-Moody algebra part of the vertices aod adjacent segments are removed, then a Dynkin scheme of a semissimple Lie algebra is obtained. •

2005

56

COROLLARY.

In relation (5.12) o£ » 0 for all i.

Proof. The complementary minors of the diagonal elements of the matrix (Aij) are non­ zero, since they are the determinants of the Cartan matrix of semisimple Lie algebras. * The gradation in G that we have so far considered is called canonical. Moreover, to each vertex of the Dynkin scheme of G there corresponds a gradation G — $ Q, called a Standard gradation. The standard gradation corresponding to a vertex c^ is characterized by the following property: em£Qu /«tC.,, and the remaining canonical generators belong to Go. Proposition 5.10.

Let G — § Gy be the standard gradation corresponding to the vertex

c m of the Dynkin scheme.

Then 1) if i > 0, then G,<=2 G> ■ 2) If i < 0, then

Go is a semisimple Lie algebra with Weyl generators ej, fi, hj, i * m. 2 O1

is generated by the elements e£, i * m. 5) The algebra

3)

4) The algebra GoO

O0fl2^

;>0

G,c=2 °'-

is generated by the

><0

elements f^, i * m. Proof. It is clear that for any «€N the space G Q (respectively, G- n ) is generated by elements of the form [ej lf... .ei^] (respectively, [fj, fjkl)> "h'te m occurs among the numbers ji,...,jk precisely n times. Go is generated by elements of the form [ej, 'jk'» wnere m and tne [fjj fjk^' J* * J k * °> °y elements hj . From this we obtain asser­ tions 1), 2 ) , 6 ) v and 5 ) . To prove 3) it is necessary to use Proposition 5.9 and further note that because of Proposition 5.9 hjj, can be expressed in terms of the elements hj , j * m from relation (5.12). ■ Remark. It follows from assertion 3) of Proposition 5.10 that the Dynkin scheme of G 0 is obtained from the Dynkin scheme of G by deleting the vertex c D and the edge contiguous to it. 5.3. If a is an automorphism of finite order of a simple Lie algebra % such that C~1*o is an inner automorphism, then, according to Proposition 5.5, there exists an isomor­ phism L(%, C)^U.(%, o). We shall indicate the explicit form of this isomorphism in the case where o —C•«•**, jcgO». We recall that

£(*, ^-{/enix.x-'ji/U^j-oc/W)},

(5 ,3)

-

where n is the order of o. We make the substitution X — e " . We denote by T the algebra of

* functions g:C->-*

of the form g(u) — 2'"'"'*'/• * * " ' »/6C, V/&L

Then

/-i

£(«.«)-(«(«))}-

(5-14)

In exactly the same way L (*. 0-{/€*|t, t''I I/ ^ « T J - C(/(C))J -{ge7"|«(u + 1)-C(g(u))}, where C « e * . It is not hard to verify that the mapping T;T -*■ T given by the formula

isomorphically maps

*(*)-«. «(«)-«""«(«), L(%, C) onto /.(ft, o).

(5-'5)

Proposition 5.1t. For each vertex cn of the Dynkin scheme of G there exists exactly one element x*G°, possessing the following property: if we set Oj, - C'e a d x and define T by formula (5.15), then T takes the standard gradation of the algebra G~t(«, C), corresponding to the vertex c m into a gradation of the algebra L(U, am) in powers of X. Proof.

Since the elements hj, j * m form a basis in G°, the desired element x can be

represented in the form J t - 2 * A i xjgC.

We recall that «»—«»C —e^expf-j-i), /» — /»C' —

/ » « p ( — j — J , where e», /"«€■. The element x possesses the property that exp(u-adx)efc does not depend on u for any k * m. From this we obtain the system 2006

57

ZAl.xi--¥,-0
k + m,

(5.16)

for determining xj . Since this system has exactly one solution (see the proof of the corol­ lary to Proposition S . 9 ) , the uniqueness of x has been proved. We shall now prove that the element j c — ^ ■*/*>• defined from the system (5.16) is the desired one.

We set um - C*ea<*x and

l*m

define L(%. o„) by formula (5.14) [it is not possible to define £(•, a,) by formula (5.13) until it has been proved that o n has finite order). By the construction of x we have x(ej) ■ «j for j * m. Moreover, it is easy to see that t(fj) - fj for j * m,_T(hj) - hj for all j , and there exists cf-C such that t(e m ) - exp (cu)« n , t(f m ) - exp (-cu)f„. We shall show that c has the form

— , /tf-N. For any

y6G*

the element z - y C h belongs to G*1.

Since z lies in

the subalgebra generated by the elements ej, it follows that T ( Z ) lies in the subalgebra

» generated by the elements t(ej) and hence has the form

^ exp(/cu)>9/, "o,0t.

On the other hand,

1-4

x(z) — z—yexp(2niu). («m) - ec-im, all j.

Therefore, c has the form

— , ixgN.

It is not hard to verify that o m x

o m («j) - «j for j * m, o m (f m ) - e~ c f n , o„(fj) - fj for j * m, o B (hj) » hj for

From this it follows that o m has order n.

Since x(em) — exp(-j-J)i m —Xe m ,

*(/„) —X~'/«,

and the images of the remaining elements ej, fj, hj under the mapping T do not depend on X, it follows that x takes the standard gradation £(H, C), corresponding to the vertex c m into the gradation of /.(If, am) in powers of X. ■ COROLLARY,

dim Gj < «.

The automorphism o m ■ C*e a< * x , where x is the same as in Proposition 5.11, we shall call the standard automorphism corresponding to the vertex c m , while the realization of G in the form £(S, 9 M ) we call the standard realization. We make several remarks concerning a practical way to find the standard realization G. If ■ is realized as a subalgebra in M a t ( k , C ) , then exp(uad.t)L(«, C) - e " L ( « , C ) « — . If, more­ over, G°czDiag, then eux is a diagonal matrix of the form dlag(C**,..., C*») (we recall that t — e's~). Thus, i(«, o«) — t/(C)£(«, Q£/-'(C), where t/(C) — dIig(C» C"«) , and the numbers m n K can be found uniquely from the following conditions: 1) the matrix U(;)e£U~'(0 does not depend on C for i * m; 2) £/(C)6W , where H — {e'|y€C>. The automorphism o,,, is given by the formula am(X)—U \eTIC(X)U\e'T j . Finally, X and c. figuring in the definitions of £(«, o„) and L(%, C), are connected by the relation X - C n ^ n ^ where n is the order of am. 5.4. The height of the Kata—Moody algebra £(<,?) is the order of the automorphism of the Dynkin scheme of V, defined by m. This subsection is devoted to Kats—Moody algebras of height 1. Let V be a simple Lie algebra with Weyl generators Xi, Y j , Hi, i - 1,...,r. It is easy to see that on K there exists exactly one gradation ■ —ffi%l such that X£V. K|6*"'. W | € * for all ie (see the end of part 5.1). We set S — { 7 6 2 1 * ^ 0 } . The largest and least elements of S we denote by k and I, respectively. Proposition 5. 12. 1) / — — A ; 2) dtolt*—dhn*"»—1: 3) there exist nonzero elements Xjt *-», Y,0V, H£H such that [X 0 , Yol - Ho, [Ho, Xo] - 2Xo, [Ho, Yol - -2Y 0 ; 4) the center of the algebra •, generated by the elements Yi Y r is equal to «"*. This proposition follows from the theorem on the existence of a maximal root (see [3], Chap. 6, Sec. 1, Proposition 2 5 ) . The number h - k ♦ 1 is called the Coxeter number of the algebra • . We set C = exp(-j--tda) .where the element off), is such that [a, Xj] - X j , [a, Yj] • —Yj , 1 < j < r (the existence and uniqueness of a follow from the nondegeneracy of the Cartan

2007

58

/, — K^"1, A, — //,.

matrix of % ) . We define elements e„ / „ A,€*IC t~'I by the formulas e,— X,t» 0 < i < r. It is easy to see that e„ f„ h£L(*.C) for all *6{0 r}.

Proposition 5.13. 1) The automorphism C ia a Coxeter automorphism (in particular, the Coxeter number of Kats—Moody algebra of height 1 is equal to the Coxeter number of the corre­ sponding simple Lie algebra). 2) The elements e£, f£, h^, 0 < i < r are canonical generators of !(«, C). This proposition is essentially proved in [21] in the proof of Lemma 22. Suppose that the canonical generators of a Kats—Moody algebra G of height 1 are chosen as in Proposition 5.13. Then the vertex of the Dynkin scheme of G having index 0 is called special. It is clear that if the special vertex and the edge contiguous to it are eliminated from the Dynkin scheme of G, then the Dynkin scheme of the corresponding simple Lie algebra is obtained. In Table 2 the special vertices of the Dynkin scheme of the Kats—Moody algebras of height 1 are represented by black circles. Proposition 5.14. The standard automorphism of special vertex is the identity automorphism.

8

(see part 5.3) corresponding to a

Proof. We recall (see the proof of Proposition 5.11) that the standard automorphism corresponding to a special vertex is equal to C*e a< * x , where x&fi is found from the following 2ntu

condition: e*'6x-e

* X * does not depend on u for all k * 0.

it follows that x=

Since [a, Xk] ■ Xk for all k * 0,

j-a. ■

Thus, the standard realization of a Kats—Moody algebra of height 1 corresponding to a special vertex is %[K, A" 1 ]. In correspondence with part 5.3, the isomorphism T:£(9J, C)-»8I|X, X"1! is given by the formula t(f) «= f, where 7(X)««exp( — lnC*ada)*/(t), t^X1'*. The canoni­ cal generators of *[X, X~l) are XA"0, X~'K0, M0, Xt, Yt, H\, \X*. 5.5. Let V be a semisimple Lie algebra of rank r; let Xit where i - 1,. . . , r, be the ele­ ments of the system of Weyl generators of M ; let V — © W be the gradation of % introduced r

in part 5.4.

We set

/-=VA" ( .

'^*

It is clear that the operator ad I maps

%t

into %l*x.

«-i

Proposition 5.15. If j < 0, then the operator then the operator ad/:%•-*>/*' is surjective.

s6I:W~*-WH

is injective.

If j ^ — 1,

A proof is given in [58]. Definition. The number j is called the exponent of the algebra M if the operator ad I: %-j-i-+%-/ is not surjective. The difference dim*"■' — dlmtt"'"1 is called the multiplicity of the exponent j. It is shown in [58] that the definition of the exponent presented above coincides with the definition of Bourbaki [3]. It follows from Proposition 5.15 that if the algebra V is simple, then all exponents belong to the segment [1, h — 1 ] , where^h^ is the Coxeter number of %. We note that the number of all exponents (counting multiplicity) is equal to the rank of *:

indeed, J 5.6.

(dimf-> - dim*-'"1) - dim IP - r.

Let G be a Kats—Moody algebra with canonical generators e£, fi, h j , 0 < i < r. We

r

set A — ^ e , , 9—KeradA — {it€G|[A, u\ — 0}.

The next r e s u l t i s important for our subsequent pur-

poses. Proposition 5. 16.

1) The algebra

3 is commutative.

2)

G - 3@ImadA.

A proof is given in [57] (Proposition 3.8). As B. A. Magadeev has informed us, the following converse assertion also holds: if G is a contragradient Lie algebra in the sense of [21] and G — K e r a d A S l m a d A , then G is the direct product of a finite number of Kats-Moody algebras.

2008

59

TABLE 3

—sr—

31

14} Si

osa

Exponents

&,{ 51

- * ■a

■" f ^ »«t « - , ■

5

i

A-t

*

5

3 *.. Let G—9GI /gz

i»

iM-^-UMU.** f' >m-t UA-**«-i r r» ( i !* . U - t n - f tn UI—JH-I r bf! U4.-.J»-J.»-» tn't « 5 . _ M . « »" •»♦»

i^ -l—

be the canonical gradation.

« « * « » n

■

Exponent*

WM IttW ( M W «M*» t r.(f.ttfl»a» f.9.7.«

u

It is clear that 3 —ffl3/. , where 3/=-3flO-'. /gi

It is easy to see that 3°-=0 • and for any j, dim'3'** — dim 3'. where h is the Coxeter number of G. Definition. The dimension of

The number y€[l.A—1| is called the exponent of the algebra G if 3'^Q3* is called the multiplicity of the exponent j.

The exponents of Kats—Moody algebras are presented in Table 3. We note that only algebras of type Djn n a v e a n exponent of multiplicity greater than 1 (in this case 2n — 1 is an exponent of multiplicity 2 ) . It is known (see (57], Proposition 3.7) that the expo­ nents of a Kats—Moody algebra of height 1 coincide with the exponents of the corresponding simple Lie algebra. Therefore, the exponents of simple Lie algebras can also be found from Table 3. We note (see Table 3) that j is an exponent of G if and only if h — j is an exponent, and the multiplicities of these exponents coincide. In other words, dtm3*"*dim3 -i for any J'6Z. This equality can also be derived from Proposition 5.20 below. Proposition 5. 17.

If u«3

and [u, G°] - 0, then u » 0.

Proof. By hypothesis [u, A) - 0, [u, G°] - 0. It is not hard to see that the elements A and hi, 0 < i < r generate the algebra j^G'Therefore, [u, G k ] - 0 for k > 0. We recall that C - i ( « . O — ® , V , where », = {jce*|C;c-* * x) . Here G»-*,C* . Thus, [u(C), «,|-0 for c Since « — ' © « . , it follows that la(C), « ] — 0 . It remains to note that the center of

k > 0. %

is equal to zero, since the algebra

% is simple.

■

5.7. This subsection is devoted to invariant bilinear forms on semisimple Lie algebras and Kats-Moody algebras. We recall that the Killing form on a finite-dimensional Lie algebra 8 is defined by the formula (x, y)K " t r ( a d x * a d y ) . It is easy to see that this form is invariant and symmetric. Proposition 5.18. 1) A finite-dimensional Lie algebra is semisimple if and only if the Killing form on it is nondegenerate. 2) Let % — *, X . •. X * . .where *, * . are simple Lie n

algebras.

Then any invariant bilinear form on * has the form {X, y) — 2

c

i (•*(■tfiVic(€*-, where

j—1

Xi and yi are the components of x and y in % is synmetric.

>,. In particular, any invariant bilinear form on

Assertion 1) is proved, for example, in [2] (Chap. 1, Sec. 6, Theorem 1 ) . the proof of 2) see Exercise 18 in Sec. 6, Chap. 1 of the book [2).

Concerning

Proposition 5.19. Let I be 1 semisimple Lie algebra on which a nondegenerate invariant bilinear form has been fixed. Let $. f, » be the same as at the end of part 5.1, and let ** be the same as in part 5.4. Then 1) if x(*', ffV, / + *■*(>, then (x, y ) - 0; 2) on $ the b i ­ linear form is nondegenerate; 3) the orthogonal complement of I is equal to 1. Proof. It i* easy to show that there exists an element hit) such that [h, Xi] - Xi, lh, Yi] - -Yi for all *6{1 r). Than (h, x] - jx if X&V. If X&U. !«**. then (j ♦ k)(x. y) - (fh, x ] , y) ♦ (x, [h, y]) - 0. Assertion 1) has been proved. It implies assertions

2009

60

2) and 3), since $ — »> » — QV. Let O — BO,

n—feW.

■

be a graded Lie algebra.

It ia said that a bilinear form(,) on G is coor­

dinated with the gradation if (x, y) - 0 for JCgO,, yff}„ k +

1^0.

Proposition 5.20. On a Kats-Moody algebra there exists a nondegenerate, symmetric, bilinear form which is unique up to a factor and is coordinated with the canonical gradation. This form is also coordinated with the standard gradation. Proof. Let G — I («, C), where S is a simple Lie algebra and C is the Coxeter automorphism~ On V we choose a nonzero, invariant, bilinear form. On G we define a bilinear form B by the formula B(u,«) — 2 (Bi- *-i) > "*«« u—2«£', v—^vf.1, t

t

U,v0t.

It is easy to see that

i

the form B is the desired one. Noting that B(u, v) is the free term of (u(C), v ( O ) and using (5.15), we find that in the standard realization of the algebra C (see part 5.3) the form B has the form B[£U,V, 2 "/*•']""2 ("'• *-i) • Therefore, B is coordinated with the standard gradation. The uniqueness of the form follows from the lack of nontrivial, homogeneous ideals in G which is proved in [23] (see Exercise 18 in Sec. 6, Chap. 1 of the book [2]). 6.

ANALOGOUS OF THE KdV EQUATION FOR KATS-HOODY ALGEBRAS

6.1. Let G be a Kats-Moody algebra, let e£, f£, hi be its canonical generators (i 0,...,r), and let G — &QI be the canonical gradation (we recall that ej&O1, fjSQ-'. kg/OO). We fix a vertex c„. of the Dynkin scheme of G.

Let G — &Q,

be the standard gradation

1 0

corresponding to this vertex.

We set C_O 0 , $—G», » — G , n 2 ' ' " — O 0 n 2 ° ' -

"* "call (see

Proposition 5.10) that C is a semisimple Lie algebra, $ and I are its Cartan and Borel subalgebras, and ei, fi, hi, where i - 0,...,m — 1, m + 1,...,r, are the Weyl generators of the algebra €. Let t > $ E ' be the gradation of t, corresponding to this choice of generators. For any i > 0 we set 1,_(&-'. Thus, » — ®l„ ft —»0, »—©»i.

Then $ —1°. » — ©*'.

[«„, •) — 0. Indeed, the algebra

We note that

» is generated by elements fj, i * m which commute with e m .

We consider the operator

T

where t76C"*(R. »), A — 2 * . ' ^

note that tne

°P erat °r (3.1) considered in Sec. 3 under the

(-0

additional condition trq ■ 0 corresponds to the case where Q — •!(*, C\X, VJ) and cm is a special vertex of the Dynkin scheme. All assertions of the present section constitute a generalization of the assertions of Sec. 3 to the case of arbitrary Kats-Moody algebras. If £ is an operator of the form (6.1) and 56C~(R,R) , then the operator

i-e^iS.)

(6.2)

also has the form (6.1). This follows from the fact that [», l|c» ■ (», em\— 0, [», «,lc> f° r i * m. We call the transformation (6.2) a gauge transformation, and we call the operators S and £ gauge equivalent. It follows from Proposition 5.15 that for any i > 0 the operator ad I, where / — A — em, acts from b( to »,_, injectively. For any exponent j of the algebra 15 (see part 5.5) we choose a vector subspace V ,c»; so that >;_ [/, »y.i]ffiV,. We set V — ©Vy. Since i"V-£[/, ») and the mapping ad/:a-»» Proposition 6.1.

is injective, it follows that dim V — d i m ) — dim,*—dim$ —r.

Any operator fi of the form (6.1) can be uniquely represented in the

form fi-e-^S*") . where SeC"(R. »). 8 — - £ + «"' +A, «"*6C-(R. V). ferential polynomials in q. 2010

Here S and q c » n are dif­

61

Proof. q can +

j)#

Since [S, e m ] ■ 0, we must actually represent ^ Lec

S

^ " — ^ ^", 5 =>2 5 i » w h e r e

i-q-^-l

in the form £*aS|,jT +

?!,n€C""(R. »,)• Equating components lying in «,

f

in the relation 3 — h ^ -j- / = e,as (- h?c"' + / ) . w e find that q£ + lSi + 1 , I] can be expressed , can can ~ —. * • * can can „ , , , in terms of qo ,...»qi_i» Si Si- Therefore, if q^ ,...,qi-i, Si,...,S{ have already an been found, then q £ and S £ + 1 can be found uniquely because ad I is injective. ■ Thus, the choice of the space V provides a "coordinate system" in the set of classes of gauge equivalence. 6.2. In this subsection we write out equations for the class of gauge equivalence (see part 3.1) which are naturally called generalized KdV equations. If W is a subspace of G, W=*®W>, W'czG1, then we set W*= © * \ W- = Hwi, The elements of W are series of the form 2

&=

WeW'.

w

t* &$?[ For example, if G«=*(A., X"1] , then G -*((*"'))•

Let 3. 3* be the saA» as in part 5.6. We set 3 1 — ImadA. It is easy to see that 3 1 is the orthogonal complement of 3 relative to the scalar product on G (see Proposition 5.20). It is clear that 31 —@(3*)i , where (31)1 =~1L ("!G(. Proposition 6.2. For any operator C of the form (6.1) there exists an element U Jt/,, £/,6C"(R, G"') such that the operator

2Q = etAU(Si)

So-j|- + A + //.

has the form <6-3>

WC"(R,3").

If U and 0 are two such elements, then e'w■ *-•«*? = «""\ where 7"6C"(R, 3"). U can be chosen in precisely one way so that '/6C"(R, (31)") • The U£ are hereby differential polynomials in q. M rr

We note that C"(R. 3") = 11 C"(R. 3*). Notation of the type C"(R. 3*) and C (R. 3) has an analogous meaning. Proof.

Let H—^H,,

//,6C"(R. 3"1).

Equating in the relation 2o—e ,d "(S) the components

lying in G""-, we find that Hi + [Ui»i, A] can be expressed in terms of U i , — ,Ui, Ho ,..•,Hi-i. It follows from Proposition 5.16 that any element g£G-< can be uniquely represented in the forma + [b, A ] , where a<3~', fte©"'"')1- Therefore, knowing Ui,...,Ui, Ho,....Hi-!, it is pos­ sible to determine £/, ,6C~(R, (3"'"')1). //|6C~(R, 3"1) uniquely, etc. Since 3" — 0 (see part 5.6), it follows that Hf_C (k, 3")Suppose that 2 0 — e""(i)

and 8o-e" 5 (S) have the form ead\l.e-a
. e adT f where T —

^T„

1

7",eC"(R, G-0 . We must verify that 7"^C"(R, 3" ). Equating in the relation £o — C* (SW com­ ponents lying in G"i, we see that if 7", 7\6C~(R, 3), then |A, 7",,,]eC"(R, 3) and hence TM£ C-(R,3-')»

„,

For any operator 8 of the form (6.1) we set Zg = {AfeC"(R. <5)| |Af, 8| - 0 ) . LEMMA 6.3. Zg —tf-,4y(3) » where U is the same as in Proposition 6.2. Proof.

It suffices to show that if S 0 is an operator of the form (6.3), then Zg^-«3n

Let [M, Sol—0, M— 2 that MjgC~(H, 3").

M„ Mfifj* .

Equating to zero the component of Gn*' in [M, Sol, we find

Equating now to zero the component of G n , we obtain — Af» — [Ati.i. A]. Since

the l e f t side of t h i s e q u a l i t y belongs to we have

j — Af„—0,

i.e.,

MjSf.

C~(R, f),

while the right side l i e s in C"*(R, (3*)1).

We then apply analogous considerations t o

M—M£Z^

2011

62

Let

S1* g W .

g— ^

Since O ' — © (G'nOy), *nd the spaces Gj are finite-dimensional, g

can be represented uniquely in the form

*«'

*•*

T

•

2

£j » where

gfiQr

We set

fc—i<j»

£* = iffl»

. _

n_ = o _ g + I g' — g—g*' It is easy to see that g+—g*€». LEMMA 6.4, Let AfeZg. Then [Af., 2) and [AT, fi| belong to C~(R, »). Proof. We shall prove that \M.t £J6C*"(R, *)• The fact that [AT, £]eC"(R, ») is proved similarly. We must verify that the element [M+, £| is nonpositive in the canonical gradation and belongs to C°°(K, G o ) . We have \M., £] — — \M_, £]. The right side of this equality does not contain positive components relative to the canonical gradation, since in this gradation M_ is negative (see Proposition 5.10) and £ — J J -

contains no components of degree greater

than 1. In the standard gradation the right side is nonpositive, while the left is nonnegative. ■ «*t

Let U be the element of Proposition 6.2. For any w63 we set
is a self-consistent t ion

equation for ?6C"(R2, 1).

■§—

Together with Eq. (6.4) we consider the equa-

If («)..«].

(6.5)

which is also self-consistent. In analogy to what was done in Sec. 3, it can be proved that Eqs. (6.4) and (6.5) pre­ serve gauge equivalence and lead to the same equation for the class of gauge equivalence which we call the generalized KdV equation corresponding to the algebra G and the vertex Cm. If an operator

2

of the form (6.1) satisfies Eq. (6.4) [or Eq. (6.5)] and £**• —5j-+fl*" + A

is the operator of Proposition 6.1, then q —s—»•••],

satisfies an equation of the form

which is a coordinate realization of the generalized KdV equation.

pair for this equation relative to q

- \ I - — f (g"", An (L, A ) -

can be constructed as in part 3.2.

It is clear that Eq. (6.4), and hence also the generalized KdV equation, does not change if an element of 3" is added to u. Therefore, it may be assumed with no loss of generality that u*3*. Remark 1. We write the generalized KdV equation corresponding to an element the form of the system of equation ^ - f . ^ . i ^ L , . . . ) ,

u£3". in

(6.6)

where i runs through the set of exponents of the algebra €J, and q C a n and q £ a n are the same as in the proof of Proposition 6.1. It can be shown that the polynomial Fi is homogeneous of degree i + 1 + n if it is assumed that 3Jq£ an /3xJ has degree of homogeneity i + j + 1. From this it follows that the order of the system (6.6) does not exceed n + s, where s is the difference between the largest and least exponents of the algebra 9 . Generally speaking, the order of the system (6.6) depends on the choice of q c a n . Remark 2. If two vertices of the Dynkin scheme of G go over into one another under an automorphism of this scheme, then the series of generalized KdV equations corresponding to them coincide. 6.3. The following assertions are proved in the same ways as the analogous assertions in Sec. 1. Proposition 6.5.

We consider the equations T T - — j

Then the mixed derivatives 2012

375^- and

■3757. computed by means of these equations coincide.

63

An analogous assertion holds also for the equations Proposition 6.6.

-57 —[7(a)*. £1 and

-^- —[?(«)*, £|. ■

Let H be the same as in Proposition 6.2, // — \ / / j , A/^C""(R, 3"*).

Then

1—1

Hi are densities of conservation laws for Eqs. (6.4) and (6.5). Up to total derivatives these densities do not depend on the arbitrariness in the choice of U (see Proposition 6 . 2 ) . " Remark 1.

Since

«cQ", the densities Hi are gauge-invariant up to total derivatives

(i.e., if the operator fl =« ^ 4-^ + A

is replaced by tfad5(£), S(x)6», then a total derivative of

a differential polynomial in q and S is added in H i ) . From this it follows easily that it is possible to add a total derivative to Hi in such a way that a gauge-invariant differential polynomial in q is obtained as a result. Remark 2. According to part 5.6, Hi * 0 only if the remainder on dividing i by the Coxeter number of C is an exponent of this algebra; the number of scalar conservation laws corresponding to this exponent is equal to its multiplicity. Later (see Proposition 6.12) it will be proved that the densities of the conservation laws obtained are linearly indepen­ dent modulo total derivatives. Remark 3. It is not hard to show that if the element U (see Proposition 6.2) is normal­ ized by the condition CfcC~(R, (3"1")"). then the Hi is a homogeneous (in the sense indicated at the end of part 6.2) differential polynomial in q of degree of homogeneity i + 1. 6.4.

In this subsection we consider generalizations of the modified KdV equation.

LEMMA 6.7.

The relat ion (6.4) admi t s the reduct ion £ - £ + <7-f-A, ^C-(R.Jb).

(6.7)

Proof. Let £ be an operator of the form (6.7), and let AffZg. It must be shown that [AT, £|eC*(R. $ ) — C ( R , GO). We have [M*. £ [ — — |Af". 8J. The left side of this equality does not contain components that are negative relative to the canonical gradation, while the right contains none that are positive. ■ For any u*3* Eq« (6.4), where e has the form (6.7), we call the generalized modified KdV equation corresponding to the algebra G (it does not depend on the choice of the vertex Cm, since it is defined in terms of the canonical gradation). The mapping u assigning to each operator fc of the form (6.7) its class of gauge equiv­ alence we call a generalized Miura mapping. It is clear that u takes solutions of the gen­ eralized mKdV into solutions of the corresponding generalized KdV. It is obvious (see Proposition 6.6) that Hi are densities of conservation laws for gen­ eralized mKdV. It is not hard to verify that with a suitable normalization of U (see Remark 3 of the preceding subsection) Hi is a homogeneous differential polynomial in ^6C"*(R, $) of degree of homogeneity i + t if it is assumed that degq(J) ■ j + 1. Proposition 6.8. -J? — j - /(?,. • >,ql*~}))-

Let

1163". fl>0. Then the corresponding generalized mKdV has the form

If it is assumed that q

a homogeneous polynomial of degree n.

J

has degree of homogeneity j ♦ 1, then f is

Its part linear in q is equal to (—ad A)

[u,q

].

We mention that (—ad A ) ~ n here and below is considered as an operator on 3 (it follows from Proposition 5.16 that the operator ad A acts bijectively on 3 1 ) • The correctness of the n n-1 expression (—ad A ) ~ [ u , q ( ' ] can be verified as follows. Since 3 ° ~ 0 (see part 5 . 6 ) , it follows that J&C3 1 . On the other hand, [adu, ad A] - 0, and hence |o, 3 x ) c 3 x . Thus, \u, $ l c 3 x , so that the expression (-ad A)" n [u, q ( n _ l ) ] makes sense. Proof.

Let 9 ( a ) — 2 ^ " A,^Cm(U,

0%

Equating in (6.4) the components in G° and using

the commutativity of G ° , we obtain ^ — — —- AQ.

It remains to prove that Ao is a homogeneous

differential polynomial in q of degree n and to find its linear part. same way as in the proof of Proposition 1.4. ■ Proposition 6.9. are nontrivial.

This is done in the

The generalized mKdV and KdV corresponding to a nonzero element «63*.

2013

64

By definition, nontriviality of the generalized mKdV means that its right side as a dif­ ferential polynomial in q is not equal to zero. Nontriviality of the generalized KdV means that if this equation is considered as an equation for q c a n , then its right aide it nonzero. Proof.

In the modified case it suffices to use Propositions 6.8 and 5.17. We shall now

show that triviality of the generalized KdV implies triviality of the modified equation. We represent the operator fi in Eq. (6.4) in the form pose that — ^ - — 0 .

S —e,d*(8e,B) (see Proposition 6.1). Sup­

Then .£?._[/?, 2] _ [ %t -2.—J-/4-17I, where R is a differential polynomial in

q with values in •. Suppose now that ?fcC"(R, $). Then _£L€C"*(R. $) fore, p\\R%-i--ir!

+ qY\—0,

(see Lemma 6.7). There­

where P is the projector *-#•« such that KerP —Jb.

We rewrite

the equality obtained in the form

4§—*[(*./» + [«.«]•

(6.8)

We suppose that as a differential polynomial in q [where q(x)£§ ] R has order k, i.e., R de­ pends on q( k ) but not on q ^ k + 1 \ q^k*2^, etc. Then the left side of (6.8) has order k + 1, while the right side has order no more than k. Therefore, R does not depend on q. From this and (6.8) it follows without difficulty that R ■ 0, i.e., the generalized mKdV is triv­ ial contrary to what has been proved. ■ We note that together with the generalized modified and unmodified KdV equations it would be possible to consider a "partially modified" equation corresponding to an arbitrary proper subset S of the set of vertices of the Dynkin scheme. For this in the definition of the unmodified equation it is necessary to replace n and b by algebras n, and b„ where n. is generated by elements fi corresponding to the vertices in S and »,de,-'n,©$. The modified (respectively, unmodified) equation is obtained if S ■ 0 (respectively, S consists of all vertices except one). 6.5. The remainder of the section is devoted to the Hamiltonian formalism for general­ ized KdV equations (including modified equations). In this subsection we define the corre­ sponding Hamiltonian manifolds and in the following subsection we prove that the generalized mKdV and KdV are Hamiltonian. In the case of the unmodified equation the manifold on which it is necessary to introduce a Hamiltonian structure is the set of equivalence classes of operators fi of the form — — \ - q + A, f6C"*(R/Z, I), but it can equally well be taken to be the set of equivalence classes of operators 2 of the form — — \ - Q + l

(see the beginning of the

ax

proof of Proposition 6.1). In the modified case the manifold on which it is necessary to introduce a Hamiltonian structure is C"(R/3, $). Thus, both manifolds are defined in terms of the semisimple algebra ©, and not the Kats-Moody algebra G. Therefore, in the present subsection we shall assume that 9 is an arbitrary semisimple Lie algebra (not related to any Kats—Moody algebra). Thus, let © be a semisimple Lie algebra with Weyl generators Xi, Y{, Hi, 1 < i < r. We denote by a and Jb , as always, the subalgebras generated by the elements Yi,...,Yr and r

Hi,...,Hr» respectively.

We set )_$$■,

/ — £.Xt.

We consider operators

2

of the form

1—1

_£ \-q+ /, 06C""(R/Z, *).

We call two such operators 2* and & gauge equivalent iffi—c*"(fi) ,

where S$Cm (R/Z, ■). It is clear that Proposition 6.1 remains in force if fi is replaced by 2 . We denote by Jf(6) the set of classes of gauge equivalence of the operators St. We call the mapping u:C""(R/Z. Sj)-*-J((6), assigning to a function $€C"(R/Z, $) the equivalence class of the operator d/dx + q + I the Miura transformation. On 6 we fix a nondegenerate, invariant, bilinear form (according to Proposition 5.18, this form is always symmetric). We extend it to a bilinear form on C " (R/Z, 6) by the formula (u,v)~

\(u{x),v(x))dx,

(6.9)

*7z where «. t>6C~(R/Z, 6). The gradient of a functional /: C~(R/Z, Jb)-*-C at a point ?€C"* (R/Z, Jh) is a function gttd,l£Cm(RfZ, $) such that the relation (3.15) is satisfied for any A^C" (R/Z, $). The gradient is uniquely determined by this condition, since the scalar product on Jb 2014

65

is nondegenerate.

We define a Poisson bracket on C (R/Z. Sj)

by formula (3.28).

As in Sec. 3, on Uf(0) we define two Hamiltonian structures. Here the first structure will depend on the choice of the element e in the center of St. If I is a functional on -*(0) [i.e., a gauge-invariant functional on C**(R/Z, I)], then gradf/ denotes any element in C~(R/Z,6) satisfying (3.15) for all A€C"(R/Z, I). From assertion 3) of Proposition 5.19 it follows that grad,/ is defined up to the addition of elements of C*"(R/Z,»). We define the first and second Hamiltonian structures on Jf(0) by formulas (3.16) and (3.17). Just as in part 3.6, it can be verified that {•, • h and {*, • h are well defined and are coordinated Poisson brackets. Moreover, just as in part 3.8, it can be proved that the mapping u: C~(R/Z, $)-*-jf($) is Hamiltonian if the second Hamiltonian is considered on uf(0) . The manifold denoted by Jt, in part 3.6 will henceforth be denoted by jf (|l(Jk)) [we are obliged to specify this, since the algebra •*(£) is not semi simple]. In Sec. 3 JF(|((&)) was 4-1

identified with the set of differential operators of the form

D*-\-Zi

UjD1 > so that the first

and second Hamiltonian structures on jf(|((A)) go over into the corresponding Gel'fand-Oikii structures. Analogous realizations of Jf(0) for all classical simple algebras 0 will be con­ structed in Sec. 8. Remark 1. By definition, Jf(0) depends on the choice of Weyl generators of the algebra 9 f but from assertion 2) of Proposition 5.1 it follows that this dependence is actually in­ consequential . Remark 2. Both Hamiltonian structures on Jf(0) depend on the choice in 9 of an in­ variant scalar product, while the first structure additionally depends on the choice of an element e in the center of a If the algebra
66

is a subalgebra of it, and A: is a connected Lie group with algebra «,, /€*,*. We than denote by Af(»i, «), I) the Uamiltonian manifold obtained fron the reduction of »,* by the coadjoint action of A2 on I (we consider the Kirillov Hamiltonian structure on K,* ) . It is not hard to see that the manifold Jt Ifi), equipped with the second Hamiltonian structure has the form Af(*,, t,, I): as S, it is necessary to take the algebra tf, considered in part 4.4, as «, the pre image of i under the natural mapping 8-»C"(R/Z, 9) (we note that the algebra ■, is canonically isomorphic to » X Q , and define the functional / : « X C - » C by the formula Z(f, a) - (f, I) ♦ a. All that has been said above is valid, in particular, for 9— •<(*). Thus, we have obtained a group-theoretic interpretation of the second Hamiltonian structure of Gel'fand-Dikii. 6.6. We now return to the Kats-Moody algebra G. Let 9, > , etc. be the same as in part 6.1. On G we fix a nondegenerate, symmetric, invariant, bilinear form coordinated with the canonical gradation (see Proposition 5.20). Since this form is coordinated with the standard gradations, its restriction to Go (i.e., to 9 ) is nondegenerate. We shall define the Hamil­ tonian structures on C~(R/3. t>) and Jf(®) (see part 6.5) using just this bilinear form on 9 . The bilinear form on G obviously extends to G and then to C~(R/Z, G ) [see formula (6.9)). For any uf3 we define the functional J»\ : C-(R/Z, »)-»-C by the formula M. («) - C («) • «)• where H(q) is defined by formula (6.3). If u«3~. then Jt. — O, and hence it may be assumed with no loss of generality that u€3*- We note that £f« does not depend on the arbitrariness in the definition of H (see Proposition 6.6). From Remark 1 following Proposition 6.6 it follows that 3$M is gauge-invariant and hence can be considered a functional on J119). Proposition 6.10. The generalized KdV corresponding to an element u69. is the Hamil­ tonian equation corresponding to the Hamiltonian Mu and the second Hamiltonian structure. Proof ■

Let m(u) be the same as in formula (6.5), < P ( B ) — 2

as in the proof of Proposition 1.9, it can be verified that for take Ac.

It remains to show that I A0, ^—(-?+/]

•*'• A)€C"(R/Z, G,) ■ Exactly gradJf. it is possible to

—|V(u)., j j +fl+ A I. This equality follows

from Lemma 6.4 and the fact that the projection of q + A onto Go is equal to q ♦ I.

■

Since the functionals Mu are conservation laws for generalized KdV (see Proposition 6.6), the next result follows from Proposition 6.10. COROLLARY. {*,, M;},—0

for any

a, u Q .

We denote by j t , the restriction of 3$, to C"(R/Z,$). From the equality {%., 3t;).~0 and the fact that the mapping u:C"(R/Z, $)-* JT(D) is Hamiltonian it follows that {3t,.H;) — 0. Proposition 6.11. The generalized mKdV corresponding to an element tonian equation corresponding to the Hamiltonian ~Mm. Proof.

Let f ( u ) — ^ , A' , where AWfRlZ,

C).

In the proof of Proposition 6.8 it was

shown that the generalized mKdV has the form dq/dt - — ( A 0 ) ' . the proof of Proposition 1.9, it can be verified that question has the form

U63- is the Hamil-

On the other hand, just as in

g r a d ^ , — .A0.

Thus, the equation in

~ — —(gradJ^s)', and this is a Hamiltonian equation.

■

The next result follows from Propositions 6.9-6.11.

Proposition 6.12.

If u&'.u + O. then X.-t-O. X. + 0. ■

The equations considered in Sec. 3 are Hamiltonian relative to not only the second but also the first Hamiltonian structure. For generalized KdV this is not true, generally speak­ ing. A counterexample is provided by generalized KdV corresponding to the algebra A£ ) (see Sec. 9 ) . We shall show, however, that the assertion regarding the Hamiltonian character relative to the first structure is valid for generalized KdV corresponding to a Kats—Moody algebra of height 1 and a special vertex of its Dynkin scheme (see part 5 . 4 ) . We recall that the standard realization of a Kats-Moody algebra G of height 1 corre­ sponding to a special vertex has the form «(X,X"'] , where V is a simple Lie algebra. In this realization G, — t U » and, in particular, « — * . Let Xi, v i , H£, where 1 < i < r, be the Weyl generators of the algebra t , and let Xo be the same as in Proposition 5.12. From the 2016

67

explicit form of the canonical generators of H[X,X"'| 1 + AXo, where

/ — ^X,.

(see part 5.4) it follows that A ■

For the element e contained in the definition of the first Hamil-

tonian structure we take Xo (we recall that according to Proposition 5.12 the center of a Is equal to CXo). The following result is proved just as analogous assertions in Sees. 1 and 3. Proposition 6.13.. In the situation described above the generalized KdV corresponding to an element «63. is the Hamiltonian equation corresponding to the Hamiltonian 3H\» and the first Hamiltonian structure. Moreover, [3^m, 39;}i — 0 for any u, B£§. 7.*

SCALAR (L, A)-PAIRS FOR GENERALIZED KdV EQUATIONS

7.1. In Sec. 3 the scalar Lax equation (2.1) was interpreted as the equation for the class of gauge equivalence for Eq. (3.8). In Sec. 6 for an arbitrary pair (G, c m ) , where G is a Kats—Moody algebra and c m is a vertex of the Dynkin scheme of G, an analogue of Eq. (3.8) [Eq. (6.5)] was constructed. The equation for the class of gauge equivalence corre­ sponding to this analogue was called a generalized KdV equation. A generalization of the generalized KdV corresponding to (sl(*. C[A., X~'l)» cQ) , where co is a special vertex of the Dyn­ kin scheme, is the scalar Lax equation (2.1) in which the additional condition ufc-i ■ 0 is imposed on the operator L (regarding the possibility of such a reduction see the remarks following Propositions 2.3 and 3.7). It is possible to not consider vertices of the Dynkin scheme of sl(A, C[X, X~'l). distinct from co, since they go over into CQ under automorphisms of this scheme (see Table 2, type A ^ x O . In this section for generalized KdV corresponding to classical Kats—Moody algebras dis­ tinct from «I(ft, C[Kt X"1]), we will find realizations analogous to the scalar Lax equations. It turns out that a generalized KdV corresponding to a classical Kats-Woody algebra G dis­ tinct from *I(*. C[k, X']]), and a vertex c m of its Dynkin scheme can with some conventions (see parts 7.3 and 7.4) be written in the form - j p — A2Lt—

LXAU

(7>])

-

"3T ™ A\Li — L2A7, 2l + \

where Li and L2 a r e scalar pseudodif ferential d.r ft,6C, * —ord Z-i + ordij. P n , Q n , and R n ) :

21+t

s y m b o l s , A { ■- V ft,(/.2i|)+* , A 7 — V £,(1,/^)+* ,

* ' Here the operators Li are of three types (we denote these types by

i) p.-D^'+J^wr'+o^a^. 2) Q„ - O " + 2 <M*)0*+ £>%,(*)), d.f

*-*

3) /?„=-D*" + 2 (u({x)D^ + D"-iul(x)) + u
In order to determine the types of operators Li and L 2 corresponding to a given pair (G, c m ) it is convenient to use the language of Dynkin schemes. The Dynkin scheme for G after re­ moving c m decomposes into Dynkin schemes of two simple Lie algebras of types Bn» Cn, and Dn (see Tables 1 and 2 ) . To one of these algebras there corresponds the operator Li and to the other the operator L2• To the algebra Bn there hereby corresponds an operator of type P n , while to the algebras C n and D Q there correspond operators of types Qn and Rn. A detailed proof of the assertion formulated above will be presented for the cases (A2n , c m ) and (D* 1 ', c m ) , where c m is a vertex of general position. In the remaining cases the proof is analogous. We note that if Li and L2 satisfy (7.1), then the operator .. ._-, m-' ««f tion ^=- — (Ai.Z-I , where At — 2*b,L * , * — o r d £ .

i — l^C%

satisfies the equa-

Therefore, the system (7.1). where Li and L 2

have type P n or Qn, is in a certain sense a reduction of the scalar Lax equation.

Special 2017

68 cases of such reductions were considered in [41]. (2} 7.2. Suppose that the algebra G of type A 2 n is realized in the form Z.(*l(2rt-f 1), a„), 0 < m < n, where o m is the standard automorphism corresponding to the vertex c^ (see Appen­ dix 2 ) . Then the operator 2. corresponding to the pair (G, c m ) [see formula (6.1)] has the form

*-,4 + (o'£) + A. where the blocks #i(x) and a2(x)

*

1) and »»{2m); A — / . ^

(7.2)

are upper triangular and belong, respectively, to *(ty — 2m-\-

(the form of ef is indicated in Appendix 2 ) . We denote by Wi

1-0

the set of columns of the form (ui, U2» ■ •. ,U2 n 4i) c * where u,(X)£&((\~l))t

*—1

2/t-J-l , whereby tat

u£(X) - U£(-X) for i < 2n - 2m + 1, ui(X) - -ui(-X) for i > 2n - 2m + 1. We set (a fi)eG * where afMst (2« — 2 m + 1, C|X, X~% 66Mat(2w, C\K

JP2—XIF,.

If

V'J), then the blocks a and 6 contain

only even powers of X, while the blocks B and y contain only odd powers. Therefore, Xffific W£ for any X&G. Moreover, £(W,)cU?V By means of the operator 2 we introduce the struc­ ture of a B((D"*1))-module on Wi in the same way as this was done in part 3.4. The following result is proved by direct verification. LEMMA 7.1. Any vector in Wj whose components contain X only in nonnegative powers can be uniquely represented in the form A ^ i , where A£B[D], ^, — (1, 0,.. .,0)', y 2 — (Q,.. .,0. 1,0 It follows from Lemma 7.1 that there exist uniquely determined operators Lx, L£B\D\ such that Li*^j - X^2, hz'i>2 ■ A^i. It is easy to see that the orders of these operators are equal, respectively, to 2n — 2m ♦ 1 and 2m, while the leading coefficients are equal to one. It is not hard to verify that Li and L2 do not change under gauge transformations of the operator fi. LEMMA 7.2. Proof. ai(x)

Lj ■ - L I , L J - L2.

It is proved in analogy to Proposition 3.14 that Li ■ —(A(Pi))*, where Pi ■

+ \ \ "'."?]•

According to Lemma 3.13, (A(Pi))* - A(pJ).

On the other hand, ?J - - d i a g O ,

V0...1 D) -1

(-ir'J^dlagO, - 1

(-1)1*1). whence A (Pj)~

- A (/>,). A ?)-A ,). ■

We have thus constructed a mapping from the set of classes of gauge equivalence of oper­ ators 8 of the form (7.2) to the set of pairs (L,, L 2 ) , where Li and L a are oper­ ators of type Pn-m and Q m » respectively (see part 7.1). In Sec. 8 it will be shown that this mapping is bijective (see Propositions 8.2 and 8.4). It is not hard to verify that elements of the form A 2 l + 1 , where 2/+1GZ is not divisible by 2n + 1, generate the centralizer 3 of the element A as a vector space over C (see Appen­ dix 2 ) . Proposition 7.3.

Suppose that an operator S

of the form (77^) satisfies Eq. (6.5),

1

where B — ^btA***1,

btGC.

Then the operators Li and L 2 satisfy the system (7.1), where Ai -

Proof. Throughout the proof B will denote a ring of functions of x and t. On Wi, i 1, 2 we introduce the structure of a B[D, D t )-module so that the operator of multiplication by D t is equal to

J^—
This is possible since 1) r^L_n>(«)^ J>1 _ 0 ,

2) 9(u).6G , and the

elements of G take Wj into itself. We shall find operators A£B[D], «'—1,2, such that D t - * i " Ai-ifi- Since D,-Vt — —9(«),^i , according to Lemma 7.1 the A£ exist and are unique. We have

(».-(oW,;)-°- G!m-W 2018

Hen

«

_69

and hence the operators L£ satisfy the system (7.2). In order to express K\ in terms of Li and L2 we note that LzLi-ti - \*ti. From this, as in the proof of Proposition 3.16, it foli

lows that |. Further, following the proof of Lemma 3.17, we find that Ai * n+i —^ 6j(£jii)++l. The formula for A2 is derived similarly. ■ For algebras of types C Q ' and D n the realization of Che generalized KdV in the form of a system (7.1) is found in exactly the same way. In the case of the algebras B^ 1 ', Ajjjlj, L and D n ) the arguments are somewhat more involved, since det A ■ 0. In the next subsection we treat the case D£ ' which is the most complicated, since Che zero eigenvalue of the matrix A is multiple (multiplicity 2 ) . 7.3.

We denote by G the scandard realization of the algebra of type Dn

corresponding

to a vertex Cm, 1 < m < n — 1 (see Appendix 2 ) . We note chat if ( a 5)eO. a€Mat(2/» — 2m, C(X, V'j). 6eMat(2m. C | * . V ] ) .

then „ ( X ) - a ( - X ) , o(X)-6(-X), P(X)

P ( - * ) . Y(M

Y<-*) •

He

a

henceforth write matrices of order 2n in the form [ £j , where Che orders of a and 6 are equal, respectively, to 2n — 2m and 2m. The operator S, corresponding to che pair (G, c m ) has the form 2 — ~7T^"('o matrices a{(x)

«.) + ^'

wher

e

A— 2 * '

(the

£orm

"t e£ is indicated in Appendix 2 ) , the

are upper triangular, and <Ji6'(2rt — 2m), 0,6'(2m)-

We shall formulate some properties of the matrix A that we need below. LEMMA 7.4.

1) The eigenvalues of A are rooCs of degree 2n — 2 in X

and zero (having

multiplicity two). 2) BftX-'))5" — Kef A © l m A . 3) The matrix P — X - W " ' is the projector onto ImA. P does not depend on X, and the first and (2n — 2m ♦ 1)-th columns of P have the form ^, — (1,0

0)' and 9, — ( 0

0, 1, 0, ..., 0)', respectively.

4) The centralizer of A in G is

lM-7m

generated as a vector space over C by the elements A"*1

and XV'F, l£Z.

Here A* for k < 0 is

defined by the formula A*-X-»'A* +,fa " n ^ r » 0 ; *t, F — -j- («.-«.!■-• + <»-«+! ,li-«+l + ( — Vp*U-m.m-m + l—\y,eu-m+i.~-m+,)—e^m.1.-m,t+l—\)m"eu-m,^m+,—

+

— J (««-«+!.!«-» + (—l)*«2»-«+l.»-«). 5) For k < 0, A** does not contain p o s i t i v e powers of X; the f i r s t and (2n — 2m + 1 ) - t h columns of A^ contain only s t r i c t l y negative powers of X. ■ We denotebyWj the s e t of columns of the form ( u l f . . . . u j n ) ' such that l , . . . , 2 n , u£(X) - u i ( - X ) for i < 2n - 2m, uj.(X) - - u i ( - X ) for i > 2n It i s c l e a r that B((X"1))*-W,(BWt, 8(W,)cJT„ X<W,)<=W, for a l l X € 0 .

K,6£((V'))

2m. We s e t

According to P r o p o s i t i o n 6 . 2 , there e x i s t s a s e r i e s T such that the operator has the form

for i 01 Wt—XW,. &<, — TSiT~'

*.--£■+A f gM-^^+gi*-*"*^. where

/ „ gjiB. LEMMA 7.5.

We shall need the following properties of T. 1) T(Wi) - Wi.

2) T(*i) - *i is a series in strictly negative power* of X.

2019

70

Proof.

Since T - e u , where UeQ, it follows that T(Wj) - W£.

Since U&J-,

it follows

that T does not contain positive powers of X while the free tern of T has the form I Q _ j . where <*j are upper triangular matrices with ones on the main diagonal. assertion 2 ) . ■

From this we obtain

We set IV,' — T - ' O ^ n l m A ) , If',"-7-"1 (W, n KerA). Since B(Qr')f — K e r A © I m A — W,BW, and A0F,)<=XP, , it follows that IT, — (W, f| Ker A)®(K, f) 'm A) , and hence IT, — W/9W,". It is clear that 2(W,')cW/, 8 (ir,*)clp," . Moreover, it is not hard to show that the operator C is invertible on wj. On w[ we introduce by means of the operator 2 the structure of a BCtD"*1))module in the same way as this was done in part 3.4. LEMMA 7.6. Any element of W^ can be represented uniquely in the form A-ij, where A 6 B((D - 1 )), and ij»j is the projection of \fi^ onto \>[. Proof.

It suffices to verify that any element of Wt{]lTn\

can be represented uniquely

*

in the form

2

ffit'PT^,,

ffcB.

where P is the projector onto Im A such that Ker P - Ker A.

This can easily be seen directly. It is only necessary to note that PTi^ ■ i^j + Ri, where Ri is a series in strictly negative powers of X (this follows from assertion 2) of Lemma 7.5 and assertion 3) of Lemma 7.4). ■ According to Lemma 7.6, there exist pseudodifferential symbols Li and L2 such that Li* ' X*2, L2 - *2 - X*l. LBiMA 7.7. Lj and L2 have the forms Rn- m and Rgp respectively (see part 7.t).

il

Proof. In analogy to the way this was done in the proof of Lemma 7.2, it can be shown that I* - —Li, and Lj - A(Pi), where /to t I.0 1. V t'*ilx>'

I

0 0 0

••• \ ?

0

V\\. • * 3

'0*3/

It remains to find the order and leading coefficient of Li and also prove that (Li>_ has the form /£)"'/i /6B. Let, say. i - 1. We recall that Pi has dimension 2n — 2m. It is convenient to go over from the matrix Pi to the matrix Px-*SPxS~l, 2eH-m+itn-m+i-\-en-m-„-m+i-r-^e„-m+i^-m

where S — £ " — «■£.._»,,,._„,—

and then in Pi permute the columns with indices n — m and

n — m + J_. We denote the matrix obtained as a result of this by Pj. It is clear that A(Pi) ■ A(Pi). Pi has the form T', -= ?{j:)-r(f-«.-«.*-«- ^-«+i,,-«+i + ««-«,..„+, + *..a+It((.m)Z) + ^

el+l./— «i,_w+i.i,_m , where the matrix q(x) is upper triangular.

deduce that (Li)+ has the form D7*'2*"1 -f £u,(x)Dl, Let £,.*.[ 4 " jl"),

From this it is not hard to

2n—2m

'—'

and (Li)- has the following structure.

where Aij are blocks of order n - m.

and A(A 2 2 ) in the form ^ D'b,-

-1

Then (Li)- *—<2oD bo.

We write A ( A l t ) in the form

^a,D'

It remains to note that A22 ■ d i a g O ,

1. - 1 . 1. - 1 (- 1)""" Xi4l,diag(I, - 1 , 1, - 1 (-1)"""",whence *(A27)) -(-1)-*-'(A <*.,))• and hence * 0 — (— l)"""* 1 ^- Thus, ( £ 1 ) - — < — I ) " - * ^ - 1 ^ ■ n m We set fi ■ i ~ rto, where aa is the same as in the proof of the lemma. The analogue of fi constructed on the basis of the matrix P2 we denote by f2* It is easy to see that Li and fi do not change under gauge transformations of €. Thus, we have constructed a mapping from the set of classes of gauge equivalence of operators (Lj, L2, fi, f2>, where f(6fl and Lj are skew-symmetric pseudodifferential symbols with leading coefficient 1 such that 2020

71

(Z,i)-"/»^-tf(. ord/., — 2rt—2m—1, ordLi=2m—I. Later (see Proposition 8.5) it will be shown that We note that the mapping from the set of classes of gauge equiv­ chis mapping is bijective. alence of operators 8 into the set of pairs (Li, L2) is not bijective, since, knowing Li, it is possible to recover f^ only up to sign. In part 7.1 an assertion was formulated that the generalized KdV corresponding to a classical Kats—Moody algebra distinct from A^ 1 ' reduces to the system (7.1), where Ai and A2 are expressed in terms of fractional powers of L1L2 and L2L1. In the case D ^ 1 ' this asser­ tion needs refinement: the generalized KdV in question must correspond to an element «63. having the form u —2*'A 3 '*', * ^

(we

n

°te that in all cases except D n

any element of

3

t

has this form; see Appendix 2 ) . Proposition 7.3. Suppose the operator fi satisfies Eq. (6.5), where it — y b ^ i v * 1 , /-o Then L x and L2 satisfy the system (7.1), where 4 , — (Mt).,

Mx~* — ^ftj^i,)*^ 5 .

bfC.

x

^ 2 — — 2*/

The proof is entirely analogous to the proof of Proposition 7.3. However, because the ]>i are not entirely explicitly defined the proof of the formula Pt'^i " Aj/i^i is nontrivial. It is based on the following lemmas.

LEMMA 7 . 9 . DrWtfcW/t Proof. [i.e.,

DfWfczWf.

We denote by V t the operator on W acting according to the formula V t (w) - D t *w Since [V,. fi|—0 , it follows that \TvtT~l, fio]— 0.

V | — j f — 9 ( a ) * ]•

From this it is

easily deduced that [TV t T~ , A] " 0 , and hence the kernel and image of A are invariant under TV t T~ l . Using assertion 1) of Lemma 7.5, we find that T^,T-[ (Wt)czXPt whence the lemma fol­ lows. ■ LEMMA 7.10. Proof.

(M-ii) + - M+'i^ for any

(M^t)^—

M+^t

Afg5((£>"')).

— (^♦(—Af*-ifri)» —(Af.-^( —Af + -^j) + , where

$,—ty— $ , . It remains to

verify that (fi'9()„ — 0 for > < 0 , ( £ / ^ - 0 for j a O . It suffices to prove that ( f i o T ^ — 0 for y < 0 , ( Q o ^ r ^ M 0 for j > 0. We have 7*9,— PT$t, Tyt-- (E — P) T$t, where P is the same as in Lemma 7.4. Therefore from assertion 2) of Lemma 7.5 and assertion 3) of Lemna 7.4. it fol­ lows that (7*9,),— 0, (7*9,).,—♦,. Since T^eKerA, the equality (T*i)* - 0 implies that (flo'T"*]),.— 0 for j > 0. The equality (So^if,),,, — 0 , where j < 0, follows from assertion 5) of Lemma 7.4. ■ The formula D ^ * ^ ■ Aj/^£ is induced from these lemmas as follows. It( follows from Lemma 7.9 that Dt»ijii ■ ^ ( D t / ^ i ) . where *{:Wi -»■ W^ is the projector with kernel W^. By definition, 0„ Df-9,_—j*^

9/——(J*9()^ where J* — 2

*/"'A2/*'7".

Since

J # ( W V ) — 0 , it follows that J*^ —

•s*^,. Since ■&$,— — Af*9, (see the proof of Proposition 7.3), we have plying now Lemma 7.t0, we find that Dt*£i • *i(A£*iJii) ■ A i * ^ . If u has the form ^

btA7i*1

Dt$t

—ft*((Aft • 9*)J*

Ap-

, then, as before, the corresponding generalized KdV reduces

/ to the system (7.1), but unfortunately in this case we do not know an explicit formula for Ai and A2> In the simplest nontrivial case Ai - f 2 n _ 1 f i , A2 " fiD~ l f2. 7.4. In part 7.1 a rule was formulated for determining the types of the operators Lx and L 3 corresponding to a given pair (G, c m ) , where G is a classical Kats—Moody algebra and C Q is a vertex of its Dynkin scheme. However, in the case of algebras G of low rank and also in the case where c m is an extreme vertex this rule cannot be understood literally. We there­ fore present a table (Table 4) in which for each pair (G, Cm) the types of the corresponding tftf

4t(

tef

operators Li and L2 are indicated. In t h i s t a b l e i t i s understood that P0—•£>, Qo—«1. /?©—D"1. The v e r t i c e s of the Dynkin scheme of G are numbered as in Table 2 . It was noted in part 7.1 that i f Li and L2 s a t i s f y the system ( 7 . 1 ) , then the operator L—LfLi s a t i s f i e s the equation dL/dt - [ A l t L ] .

It i s clear that i f one of the operators

2021

72

TABLE 4

<£

•n»f

*K-t " » t «?

»»I

> —n>1 »

CS'

eff *

»>i —■

»Si »><

c«

kt

0«m«n 2«m«n

'«-«» «n-m

m-V

On ^n-m

><m<* m-V

0<m
<•!

tm »m •o *m

%

«. •m

•n »0

«. •« *w

«n-m B »-m

«m

Pn-m

Li belongs to type Po, Qo, or Ro (this condition is almost equivalent to C Q being an extreme vertex) the system (7.1) is equivalent to this equation. In conclusion we consider conservation laws for generalized KdV. In analogy to Proposi­ tion 3.20, it can be proved that in the case of classical Kats-Moody algebras distinct from D^ ' the densities of the conservation laws Hi considered in Proposition 6.6 up to constant multiples and addition of total derivatives are equal to res ( L i L j ) ' ^ , where k - ord(LiL2). In the case of D n '> the situation is as follows. We write the element W € C ~ ( R , 8")- considered A/A-'"+1l + 2sA _l, ' + "/ : '.

in Proposition 6.6, in the form H—2 '~°

"here

A„ gfiB

(see Lemma 7.4).

—

ti+i

Then up to a constant multiple and addition of total derivatives hj is equal to res (£,£,) * . Unfortunately, we do not know a general formula expressing gi in terms of L j , L j , fi, fz (the f£ are the same as in part 7.3). We note only that up to a multiple and total derivatives go is equal to fif2. 8.

HAMILTONIAN MANIFOLDS

M(9)

In part 6.5 for any semisimple Lie algebra 9 we defined a manifold A (9) equipped with two Hamiltonian structures. Moreover, jf(|I(A)) is by definition the manifold denoted in part 3.6 by

Jt.

We denote by

M(|t(*)) the set of operators of the form D'+\u.,D',

u£B*

In

part 3.3 we constructed a bijection F:JC(H(k))*M(tl(k)). and in part 3.7 it was ahown that the first and second Hamiltonian structures on JT(|I(*)) go over under the mapping F into the cor­ responding Gel'fand-Dikii structures on A1(|I(*)). In the present section for each classical simple Lie algebra 9 we define a manifold M(9) consisting of scalar differential [and in the case € ) — »(2/i) pseudodifferential) operators of special form' and a bijective mapping F:JC{9)-*M(9). Moreover, the Hamiltonian structures on M(9) corresponding to the first and second Hamiltonian structures on A (6) will be found. We recall (see part 6.5) that the definition of the manifold Jf(«) and the Hamiltonian structures on it involves the Weyl generators of 9 , an invariant scalar product on 9 , and an element e of the center of the algebra a. For all classical algebras 9 we use the Weyl generators presented in Appendix I and the scalar product (X, Y) - tr (XY). We shall indicate the element e each time. 8.1.

We begin with the simplest case

«-•<(*).

It is easy to see that Jf(«l(*))cJf(•'(*)).

We set M("(*))-{i)' +Vu,£)'|U,eBo • It i« clear that the bijection Uf(|l(*)) — M(«<(*)) consi-o I tructed In part 3.3 map* Jt (•'(*)) onto M(•!(*)). We denote by {•, -)i and (•, - h the Poisson brackets on M(»>(*)). corresponding to the first and second Hamiltonian structures on •*(»'(*)) (as the element e in the definition of the first Hamiltonian structure we take the matrix e ^ k as in Sec. 3 ) . We shall find the explicit form of these brackets. For this it suffices for any integral symbols JT, Y£B, ((O"')) to find {/*. /K}i and {lx, lr)%, where /jr:M(»l(*))-»C is defined by the formula Ixjf-) — Tr (XL). It is not hard to see that if a and * are functionals on jT(|l(*)) and m and * are their restrictions to •#(•!(*)), then fi, ^}, and fi, ^ } 2 are equal to the restrictions to Jt(»!(*)) of Mikhallov [65] pointed out the connection between the types of scalar differential operators and the types of the classical simple Lie algebras. 2022

73

the functionals {9. 9}i and {9, if),. It therefore follows from Theorem 3.22 that {lx. M i and (/*. M J « e defined by formulas (2.16), (2.17). 8.2. Now let «—e(2n+l) —(i46«l(2n+l)|o(^)=i4} , where a is the automorphism of il(2rt + l). given by the formula o(A) — — dlag(l, — 1 l)-i4r-diig(l, — 1 1). The Borel subalgebra in 6 will be denoted by lg, while t denotes the set of all upper triangular matrices (we recall that »g — >n®)- The meaning of the notation Hq, and • is similar. Since l^ci, '$cr», while the elements I for the algebras 8 and jl(2rt+l) coincide (namely, /-= V «,+1,,) , there is the i—i

natural mapping ?:Jl(9)-*Jl(il(2/i + 1)). Since o(») —», o(«) —», o(/) —/,it follows that o in­ duces a mapping .if (al(2/l+I))-*-*(l<(2/.+I)), which we also denote by a. LEMMA 8.1. f maps .* (8) in one-to-one fashion onto the set of elements of jf(gt (2/t + 1)). invariant under o. Proof. We denote by I, the set of matrices (aag) such that a0g » 0 for S — a * i. For any i > 0 we choose a vector subspace V,
«/./+. I- We set

V —©V,. It is clear that »a = [/, »8]SK», where V°— {A&/\o(A)~A)

. Ac­

cording to Proposition 6.1, it is possible to replace JT(9'(2rt+1)) by C~(B/Z, V) and Jf(8) by C*(R/Z, V a ) after which the assertion of the lemma becomes obvious. ■ We henceforth identify Jt(>(2n + \)) with its image in Jt(il(2n +1))- We recall that the bljective mapping constructed In Sec. 3 F: JT(|I(2n+ 1)) —Af(«I(2n +1)) assigns to a class of gauge equivalence of the operator d/dx + I + q the operator L ■ —(A(P))*, where P ■ I ♦ q ♦ diag (D, 0,... ,D) (see part 3.3). If q is replaced by o(q), then P is replaced by—diagO, —1 1)-PT-diag (1, —1 1), and hence (see Lemma 3.13) L is replaced by -L*. Thus, the mapping icoF-':/W(l<(2rt + l))->-Af(l<(2)t +1)) takes L into -L*. From Lemma 8.1 we therefore obtain the following result. dtr

Proposition 8 . 2 . {£6Af(«'(2tt + l)) | £ • - - ! } Remark 1. It is verse to it are given

F maps Jl(>(2«-)-l))

in one-to-one fashion onto the s e t M(• (2r»+1)) =-

. clear that the mapping F:JC(»(7n-\-\))-*MlfC2n-\-\)) by differential polynomials.

and the mapping in­

Remark 2. Let 4 be the Cartan subalgebra of «(2*i-f 1). Composition of the Miura trans­ formation u:C~(R/Z. !))—■£(o(2«+l)) and the mapping F:Jt(>C2n + l))->-M(°(2n + \)) takes the matrix dlag(/, /„, 0. — / . —/,) into the operator I — (D + /,) • • • (D + f„)D(D—/„)... (D- fO. We denote by {•, *}i and {•, »}2 the Poisson brackets on Af(»(2fi-f-1)), corresponding to the f i r s t and second Hamiltonian structures on J[{*{2n+1)) [for e we take (e x 2 n * e 2 2n+i)/ 2 ) . We shall find the e x p l i c i t form of these brackets. For t h i s i t s u f f i c e s for any i n t e ­ gral symbols X, K6flo((D"')) to find Ifx, Mi and (/», I,), , where lX-M(»(2n+1)) — C i s defined by the formula lx(L) = 7t (XL). Moreover, i t may be assumed with no l o s s of g e n e r a l i t y that X* - X, Y* - Y [indeed, the formula Tr Z* - -Tr Z v a l i d for any ZtBMD'% implies that i f L* - - L , then T r ( X £ ) - T r ( - ^ ^ - - Z . ) J . Proposition 8 . 3 .

Let £ C M(»(2»+1)). X, Y^B^D-% ordA'<0, ordK<0. X'-X,

Y'-Y

. Then

l'x. lrh(l)-T'C-(YDX-XDY)), Vx. lrh{i-)-Ti((LYUX-XHYL)J. We note that since L* - -L, X* - X, Y* - Y, it follows that lt(L-(YDX (LYDX), Tr lif.Y)JJC - XL (YL) J - 2 Tr ((LY).LX).

(8.t) (8.2) — XDY)) — 2Tr

x

Proof. From formula (3.17) it follows that if the functionals I,, l,:Jl(«'(2n+l)-*C are invariant under a and Z\ and Z2 are their restrictions Co Jf(*(2n-(-1)), then {'1, £2)2 is equal to the restriction of Hi, £2)2 to JT(• (9n +1)) [to see this it suffices to note that if o(q) q, then for a suicabl* normalization of Che gradienc o(grady,) — grsd,/, , whence gttdfl,gC"(R/Z, O(2B+J1)) and hence grad,/,—grtd,/, ]. Therefore, if 9, and 9, are functionals on yW(|!(2n + 1)) such that ?,(£) —9,( — £•) and 9, and 9] are their restrictions to Af (• (2»+1)), then 2023

74

{9i. ^jk i s e qu«l to the restriction of {9i, »,}, to jW(»(2n + l)). Setting now *,(£)— Jt(XL), 9,(i) —Tr(Ki) , we obtain (8.2). Since the element e for the algebras >(2n-fl) and (1(21 + 1) do not coincide, equality (8.1) cannot be proved in a similar way. It is not hard, however, to deduce (8.1) from (8.2) by arguing in the same way as at the end of the proof of Theorem 3.22. Here we use the following simple assertion: if to the operator d/dx ♦ 1 ♦ q(x), where q(x) is an upper triangular matrix in »(2« + l), there corresponds LGM(»(2n+\)), then to the operator d/dx ♦ I ♦ q(x) — e there corresponds L ♦ D. ■ 8.3. We consider the case 6—»Pf2l)—Me»l(2«)|o(.A) —.A)i where o is the automorphism of |I(ai), given by the formula a(A) — — ateg(l. — 1 1, — l)-i*r-dlag(l, — 1 1, — 1 ) . Lemma 8.1 also holds in this case. In particular, JT(sv(2n)) can be considered a subset of Uf(|l(2l))> The following assertion is proved in the same way as Proposition 8.2, 8.3 [and even somewhat more simply since the elements e in the definition of the first Hamiltonian structure for the al­ gebras ff(2n) and i<(2n) coincide]. Proposition 8.4.

1) The mapping F:UT(«l(2ri)-»A1(|I(2n)) maps Jf(»>(2n)) in one-to-one fash-

ion onto the set At(•»(2*) — (i6M(l'(2n)) | £• — £). The mapping F: Jl (•»(2rt))-<-.M(»»(2n)) and the mapping inverse to it are given by differential polynomials. 2) Let $ be the Cartan subalgebra of <>(2R). Composition of the Miura transformation u:C~(R/Z, !})-*■ JH*i(2n)) and the mapping F:JHn(2n)) — AHt (2n)) takes the matrixdlag(/ /«• _/ - / , ) into the operator !-(£> + / , ) . . . ( D + /„)(£>-/.)••• (£>-/.)• 3) We denote by {*, *}j and {*, *)2 the Poisson brackets on M*>(2n)), corresponding to the first and second Hamiltonian structures on uT(*9(2n)) (for e we take eiv2n)> For any integral symbol X6B0(D-i)) we define 1, : M(«»(2n))->-C by the formula /,(L) -ir(XL). Then for any Z.€M(*p(2n)) and any skew-symmetric integral symbols X, Y^B0(D~1)) the following equali­ ties hold: {'x. /r},(£)-Tr(£.IK,

X])-2-Tr(LVX).

Vx.lrhM-TH(t-r)A*-XL(n).)-2TT((Lr).LX). Remark. The analogue found in [27, 44] of the interpretation of the first Gel'fand— Dikii bracket as the Kirillov bracket also holds for the first Hamiltonian structures on Af(*»(2n)) and M(o(2n+1)). In the case »(2n) it is necessary to consider the Kirillov bracket for the Lie algebra of skew-symmetric integral symbols. In the case e(2n+l) it is necessary to consider the Lie algebra of symmetric integral symbols with the unusual commu­ tator [X, Y]-XDY-YDX. 8.4. We consider, finally, the most difficult case C«»(2n). We denote by M(o(2n)) the set of pairs (L, f ) , where ftBi, IeBc((£)-i)), !•- — L, L.-fD-'f, ordI-2rt— 1. and the leading coefficient of L is equal to 1. In part 7.3 (see the proof of Lemma 7.7) we essentially con­ structed a mapping F : jT(o(2n))->M(»(2n)). Proposition 8.5. polynomials.

1) F is bijective.

The mappings F and F - 1 are given by differential

2) Let t) be the Cartan subalgebra of *(2n). Composition of the Miura transformation u:C"(R/Z, $)-jr(o(2n)) and the mapping F:J*(o(2ri|)-*Af (»(2n)) takes the matrix dlag(/, /•• -/. -ft) into the pair (L, f), where I_(D + / , ) . . . (D + /„) D-'{D-f.)... (D-/,). / (-1)-W>(1), where P = (D + / , ) • • • (D + /»). 3) We denote by {•, • )i and {•, • >2 the Poisson brackets on M(»(2n)), corresponding to the first and second Hamiltonian structures on UT(°(2n)) [for e we take ( e 1 ( 2 n - 1 + e 2 , 2 n ) / 2 ] . For any integral symbol X£Bt((£>-')). such that X* - X we define lx:M{>(2n))->C by the formula U(£. /)-Tr(A'I). Moreover, for any s6B0

we

set X,(£, / ) — \ S{x)/(x)dx.

Then

(lx.lr)i(L.f)>-,n<(LYDX). Vx,K)>-0,

{K. U — j - $ s(x)t"(x)dx;

{'x. lrh(L, / ) - 2Tr ({LY),LX), {).., /,},(£, f)-Tt(LXfD-'s),

^..M 8 (i./)-T T '(" D " J )2024

75

We shall only outline the proof. Assertion 2 ) , of course, is proved by direct computa­ tion. The difficulty in the proof of the remaining assertions as compared with the analogous assertions for o(2n+I) (see part 8.2) is connected with the fact that the matrices I for o(2n) and gl(2n) do not coincide, and hence there is no natural mapping Jf(o(2n))-*-Jf(gt(2n)). In order to overcome this difficulty, we define a new manifold Jf', in which Jf(e(2n)) is imbedded in a natural way. For this we consider operators ft of the form

*-£

+ r+i. ?ec-(R/z,e),

(8.3)

where I is the sum of the Weyl generators Xi of the algebra o(2re) (see Appendix 1) and 9 is a set of matrices (aag) of order 2n such that Sag - 0 for a > B and (a, 8) * (n ♦ 1, n ) . We denote by 91 the set of matrices (a a g) of order 2n such that a 0 g ■ 0 for o > 8 and, more­ over, aa [,*! - 0. It is easy to verify that if 2 has the form (8.3) and 56C"(R/Z, S ) , then esie~s also has the form (8.3). Such transformations we call gauge transformations. The sec of classes of gauge equivalence of operators 2 of the form (8.3) we denote by JL' . We recall that • (2n) =» {A&S (2/t)| a (A) =- A \ , where o is the automorphism of «<(2n), given by the for­ mula o(/4)--diag(1. - 1 (-1)"-'. (-!)"-'.(-')" 0 <*r.di«g(l, - 1 (-If1. <-ir'. (-1)* !)• Since o(/)=«/. o(8)«=$, o_[3i)»=9l, it follows that c induces a mapping Jt'-*-JC', which we also denote by a. In analogy to Lemma 8.1 it can be shown that the natural mapping from Jt(°ipi)) into the set of elements M', fixed under a is bijective. We denote by M' the set of qua­ druples of the form (L, f, g, h) , where /. g, A(=fl0. and L is a pseudodif ferential symbol of order 2n — 1 with leading coefficient 1 such that L_ ■ f(D + h) - 1 g. We define the mapping F':Jt'-*M', assigning to the class of the operator d/dx ♦ I ♦ q(x) the following quadruple (i./.g.*): «) i - A ( P ) , where P= dla?(D, D

D) + q + f,

b) *=--£-(?„. . + ?„+i. .+ i)-?.+l. . -

-r-tjit. n+i, where qi,j are the elements of the matrix q; c) let Aij be the same as in the proof of Lemma 7.7; we write i ( * n ) in the form 2 a , ( & + *)'• •i«t

d*r

and

M A 2 2 ) in the form 2 ( ^ + *)'*j • i

'

then /S-X^OQ, g — ( — I f i * ^ . It is not hard to verify that F' is well defined and is bijective, whereby the elements of the matrix q corresponding to the quadruple (L, f, g, h) can be chosen in the form of differential polynomials in f, g, h and the coefficients of L; the mapping F'-o-(F')~':M' -► M' acts by the formula (L, f, g, h) ■» (-L*, g, f, - h ) . From this we obtain assertion 1) of Proposition 8.5. On Jt' we consider the Poisson bracket given by formula (3.17) and carry it over to M' by means of the mapping F'. For any integral symbol X^BQ((D':)) we define lx:M'-*C by the formula lx(L, f, g. A) — Tr (XL). Moreover, for any s£B0 we define functiooals
u) - D»+2(uD + D«); f,(D*+u, D'+v)-(D'

+ (tt + v)D+D(u + v)+(u-v)D->(u-v).u—v);

D*u + 0)—D» + 2(O£), + £>»U) + 2 ( J D + D S ) . where s — «' — «* — v; /i(D*

2(uD' ♦ D»u)+2(i£> +fls)-4trD-'»,2lw).

+ uD'+D'u+v

/4D*+uD*'+

+ wD + Dw) — (D,+

s-u'-u"~v.

We call a mapping of Hamiltonian manifolds f :Mi •* M2 almost Hamiltonian if for any func­ tional m, $:Mt-*C {/*«>, /*♦)-X-/*(f. ♦)• where \ is a number not depending on «> and ♦ (we recall that f is Hamiltonian if X - 1). It is clear that the mappings fi, 1 < i < 4 arc almost Hamiltonian relative to the first and second Hamiltonian structures. They are not Hamiltonian, since the isomorphisms »1(2) -io(3), »1(2) X»I(2)-i«(4) , etc. preserve the scalar product only up to a constant multiple (in addition, under these isomorphisms the elements e in the definition of th« first Hamiltonian structure go over into one another again up to a multiple).

202S

76

8.6. If 9 T is the automorphism of the semisimple Lie algebra x induced by an auto­ morphism |, of its Dynkin scheme (see part 5.1), then ,(•) _ s , and hence •, induces a mapping 5,:Jf (•)-► JT(I). It can be shown that if « is a simple algebra, then a, preserves the scalar product, and hence the mapping S T is Hamiltonian relative to the second Hamiltonian structure (relative to the first structure it is almost Hamiltonian). Among the Dynkin schemes of the classical Lie algebras An and D Q have nontrivial automorphisms (see Table 1). It is not hard to verify that to the nontrivial automorphism of An there corre­ sponds the mapping Af(»l(ii+l))-*/to(»l(B+l)), acting by the formula L-*(— l)**'!*. while to the automorphism of the scheme of Dn changing the places of en and cn-1 there corresponds a map­ ping M{»
is the set of all operators of Bo[Dl of the form

£ > ' + « Z ) » + £>»« + ( ^ - 2 B » J D

S

+ D»(-£ —

2u") ♦ vD + Dv. 9.

EXAMPLES OF GENERALIZED KdV and mKdV EQUATIONS

In this section we present some examples of generalized KdV and mKdV corresponding to classical Kats-Moody algebras distinct from »!(*, C[X,X"']. The generalized KdV are hereby considered as equation for «**° (see part 6.2). The form of e°™ is chosen so that the sim­ plest equation of the corresponding series has minimal possible order. Table 5 is devoted to generalized mKdV for which the type of the algebra G and the Hamil­ tonian H is shown that leads to the simplest equation of the series [the generalized mKdV corresponding to the Hamiltonian H(ui

uk) has the form -37^~"s—1—,i«], .■.,* ]•

We further indicate the Hamiltonian for mKdV corresponding to D ^ 1 ' and having the form ■ 3 P ~ I ' ( T . £]. where F is the same as in Appendix 2. H in this case is equal to —4(u,'tf4'-fU.'tt>tt4— Ut'U,U,

— I^UJUJK,).

Examples of generalized KdV are presented in Tables 6 and 7. Table 6 is devoted to equations corresponding to pairs (G, c m ) such that after removal of the vertex c m of the Dynkin scheme G decomposes into unconnected points. In this case the algebra 9 (see part 6.1) is isomorphic to the direct product of several copies of sl(2) . Therefore, for suitable choice of the "coordinate system" ui,...,u K the generalized KdV corresponding to the Hamil­ tonian H and the second Hamiltonian structure has the form

- £ - _ A,(D«+2(«1D + 0«1))-Si7. '-1

*•

<9-'>

where X|£C. We mention that the operator D 3 + 2 ( U J D + Du^) corresponds to the second Hamil­ tonian structure on M(*l(2)), and the occurrence of the factors Xi is connected with the fact that the isomorphism 6^.»1(2)X •• • X**(2) does not preserve scalar products. The simplest example of an equation admitting the form (9.1) is the KdV equations for which k - 1, X\ - 1, H ■ uf/2. In Table 6 for each pair (G, c m ) of this type the factors Xi and Hamiltonian H are presented that correspond to the simplest equation of the series. The simplest generalized KdV corresponding to algebras of types A?*, A™, B"* are pre­ sented in Table 7. We recall that each generalized KdV possesses the Lax representation ^ L — jj^c-ii^ gcM], w here £ C M — JJ- + A-fg CM

(see part 6.2).

For each equation Table 7 shows the

can

form of the matrix q . The matrix A is equal to the sum of all canonical generators e j . The explicit form of the ej is indicated in Appendix 2. Remark 1. The equations of Table 7 corresponding to the pairs (Af ) , c0) (A-?\ ct), ( A f , after linear changes of unknowns and scale transformations can be written in the Hamiltonian form (9.1) (see Table 6 ) . The remaining two equations, which are of a rather simple form, have a very complicated Hamiltonian form. Remark 2. Tables 6 and 7 contain all the simplest generalized KdV corresponding to al­ gebras G of ranks 1 and 2 except the equations corresponding to (A;2', Co) and ( A j 2 \ C i ) . In

2026

77

TABLE 6 Cm »i Cf * , • <

» JJ Cf a,.f j j C| JJ C| •W^i Vl.lt-*

TABLE 5

1*-

»f*»I-tu,«, tluf ♦ « ! -f?UiUt uf.u{-«}-»li,U t .«U,Uj«fcl ; U| !

1

'I

-T7JT

M

JfuJJi -MnJ J«j)« -Uj

sftsfcy*"'

1.-/ li«2t-l

u, »u;-(u,u,

•? 'C|I *,.!,< li"»

•r

feeaiaaafestay^

»7

UfU]-U,U|-U2U|*Uj *|f-1t t U]«-U,Uj*Us1l«-U]U«J»

'2

TABLE 7 6 A.

i

,*■

Simplest generalized KdV

•'

SX*tOfL*Ux

l

i

txxxx

T

*$U*U?

-«<«..« ♦ « I . J H "

"1.5"' ""U^"''^*"?"

S^c.

IT:. these two cases ic is possible to choose 2 C U so that the simplest equations of the series have third order. However, due to the complexity of the formulas, we note here only that both these equations have the form

v, - a.o,,, -f a,u,u„+GVI'U,+a« {uv,—vti,), where the coefficients a£ appropriate to each equation belong to Q[>ol. 10. TWO-DIMENSIONALIZED TODA LATTICES 10.1. In this subsection we recall the definition of the two-dimensionalized Toda lat­ tice corresponding to a Kats-Moody algebra, and for it we present the Zakharov—Shabat repre­ sentation found in [61]. In the next subsection, following [65, 72, 12], we consider local conservation laws and the connection of the Toda lattices with generalized mKdV. Let C be a Kats—Moody algebra with canonical generators e£, f£, h{, 0 < i < r. On G we fix a nondegenerate, invariant, symmetric, bilinear form coordinated with the canonical gradation G — 90'

(see Proposition 5.20).

We set ft — 0 ° .

We recall that the elements hj

generate 6 as a vector space, and there is exactly one linear relation (5.12) between then. For any i, 0 < i < r we denote by aj the linear functional on t>, such that [h, ej] ai(h)ei for all Ag$ (the a; are called simple roots of the algebra G). It is clear that [h, fj] - -ai(h)f j. We note that aj(b.£) - Aij , where (Aij) is the Cartan matrix. We call the two-dimensionalized Toda lattice corresponding to G the equation

£_2«"' W , -A..

+(*,T)e*.

(10.1)

This name is also sometimes applied to the system of equations J^R-exp2'V/'

0
n,(jc, x)eC.

(10.2)

We shall discuss the connection between (10.1) and (10.2). There is a mapping from the set of solutions of (1U.2) into the set of solutions of (10.1) given by the formula

2027

78

$ — 2 u'*i* ^ u e

to

r*!*1^00 (5.12) this napping is not bijective.

However, for any solution

i> or Eq. (10.1) it is not hard to find all solutions ( u 0 , . . . , u r ) of the system (10.2) such r

that y a,A,—if .

Namely, for u 0 it is possible to take any solution of the equation

,!??* —

£«•<*>, and after U Q has been chosen the remaining ui are found uniquely from the relation r 2 * * ( A i ~ ♦ !*■*• *■* n o t n a r < * t o s e * that the ui found in this manner satisfy (10.2)]. Thus, Eq. (10.1) is almost equivalent to (10.2). We further present the Lagrangian form of Eq. (10.1):

1-0 The equivalence of (10.1) and (10.3) follows from the following well-known lemma. LEMMA 10.1.

For any

y6$, g,(y)— ^ ' * ' ) .

Proof. (y, hi) - (y, [ei, fi]) - (ty, ejj, f£) - o,i(y)(ei, f i ) . Setting y - hi and using the equality ai(hi) - Aii ■ 2, we obtain (h£, h^) « 2(ei, fi)» whence the lemma fol­ lows . ■

Remark.

The equation J?!|-_grad U (+),

r U (+)—2 C|«*'W £ o r any

tf

/€C reduces to (10.3)

by means of a transformation of the form ^—^-f^, T —ax, where W&< a^C* Finally, we present the Zakharov—Shabat form of Eq. (10.1): l£.S]-0, 2 - r ^ . ^ . ^ + A - ^ + f + A . £-£ where A — 2'it A — 2 / i t i-o

and

(10.4)

+ €-■ Ae»,

t strictly speaking, e~*Ae* must be understood to be

e'^^Oi).

i-o

The equivalence of (10.1) and (10.A) follows immediately from the relations [ei, fjJ ■ ^ijnjt [h, fj] - - a i ( h ) f i . Remark 1. Equation (10.4) admits the more symmetric form f e ~ * n j - 1 * " + e*"Ae-*", t*n »

£ e-w+e-+*Ae**]

-0.

Remark 2. If we interpret A, A, and iKx, T ) as elements of a "genuine" Kats—Moody al­ gebra G (see Remark 2 following Proposition 5.8), then relation (10.4) is equivalent to the system (10.2). To conclude this subsection we clarify the origin of the term "two-dimensionalized Toda lattice." — 2 «*'""*"'

The Toda lattice is properly the system of equations ifo ■ ■ (*c

was

investigated in [33, 52, 53]).

. i£Z , where U -

From this infinite system there are two

t

ways to obtain finite systems: a) require that +i<.n " *i for some n (the periodic Toda lat­ tice); b) require that *o - i^+i ■ 0 for some n (the corresponding system of equations for ^l * n is called the nonclosed Toda lattice). The two-dimensionalization consists in 2 2 replacing 3 /3t by the operator 3 2 / 3 X 3 T . The two-dimensionalized periodic Toda lattice is essentially equivalent to Eq. (10.1) for an algebra G of type A^ 1 ', while the two-dimensional­ ized nonclosed Toda lattice is equivalent to the system (10.2) for the case where (Aij) is the Cartan matrix of an algebra of type AiiJ. The history of two-dimensionalization of the periodic Toda lattice and passage from A O ) to arbitrary Kats—Moody algebras can be traced in t30, 36, 45, 61, 64, 46, 5 5 ] . Concerning the nonclosed Toda lattice and its generaliza­ tions see [28, 29, 37, 44, 6 1 ] .

2028

79

10.2. The relation between two-dimensionalized Toda lattices and generalized mKdv cor­ responding to the same Kats-Moody algebra G is based on the fact that on operator 8 of the form (6.7) after the substitution fl-9* is converted into an operator 8 of the form (10.A). Proposition 10.2. Let Hi be the densities of conservation laws for generalized mKdV considered in Proposition 6.6. Then Hi, considered as differential polynomials in tj», are densities of conservation laws for (10.1). The proof is essentially the same as that of Proposition 1.5. We recall (see Sec. 6) that Hi has degree of homogeneity i ♦ 1 if it is assumed that degj-^ — J.

Moreover, Hi * 0 if and only if the remainder on division of i by the Coxeter

number of G is an exponent. Definition.

It is said that the equation £ " / ( * . *r.*T. *>„.««. S-TT...0

HO.5)

is a symmetry for the equation

^W. 9,. 9*. 9,,. ♦,*, ¥„.- • -)-0.

(10.6)

if the derivative with respect Co c of the left side of (10.6) computed by Eq. (10.5) vanishes on substituting for

qXJI...).

This equation for

4» by abuse of language we call, as before, a generalized mKdV. Proposition 10.3. for Eq. (10.1).

The generalized mKdV corresponding to the algebra G are symmetries

This proposition can be derived from Proposition 10.2 by means of a Hamiltonian formalism (see [72]). We present a direcC proof. Proof. where

i*63**

We recall (see part 6.4) that a generalized mKdV has the form It must be shown that if (fi, £J=0, then the derivative

generalized mKdV is equal to zero.

|i —[tp(u)*, g|,

;#[fi.fi).computed by the

For this it suffices to verify that

^ ™ = [9(u)*"( fi].

LEMMA 10.4. [fi, 9 («)]=-0. Proof. that

Let U and fio be the same as in Proposition 6 . 2 .

fi~**dy(fi0),

»(«)«■ e—'u(u).

Since [fi,fij —0 , i c follows that

easy Co deduce chat fi has che form fi— j ^ + c ( x ,

T), V (X, x)fc3 •

We s e t

£— e-Mt/(S)-

ISo.fi]—0.

We r e c a l l

From t h i s i t

is

Therefore, [fi. «] —0 » and hence

12. ?(«)!-0. ■ From the lemma i t follows Chac [9(a)*, fi)— — [9(a)", fi]. The degree of the l e f t s i d e (in the sense of the canonical gradation) i s not l e s s than —1, while che degree of che righC s i d e i s noc more than - 1 . Therefore, [9(u)*> fi|6C~ (R1. G-1). a n d i f 9(«) —]R Ai * w h e r e A* 6C-(R 2 , Gl), then

[©(a)*, fil —l-^ot. *~*A**I>

On the other hand, ^ . » _ l l *

«-»A**l.

In the course of the

proof of P r o p o s i t i o n 6.8 i t was shown that g e n e r a l i z e d mKdV as an equation for q has the form dq/dt - —3Ao/3x. Therefore, as an equation for 9 i t haa the form d$/3t - —Ao • Thus, rfj* — ^ —[9(a)*,fi], as was required to prove. ■ The conservation laws considered in Proposition 10.2 are only half of the known local conservation laws for Eq. (10.1). In order to obtain the second half it is necessary, roughly speaking, to interchange x and T . The same pertains to the symmetries considered in Proposi­ tion 10.3. 2029

80

APPENDIX 1 CLASSICAL SIMPLE LIE ALGEBRAS In this appendix for each classical simple Lie algebra 9 we list the system of Weyl generators Xi, Y^, H j t 1 < i < n (the numbering of the generators corresponds to the number­ ing of the vertices in Table 1 ) . In all cases the realization and the system of Ueyl genera­ tors are chosen that the Borel (Cartan) subalgebra of 9 is equal, respectively, to the inter­ section of 6 with the set of all upper triangular (diagonal) matrices. We recall that X T denotes the matrix obtained from X by transposition relative to the secondary diagonal. Type Aj,, n > 1 6 —»l(n + l) —{^eMat(n-(-l, C)|tr A— 0} . System of Weyl generators: X, — e,+,.i,

K,— «,,,+,, H,—

Type B„, n > 1 -

0*1

-*f

e - » ( & J - l ) - H A € M a t ( 2 r i + l , C ) | i * — — - S i T S - ' , S—dlag(l. - 1 - 1 . 1). System of Weyl gener­ a t o r s : XJ^-^J+I.J -\-e2n+?-i,if>-\-i, ' Ki — tfj.j+i + tfaTH-/.«*+*-/. # i ™ —«I.I -r-tf.+i.i+i—«2a+i-i.iM-i-.+ tfta+s-i.to+i-d * —1.2 « —1; Xa"=«*+,.n-t-«*+2.rt+i. J'.— 2(tf«,(,+i + e„+\,H+?), //„—2(«?.+».«+t—««.»)• Type C n , n > 1 « — »»<2/t) — {>4eMal(2rt, C ) [ * — — S ^ ^ - ' , 5 — dlig(l, — 1 1. — 1) . System of Weyl gen­ e r a t o r s : X,—et+i.t -f«2«+]-/.a«-i. ^j^*i.i+i -¥^u-t.u+i—t* W I ™ - * ( , I + 3 ©-•(2n)-{AeMat(2fl, C ) | J 4 - - 5 A r 5 - ' , S-diag(l, - 1 <-l)"-\ (-If 1 , (-1)* !)• System of Weyl generators: A", — «i+i.i +tfta+i-j.*i-i»K(«*i?,,.-4-i + £3«-..ai+i-i. //j — — *i,j-f*.+u+i — eU-l.7*-l + €u+\-t,7*+1-t-

' —1

1 — 1 ; Xft^-g-fe.+i.a-i + C«+2.a),

K* — 2(e«-l.«-H + '*.■+»).

^«

—

-e,_i,,-i-f M -fe l+ i,, + i + «,+!,,+a. For n - 2 all formulas remain in force, but the Dynkin scheme is not connected, so that the algebra 0(4) is semisimple rather than simple. Since the Dyn­ kin scheme of e(4) consists of two vertices not connected by edges it follows that «(4)=*t(2)X .1(2). APPENDIX 2 CLASSICAL KATS-M00DY ALGEBRAS In this appendix for each classical Kats—Moody algebra we present 1) a realization of G in the form L(£, C) , where C is the Coxeter automorphism; 2) a basis of the vector space 3d**—{*6G| [A, x] — 0 } and the eigenvalues of the matrix A in the realization; 3) the standard realization of G. We shall consider part 3) in more detail. Generally speaking, for each vertex c m of the Dynkin scheme of G we ought to present the corresponding standard automorphism oM:C-*-V . In constructing the standard realization corresponding to c m we actually replace M by an algebra %m of the form Rm%Rm\ where R Q is a permutation matrix, so that o m is an automorphism of %m rather than M. R Q is chosen so that the elements of the semisimple algebra 6 — { J C 6 *« I<*m(■*)—x} have block-diagonal form. Further, we do not consider all vertices of the Dyn­ kin scheme (for our purposes it suffices that each vertex be carried by some automorphism of the Dynkin scheme into one of the vertices considered). For all vertices c m considered we indicate %mt om, C , and also the canonical generators e£f.(Mm, oM)> Due to the special choice of the matrices R^ the algebras t) and I (see part 6.1) in all cases are equal, respectively, to G f l D U g and 6fi1. where t is the set of upper triangular matrices. In this appendix ©(*) and *t(2n) are the same as in Appendix 1. The number of the canonical generators of the Kats—Moody algebras corresponds to the number of the vertices of the Dynkin schemes in Table 2.

2030

81

Type A n l } ,

n > I

Mr

«-,l(n+l),

C(X)-5XS-',

*i

•<•), where « - « T ,

S-diag(l, a

A-»+1.

System of canonical generators: fg —Ci.a+iCt/a—'a+i.iC'1! A, — < n — «•+■.«+■: «, —«I+I.IC, / , — e,.1+iC"', *,— «/+!.n-i —«/./. i — I «. Eigenvalues of A equal to Cm1, i - 0, 1 n. Basis in 3 i s formed by A', where <€Z, i i s not d i v i s i b l e by n + 1. Standard r e a l i z a t i o n corresponding to « i , n * i A . *i " e i * l , i . f o r i " 1 , - . . , n .

c 0 :Vo—S, at(X) —X.

In t h i s r e a l i z a t i o n £o ■

Type A2(n' , n > 1 «-,c(2*+l), C(J0--5X r 5-'. S-dlag(l. -a, sf

-«*-', u»), u - « T , A-4n + 2.

System of canonical generators: e„—e,.j,+1C, / 0 —•JJ.+LIC*', *i-«i.i-«i.+i,i+i: «r(«iti.i + «t>+2-l. Ia+1-lKi / , — (
Eigenvalues of A equal to C w 2 1 , i ■ 0, t 21+1 is not divisible by 2n + 1. to c0:%

Standard realization corresponding -1

— 1 , t).

et—ci+i,t

2n.

Basis in

— %, o„(X) — —QXTQ-',

Order of Oo equal to two, « —»(2n-fl).

-\-ci*+7—t. %M-I—!• '■—I

formed by A 2 1 + 1 , where

3

where Q - d i a g O ,

In this realization i,—e,.

1.+1X ;

«.

Standard r e a l i z a t i o n corresponding to cm, m — \ a: «„ —«, o . (A^ — —QXTQ~', where • s «- s "+ 1 -'■

In t h i s r e a l i z a t i o n

y -" 1

■'"" ,

/n — I;

«o"-«s«-«+>. J»-«+I - . «y —

« . — («i. i«+i - f e»,-2«+}. i«-2»+i) ^

<«♦/-

n—m.

Type A(2n>_1( n > 2 «-.l(2s),C(X)

«*■-« , - a * - ' ) . u-*" 5 ". A - 4 * - 2 .

SJC\S-'. 5 - d l a g ( l . - « . »»

System of canonical generators: e 0 —^(e,. J»_I + «J. u)(, / O " * 2 ( « J _ I . 1 + **.. i)C"', Aj—eLi + ^ j — tfb-l.2t-l—*ta.

to!

/ i " " ( ' i . ( + 1 + ' l t - 1 , Li-fl-.)*."1.

' l ™ ' ('/+!.( + «lt+I-(. l i - i ) C ,

The eigenvalues of A are equal to by A"" — ~nU(\"-')E,

i&.

" J ™ — £ ( . ( - r * l + l , (+1 —

* — 1 ; <« — «.+!.»C, / « — «/■.,+ii'1. A,— —e».« +««+I.»+I-


i s formed

(we note that t r A ^ V O

only

if 21 + 1 i s d i v i s i b l e by 2n — 1 ) . The standard r e a l i z a t i o n corresponding to —1

1, —1).

The

order of «o i s

e

( i. 2M-I -)-£?. it)X; et~€t+\,

e0:Mim^%t a0(X) w^—QXTQ"1, where Q ■ d i a g ( 1 ,

equal to two,

t + 'a«+i-(. 2«-(. /—1

0_t,(2n).

In t h i s r e a l i z a t i o n

e0—-7

ft — I; £»"**a+i. it-

Standard r e a l i z a t i o n corresponding to c„:K, — *, a„(JT) —»—Q^rQ-1, where Q«di*g(l, ( —1)«*>, (— l)"'1, (—1>"'1

—1. 1).

The order of a n i s equal to two,

i z a t i o n «j — ■ J ( « « + I . ^ I + «»+J.«)'I
l)eil(2w»). The

order

' —1

cm, m — 2, 3

is equal

6 — o(2n).

—1

In t h i s r e a l ­

n— 1; «, —
- ' . 06iI(2*-2/n), B-dlag(l. of o n

x

-1

<„ — %,

oa(X) — — QX'Q''.

(-1)-". (-I)-'.

to four, « - ! ( o ° ) . a€»»(2n—2m), »€.(2fit)}.

In

*The fact that •*>o(2>i — im+l)x«p(2m) follows from the general theory of Kats-Moody algebras (we recall that the Dynkin scheme of A is obtained from the Dynkin scheme of G by removing the vertex c m and the edges continuous to it). 2031

82

this

realization

•

m—1; « „ — ( « i . s » + « j . - i i i + i , ii_j«)X: *«,„/ — «/+i. / + « > » - i « + i - / . J » - J « I - ; .

c

£0— -K (tf&,-«,+l. U-m-l + ein-m+l. im-m)'* C) —■ Clit-m+l-J. 2it-M-J + £l«-*+l+7.24i-«-W. / * "

./ — 1

»—at—1:

*« —

n-m+i,n-m' Type B n 1 } , n > 2 4.1

*■'

«-=o(2n-j-l). C (JO-SATS"1. S = d i a g ( l . o, u*

o*- 1 , 1), a-*" 1 ", h-2n.

System of canonical generators: eo—-j (tfi. 2J> + « : . 2J+I)C./o—=2(tfi,, i + tf^+i. 2)C"1, Ao«—+1-I.2.+ 1 - ; + « 2 « + 2 - / . 2.+2-I.1— 1

rt—

1; C = (««+l.« + e«+2. « + l) C

/ « — 2(*«, iwl + «»+l. «+2> »"'.

*» —

2(e.+j, »+2—e...)-

The eigenvalues of A are equ al to 0, i, at

w5"-1;. A basis in 3 is formed by A"*', i€Z,

where for 1<0 A^'-C^A""*", * > 0 . Standard r e a l i z a t i o n corresponding t o £(,:«<> —M, ization

o0{X) — X,

<0 — f (.; «, — «,+,. ,-)-e2» + 2-/.»! + i-i,

« —1

8 — o (2n + 1). In t h i s r e a l ­ «•

n: « „ — {Xei<(2n+l)|X — — RX1 Rr%

Standard r e a l i z a t i o n corresponding t o c.,, m = 2

where / ? _ ( ° ° ) , a - d l a g ( l , - l - 1 , l)e«l(2n-2m + 1), p_diag(-l,l (-1)". (-1)". (—1)"*' —l)«l(2m); om(X) — QXQ-', where Q — dlag( — 1, —1 — 1 . 1,1 I). The order of 2
o m i s equal to two, 8 —((„ ?), a€»(2fi — 2/n-f 1), &e»(2m)l.

In t h i s r e a l i z a t i o n

«2i-«+S.2»-»+l); « ; — <2i-.»+2+;,2»-«+l + / + <2»-«+2-/.2»-«+l-y, CU-2m+2.U-lM+l)k\ em.i-*ej+1./-TCU-2m+2-J.7i-7m+l-j>

j-=\,...,m—

7 - = U •••■

1;

« 0 —■j(«2«-»+2.x < -»+ e » — (<1.2*+l +

fl—m.

Type C < J ) , n > I d.-f

^

B-«»(2n), C(X) = SXS', 5 - d i a g ( l , u, »-' (a5""1). <■> — « T , A —2n. System of canonical generators: «0 — «i.j,t, / 0 — ej»,iC"', A„ — e,,, — «2«,2«; «2t+l-/.2n-/)C,

*i — («I+I.< +

/ ( — (£*.l-t-l +«2Jf-i.2*+1-j)C"1, A , ~ — * l . l + £t + l,/+!—tfld-(.2«-( + *2jt+l-J.2n+l-it f «« 1

ft— \',

1

««—««+i.ni, /.■"<».«+iC" . A, ——e„..+ e„+,,»+i. The eigenvalues of A are equal to Cw1, i ■ 0 , 1 , . . . , 2 n — 1. A b a s i s in 3 i s formed by A**1. i&. Standard r e a l i z a t i o n corresponding to c0:%<^^it oQ(X)~X, «o"=*i.2«'-'i «. — «i+i.i+e««+i-i.2«-i,

i—I

6 —*»(2rt) .

In t h i s

realization

n —1; «. — «,+, „. del

Standard realization corresponding to cm. m — 1 n—1: Km->{X£t>(2n)\X——RXR~'}, where /?-*(£*). o-diag(l, - 1 1, - l)68l(2B-2m), p-diag(l, - 1 , .. .rr. — l)£|I(2m); o„(*)QXQ"1, where Q —dlag(—1, — 1

—1.1, 1

1). The order of a m is equal to two, « — ((<>>)•

Vi-2m

a€»>(2/i —2m), »e»»(2m)). In t h i s r e a l i z a t i o n ■i-1 m —1; e „ — («!.!. +«2.-2«+i.2«-2m)>-;

e „ - e2»-«+i.2»_i»; *, — «2— «+i+;.ai-«+/ + «!.-m+i-;.2,-«-;, e».; "= 'in.i +
C* ~ £
Type D ^ 1 ) , n > 3 d.i « - ° ( 2 n ) , C(X) = 5 ^ 5 " , S — d l a g ( l , u

w«-', w»-', u"

SI w2"-3, 1), a—e* , A - 2 n — 2 .

System of canonical generators: «0— -^(«i.2»-i + <2.2«){. / o — 2(e»,_i,i + ej,.j)t"1, «2n-i;j«-l — *2«.2V»

*J ■"('(+!./ - i - <2»+l-l.2«-l)C,

e i . - i . 3 a - i - f eiM-i-i.to+i-',

2032

' —1

/(""('l.i+1 +

ftfl-l.aw+l-JC"1,

n — 1 ; «. —-jfe.+i.n-i + e.+2.«)C,

A„—e,,,-|-<j,—

A , ^ — * ( . ( + *(+!./+! —

^"« —= 2<«„_,.„_n -1- e;-", A.—

83

The eigenvalues of A are equal Co 0 (of multiplicity two), C, taC. 3

is formed by A21*', i£Zt

where for i < 0, A3**' — C"**AJI*l*w, A » 0

Standard realization corresponding to CQ-,%^ -= K, o0(X)=*X,

Standard realization corre sponding to c„, m-=2,3

GI*'*"^ . A basis in

and ;*-i**/?, igl.

6 —o(2/i).

In this realization

« - 2 : * „ - LYe»l(2/t)| A ' - — /?Aftf-'},

where /? — (f j). a — d1ag(l, — 1 , (—1)«-«-', ( —1)«—> l)6ll(&t — 2m), B — diag( — 1 , 1 (-I)".)-!)" -l)€«l(2m); am[X)-QXQ-}, where Q-dltg(-l, - 1 - 1 , 1, 1 1). order of o m is equal to two, 6 — I" X a€°(2» — 2m), *6«(2/n)| .

The x

In this realization e 0 — y

(«!«->i+i.ii->.-i + «n-«+j.!n-«i) i ef — eii-»,+i+/.2»-«,+/ + ei,-»!+i-/.i,_„_;, «;»-2*+i.2«-i«)A;

Here

y » I , . . . . m—1;

«„ -=(ei.i,+

em.) — e;-+i,/ + ej«-2«+i-/.2«-j«-;)' . / « - ! , . . . , / i — m — 1; *» —-5 («,_„+,.„-*-[ + «„_«+j.»-«). Type D ^ J , n > 2

l

*-{-*'e« (2«+2)|A' e,+J.»+i,

C(^0 —5^5"',

dii

T

RX R->),

fl-dlag(t,

S — dlag(u, a1

System of canonical generators: eau-i.ii+j);
-1

(-I)'",

0, 0, (-1)"* 1

o>», — 1, 1, w**,
l) + ( - l ) » « . + , . » + 2 <■> — e T _

A —2« + 2. ,

e0 —(«I..+J + «.+MI+2)C, / 0 — 2(«„+^l^-«!,+^.„+:)C" , A„ —2(«,.,—

/ I ~ ( « I . I + > + «J«+S-I.3«+J—iK"'i


w2"*'),

*i = —«i. 1 + « I + I . 1+1 —

«— I; «„-=(e»+i.«+«»+3.«+i)5, /» = 2(«»,.+,+ e, + ,.„ +J ){-', A„—2

(—«..«+*«+).«+J)-

The eigenvalues of A are equal to Cw 1 , i * 0,...,2n ♦ 1. A basis in 3 is formed by

A2"', ieZ. Standard realization corresponding to c„, m—0,1 where /?-(Jol- <» —dlag(l. — 1 where Q — dlag(— 1,— 1

2m+l). *e»(2m+l)}.

n: %m — [XT:tt(2n + 2)\X— —RXTR~'} ,

l)€H(2«-2m+ 1), B-dlag(-l, 1

—I, 1, 1

-l)e«l(2nt+l); o«(J0-QA-Q-'.

1) . The order of om i s equal t o two, ®""{(o »)• fl€'(2n —

In t h i s r e a l i z a t i o n «^ = «!,_„+!_/.j»_„ + ,_ ; +eu-m+*+i.n-m+i+j, j — 0. 1

C«=(£l.2*+J + *'4t-2«i+?.2»-2«i+l)*;

m— 1;

**+y "== * > + ! . ; + *li-2«+2-/.l«-2m+l-/. y — 1 , . . . . It— "1. LITERATURE CITED

1. 2. 3. 4. 5. 6. 7. 8. 9. ■0.

V. I. Arnol'd, Mathematical Methods of C l a s s i c a l Mechanics [ i n Russian], Nauka, Moscow (1979). N. Bourbaki, Lie Groups and Algebras [Russian t r a n s l a t i o n ] , Mir, Moscow (1976). N. Bourbaki, Lie Groups and Algebras. Lie Algebras. Free Lie Algebras [Russian trans­ l a t i o n ] , Mir, Moscow ( 1 9 / 2 ) . N. Bourbaki, Lie Groups and Algebras. Cartan Subalgebras, Regular Elements, Decomposable Semisimple Lie Algebras [Russian t r a n s l a t i o n ] , Mir, Moscow (1973). I. M. Gel'fand and L. A. D i k i i , "Fractional powers of operators and Hamiltonian systems," Funkts. Anal. P r i l o z h e n . , 10, No. 4 , 13-29 (1976). I. M. Gel'fand and L. A. Dlk~ii, "The resolvent and Hamiltonian systems," Funkts. Anal. P r i l o z h e n . , _H_. No. 2 , 11-27 (1977). I . M. Gel'fand and L. A. D i k i i , "A family of Hamiltonian structures connected with i n t e grable nonlinear d i f f e r e n t i a l equations," Preprint Mo. 136, IPM AN SSSR, Moscow (1978). I . M. Gel'fand, L. A. D i k i i , and I . Ya. Dorfman, "Hamiltonian operators and algebraic structures connected with them," Funkts. Anal. P r i l o z h e n . , J3_, No. 4 , 13—30 (1979). I. M. Gel'fand and L. A. D i k i i , "The Schoutten bracket and Hamiltonian operators," Funkts Anal. P r i l o z h e n . , J^, No. 3 , 71-74 (1980). I . M. Gel'fand and L. A. D i k i i , "Hamiltonian operators and i n f i n i t e - d i m e n s i o n a l Lie a l ­ gebras," Funkts. Anal. P r i l o z h e n . , 15, No. 3 , 23-40 (1981).

2033

84

11. 12. 13. 16. 15. 16. 17.

18. 19.

20.

21. 22. 23. 24. 25. 26. 27.

28. 29.

30. 31. 32. 33. 34. 35. 36. 37.

38.

I. M. Gel'fand and L. A. Dikii, Hamiltonian operators and the classical Yang—Baxter equation," Funkts. Anal. Prilozhen., J£, No. 4, 1-9 (1982). V. G. Drinfel'd and V. V. Sokolov, "Equations of Korteweg—de Vries type and simple Lie algebras," Dokl. Akad. Nauk SSSR, 258, No. 1, 11-16 (1981). A. V. Zhiber and A. B. Shabat, "Klein-Gordon equations with a nontrivial group," Dokl. Akad. Nauk SSSR, 247, No. 5, 1103-1107 (1979). V. E. Zakharov, "On the problem of stochastization of one-dimensional chains of non­ linear oscillators," Zh. Eksp. Teor. Fiz., 65_, No. 1, 219-225 (1973). V. E. Zakharov and S. V. Manakov, "On the theory of resonance interaction of wave packets in nonlinear media," Zh. Eksp. Teor. Fiz., 69_, No. 5, 1654-1673 (1975). V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Fitaevskii, Theory of Solitons: the Method of the Inverse Problem [in Russian], Nauka, Moscow (1980). V. E. Zakharov and A. V. Mikhailov, "Relativistically invariant two-dimensional models of a field theory integrable by the method of the inverse problem," Zh. Eksp. Teor. Fir., 74., No. 6, 1953-1973 (1978). V. E. Zakharov and L. A. Takhtadzhyan, "Equivalence of the nonlinear SchrCJdinger equa­ tion and Heisenberg's ferromagnetic equation," Teor. Mat. Fiz., 3_8, No. 1, 26-35 (1979). V. E. Zakharov and A. B. Shabat, "A scheme of integrating nonlinear equations of mathe­ matical physics by the method of the inverse scattering problem. I," Funkts. Anal. Prilozhen., 8_, No. 3, 54-56 (1974). V. E. Zakharov and A. B. Shabat, "Integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem. II," Funkts. Anal. Prilozhen., J 2 . No. 3, 13-22 (1979). V. G. Kats, "Simple irreducible graded Lie algebras of finite growth," Izv. Akad. Nauk SSSR, Ser. Mat.. 22., No. 6, 1323-1367 (1968). V. G. Kats, "Infinite Lie algebras and the Dedekind n-function," Funkts. Anal. Prilozhen., 8, No. 2, 77-78 (1974). V. G. Kats, "Automorphisms of finite order of semisimple Lie algebras," Funkts. Anal. Prilozhen., 2 . No. 3, 94-96 (1969). E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill (1955). V. G. Konopel'chenko, "The Hamiltonian structure of integrable equations under reduction," Preprint Inst. Yad. Fiz. Sib. Otd. Akad. Nauk SSSR No. 80-223, Novosibirsk (1981). I. M. Krichever, "Integration of nonlinear equations by methods of algebraic geometry," Funkts. Anal. Prilozhen., U_, No. 1, 15-31 (1977). D. R. Lebedev and Yu. I. Manin, "The Hamiltonian operator of Gel'fand—Dikii and the coadjoint representation of the Volterra group," Funkts. Anal. Prilozhen., 13, No. 4, 4046 (1979). A. N. Leznov, "On complete integrability of a nonlinear system of partial differential equations in two-dimensional space," Teor. Mat. Fiz., 42^, No. 3, 343-349 (1980). A. N. Leznov and M. V. Savel'ev, "Exact cylindrically symmetric solutions of the clas­ sical equations of gauge theories for arbitrary compact Lie groups," Fiz. El. Chastits At. Yad., J2> No. 1, 40-91 (1980). A. N. Leznov, M. V. Savel'ev, and V. G. Smirnov, "Theory of group representations and integration of nonlinear dynamical systems," Teor. Mat. Fiz., 48, No. 1, 3-12 (1981). I. G. Macdonald, "Affine root systems and the Dedekind n-function," Matematika, 16, No. 4, 3-49 (1972). S. V. Manakov, "An example of a completely integrable nonlinear wave field with nontrivial dynamics (the Lie model)," Teor. Mat. Fiz., 28_, No. 2, 172-179 (1976). S. V. Manakov, "On complete integrability and stochastization in discrete dynamical systems," Zh. Eksp. Teor. Fiz., 67_, 543-555 (1974). Yu. I. Manin, "Algebraic aspects of nonlinear differential equations," in: Sov. Probl. Mat. Tom 11 (Itogi Nauki i Tekhniki VINITI AN SSSR), Moscow (1978), pp. 5-112. Yu. I. Manin, "Matrix solitons and bundles over curves with singularities," Funkts. Anal. Prilozhen., U_, No. 4, 53-67 (1978). A. V. Mikhailov, "On the integrability of a two-dimensional generalization of the Toda lattice," Pis'ma Zh. Eksp. Teor. Fiz., 3_0, No. 7, 443-448 (1979). A. G. Reiman, "Integrable Hamiltonian systems connected with graded Lie algebras," in: Differents. Geometriya, Gruppy Li i Mekhanika. Ill, Zap. Nauchn. Sem., LOMI, Vol. 95, Nauka, Leningrad (1980), pp. 3-54. A. G. Reiman and M. A. Semenov-Tyan-Shanskii, "Algebras of flows and nonlinear partial differential equations," Dokl. Akad. Nauk SSSR, 251, No. 6, 1310-1314 (1980).

85 39.

A. G. Reiman and M. A. Semenov-Tyan-Shanskii, "A family of Hamiltonian s t r u c t u r e s , a hierarchy of Hamiltonians, and reduction for matrix d i f f e r e n t i a l operators of f i r s t order," Funkts. Anal. P r i l o z h e n . , Ut_, No. 2, 77-78 (1980). 40. J . - P . Serre, Lie Algebras and Lie Groups, W. A. Benjamin (1965). 41. V. V. Sokolov, "Quasisoliton s o l u t i o n s of Lax equations," D i s s e r t a t i o n , Sverdlovsk (1981). 42. V. V. Sokolov and A. B. Shabat, "(L—A) pairs and s u b s t i t u t i o n of R i c a t t i type," Funkts. Anal. P r i l o z h e n . , J £ , No. 2 , 79-80 (1980). 43. I. V. Cherednik, "Differential equations for Baker—Akhiezer functions of algebraic curves," Funkts. Anal. P r i l o z h e n . , _U, No. 3, 45-54 (1978). 44. M. Adler, "On a trace functional for formal pseudodifferential operators and the symp l e c t i c structure of Korteweg-de Vries type equations," Inventiones Math., 5£, 219-248 (1979). 45. 0. I. Bogojavlensky, "On perturbations of the periodic Toda l a t t i c e , " Comaun. Math. Phys., 5J_, 201-209 (1976). 46. S. A. Bulgadaev, "Two-dimensional integrable f i e l d theories connected with simple Lie algebras," Phys. L e t t . , 96B, 151-153 (1980). 47. E. Date, M. Jimbo, M. Kashivara, and T. Miwa, "Transformation groups for s o l i t o n equa­ t i o n s , " Preprint RIMS-394, Kyoto Univ. (1982). 48. E. Date, M. Kashivara, and T. Miwa, "Vertex operators and T-functions. Transformation groups for s o l i t o n equations. I I , " Proc. Jpn. Acad., A57, No. 8 , 387-392 (1981). 49. R. K. Dodd and J. D. Gibbons, "The prolongation structure of higher-order Korteweg-de Vries equation," Proc. R. Soc. London, 358, Ser. A, 287-296 (1977). 50. B. L. Feigin and A. V. Zelevinsky, "Representations of contragradient Lie algebras and the Kac-Macdonald i d e n t i t i e s , " Proc. of the Summer School of Bolyai Math. Soc. Akademiai Kiado, Budapest (to appear). 51. H. Flashcka, "Construction of conservation laws for Lax equations. Comments on a paper of G. Wilson," Q. J. Math., Oxford (to appear). 52. H. Flaschka, "Toda l a t t i c e I , " Phys. Rev., B9. 1924-1925 (1974). 53. H. Flaschka, "Toda l a t t i c e I I , " Progr. Theor. Phys., 5\_, 703-716 (1974). 54. A. P. Fordy and J. D. Gibbons, "Factorization of operators. I . Miura transformations," J. Math. P h y s . , 2±, 2508-2510 (1980). 55. A. P. Fordy and J. D. Gibbons, "Integrable nonlinear Klein-Gordon equations and Toda l a t t i c e , " Coramun. Math. Phys., JT_. 21-30 (1980). 56. I. B. Frenkel and V. G. Kac, "Basic representation of affine Lie algebras and dual r e s o ­ nance models," Invent. Math., 62_, 23-66 (1980). 57. V. G. Kac, "Infinite-dimensional algebras, Dedekind's n-functions, c l a s s i c a l MSbius function and the very strange formula," Adv. Math., 2p_» *>• 2> 85-136 (1978). 58. B. Konstant, "The p r i n c i p l e three-dimensional subgroup and the Bettin numbers of a complex simple Lie group," Am. J. Math., _131_, 973-1032 (1959). 59. B. A. Kupershmidt and G. Wilson, "Conservation laws and symmetries of generalized s i n e Gordon e q u a t i o n s , " Commun. Math. Phys., 8 1 , No. 2 , 189-202 (1981). 60. B. A. Kupershmidt and G. Wilson, "Modifying Lax equations and the second Hamiltonian s t r u c t u r e , " Invent. Math., 6 2 , 403-436 (1981). 61. A. N. Leznov and M. V. SaveTTev, "Representation of zero curvature for the system of nonlinear p a r t i a l d i f f e r e n t i a l equations XQ* Z Z * exp (KX)a and i t s i n t e g r a b i l i t y , " L e t t . Math. P h y s . , 2 . *>• * . 489-494 (1979). 62. I. G. Macdonald, G. Segal, and G. Wilson, Kac-Moody Lie Algebras, Oxford Univ. Press (to appear). 63. F. Magri, "A simple model of the integrable Hamiltonian equation," J. Math. Phys., 19, No. 5, 1156-1162 (1978). 64. A. V. Mikhailov, "The reduction problem and the inverse s c a t t e r i n g method," i n : Proceed­ ings of Soviet—American Symposium on S o l i t o n Theory (Kiev, September 1979), Physics, 2> Nos. 1-2, 73-117 (1981). 65. A. V. Mikhailov, M. A. Olshanetsky, and A. M. Perelomov, "Two-dimensional generalized Toda l a t t i c e , " Commun. Math. P h y s . , .79, 473-488 (1981). 66. R. V. Moody, "Macdonald i d e n t i t i e s and Euclidean Lie a l g e b r a s , " Proc. Am. Math. S o c , 4 8 , No. 1, 43-52 (1975). 67. D. Muaford, "An algebrogeometric construction of commuting operators and of s o l u t i o n of the Toda l a t t i c e equation," Korteweg-de Vries Equation and Related Equations. Proceed­ ings of the International Symposium on Algebraic Geometry, Kyoto (1977), pp. 115-153. 68. J . L. Verdier, "Les representations des algebres de Lie a f f i n e s : applicationa a quelques problemes de physique," Scainaire N. Bourbaki, Vol. 1981-1982, j u i n , expose No. 596 (1982).

203J

86

69. 70. 71. 72.

J. L. Verdier, "Equations differentielles algebriques," Sem. Bourbaki 1977-1978, expose 512, Lect. Notes Math., No. 710, Springer-Verlag, Berlin (1979), pp. 101-122. G. Wilson, "Commuting flows and conservation laws for Lax equations," Math. Proc. Cambr. Philos. S o c , 86_, No. 1, 1 3 1 - H 3 (1979). G. Wilson, "On two constructions of conservation laws for Lax equations," Q. J. Math. Oxford, 32_, No. 128, 491-512 (1981). G. Wilson, "The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras," Ergodic Theory and Dynamical Systems, ^, 361-380 (1981).

HIGHER REGULATORS AND VALUES OF L-FUNCTIONS A. A. Beilinson

UDC 512.7

In the work conjectures are formulated regarding the value of L-functions of mo­ tives and some computations are presented corroborating them.

INTRODUCTION Let X be a complex algebraic manifold, and let Kfi,X), HJQ(X,Q) be its algebraic K-groups and singulary cohomology, respectively. We consider the Chern character ch: Kj(X)®Q-*-® J Hj0~ (X,Q). It is easy to see that there are the Hodge conditions on the image of ch: we have ch (Kt(X))cz Q^tfjf'iX, Q)) f) (PZ/jf"' (X, C)), where W., F* are the filtration giving the mixed Hodge structure on H'$(X). For example, if X is compact, then ch (Kj(X)) - 0 for j > 0. It turns out that the Hodge conditions can be used, and, untangling them, it is pos­ sible to obtain finer analytic invariants of the elements of K.(X) than the usual cohomology classes. For the case of Chow groups they are well known: they are the Abel—Jacobi-Criffiths periods of an algebraic cycle. Apparently, these invariants are closely related to the values of L-functions; we formulate conjectures and some computations corroborating them. In Sec. 1 our main tool appears: the groups Hg>(x,Z{i)) of "topological cycles lying in the i-th term of the Hodge filtration." These groups are written in a long exact sequence

~"+**iil(X. C)-.//^(X, Z(i))

* H'm{X. Z)H!s(X, O-....

On H'g) we construct a U -product such that *,# becomes a ring morphism, and we show that fig, form a cohomology theory satisfying Poincare duality. Therefore, it is possible to apply the machinery of characteristic classes to Hg> [22] and obtain a morphism chg>:Kj{X)Q Q -*®//J>~y(X, Q(i)). The corresponding constructions are recalled in Sec. 2. Let H*£l{Xt Q (t))dX'y(.Y)8Q be the eigenspace of weight i relative to the Adams operator [2]; then ch^ defines a regulator - a morphism r^://^(X, Q(*))-► #£>(*. Q(0). lit is thought that for any schemes there exists a universal cohomology theory Ji'^^X, Z(i))t satisfying Poincare duality and related to Quillen's K-theory in the 6ame way as in topology the singular cohomology is related to K-theory; H'^ must be closely connected with the Milnor ring.] In the appendix we study the connection between deformations of ch^, and Lie algebra cohomologies; as a consequence we see that if X is a point, then our regulators coincide with Borel regulators. There we present a formulation of a remarkable theory of Tsygan-feigin regarding stable co­ homologies of algebras of flows. Finally, Sec. 3 contains formulations of the basic conjec­ tures connecting regulators with the values of L-functions at integral points distinct from the middle of the critical strip; the arithmetic intersection index defined in part 2.5 is responsible for the behavior in the middle of the critical strip. From these conjectures (more precisely, from the part of them that can be applied to any complex manifold) there follow rather unexpected assertions regarding the connection of Hodge structures with alge­ braic cycles. The remainder of the work contains computations corroborating the conjectures in Sec. 3. Thus, in Sec. 7 we prove these conjectures for the case of Dirichlet series; Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki Dostizheniya), Vol. 24, pp. 181-238, 1984.

2036

0090-4104/85/3002-2036$09.30

C 1985 Plenum Publishing Corporation

(Noveishie

89

Reprinted with permission from Physics Letters B Vol. 206, No. 3, pp. 412-420,26 May 1988 © 1988 Hsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

SYSTEMATIC APPROACH TO CONFORMAL SYSTEMS WITH EXTENDED VIRASORO SYMMETRIES Adel BILAL and Jean-Loup GERVAIS Laboratoire de Physique Thtorique de VEcole Normale Suptrieure \ 24 rue Lhomond, F- 75231 Paris Cedex 05. France Received 29 February 1988

The Toda field theories, which exist for every simple Lie group, are shown to give realizations of extended Virasoro algebras that involve generators of spins higher than or equal to two. They are uniquely determined from the canonical lagrangian formal­ ism. The quantization of the Toda field theories gives a systematic treatment of generalized conformal bosonic models. The wellknown pattern of conformal field theories with non-extended Virasoro algebra, appears to be repeated for any simple group, leading to a "periodic table", parallel to the mathematical classification of simple Lie groups.

1. Introduction It is important to organize the growing number of discrete series of conformal models, and to devise methods that allow to derive all such models systematically. The basic point in this connection is to extend the Virasoro algebra in all possible ways. In the present paper, we develop a general approach to this problem that, for any givenflnite-dimensionalLie group, leads to an extended Virasoro algebra, together with the associated families of conformal field theories. Details will be left for future publications and we shall, for the time being, only consider bosonic theories. Consider, as a warming up, the conformal models with unextended Virasoro symmetry. They are completely consistent only if the central charge C takes the values C=l-6/r(r+l),

rinteger>2,

C=7,13, 19, C=l+6(r+l)7/-,

(1.1) (1.2)

rinteger>0.

(1.3)

The first set is the celebrated series of Friedan, Qiu, and Shenker (FQS) [ 1 ]. The other two are not so wellknown at the present time. They were obtained by Gervais and Neveu (GN) [2,3], from their study of the Liouville field theory. These three sets lie in the ranges C< 1, 1 < C< 25, C> 25, respectively. The corresponding models are closely related since their spectrum of highest weights are given by the general formula [4] *' t(m,n) = ^ ( l - C ) [ m + n + ( m - f i ) V ( C - 2 5 ) / ( C - l ) ] 2 + 5 ! , ( C - l ) ,

(1.4)

where m and n are integers which vary over an appropriate range that depends on C. Although it looks superfi­ cially different, this last formula does give back the correct highest weights of the FQS series [ 1 ] upon substi­ tuting eq. (1.1) since the square root becomes rational. In general, this square root makes it clear why there are three different regions for C separated by the special values C0=l,

and C , = 2 5 .

(1.5)

' Laboratoire Propre du Centre National de la Recherche Scientifique associe 41'Ecole Normale Superieure et I l'Universit6 Paris-Sud. *' See ref. [ 5 ] for a review.

412

90

Volume 206. number 3

PHYSICS LETTERS B

26 May 1988

Remarkably the series (1.1) and (1.3) are both particular cases of the general formula of Belavin, Polyakov and Zamolodchikov [6]. C=l+6(r+s)2/rs,

r,s integers.

The FQS series corresponds to r+s=(1.6) is that it leads to (C-25)/(C-\)

(1.6) 1, while the GN series (1.3) is given by 5=1, r>0. The point of eq.

= [(r-s)/(r+s)]2,

(1.7)

and formula (1.4) gives rational and real values for all m and n. The region 1 1 any representation of the Virasoro algebra with positive highest weights is unitary, contrary to what happens for C< 1. As is well known this is due to the fact that the null states occur for positive (respectively negative) highest weights if C< 1 (respectively C> 1). These highest weights are given by formula (1.4) if one chooses m and n both positive. For C< 1, the spectrum of highest weights is given by a set of null states and the Green functions satisfy the associated differential equation. They are coulombic with a charge at infinity, as was shown by Dotsenko and Fateev [ 9 ] and Gervais and Neveu [ 4,10 ]. For C> 1, on the contrary, the highest weights do not correspond to null states [ 2,3 ]. They are given by (1.4) with m>0and n<0 [4,5,8]. No differential equation is known, as yet, for the correlators which are not coulombic. We recently derived exact three-point functions by other methods [11]. For later use, we note that C0, C, and the critical value C„„ = 26 are related by C ^ C o + C,.

(1.8)

The FQS series (1.1) may be obtained by the Goddard-Kent-Olive construction [12] which is based on the SU(2) Kac-Moody algebra. On the other hand, theGN series (1.2), (1.3) which both lie in the region C> 1, have been obtained by quantizing the Liouville field theory, that is a field theory involving one bosonic field with an exponential potential. Such a theory is a particular case of an infinite family of generalized Liouville dynamics, called Toda field theories [13], each of them being associated with a given standard Lie group. The number of fields is equal to the rank of the group, and the standard Liouville theory corresponds to SU(2) in this general picture. One thus sees that the family of theories (1.1)-(1.3) is related to the group SU(2). Our aim, in the present article, is to extend the scheme we have just outlined to groups of higher rank. This will lead to a systematic formulation of models with extended Virasoro algebras, of the type introduced by Zamolodchi­ kov [14]. The basic tool will be the Toda field theories [13] that generalize the Liouville field theory for arbi­ trary groups. This will be the subject of section 3, but it is useful to draw the general picture before plunging into the details. This is done in the next section.

2. The general picture of bosonic conformal Held theories In this section we give, without proof, the generalization of the above formulae to an arbitrary simply-laced group g. Some of these formulae have already been written down, but no systematic treatment exists to our knowledge. This will be given in section 3. We shall denote by / the rank of g and by hc its Coxeter number. The discrete series (1.1) becomes '2 See ref. 18 ] for a review.

413

91

Volume 206, number 3

C=l[l-hc(K

PHYSICS LETTERS B

+ l)/r(r+l)],

26 May 1988

rinteger>h c .

(2.1)

The corresponding highest weights are given by «. J .= [ l / 2 r ( r + l ) J { [ ( r + l ) m , - r n ' ] » u [ ( r + l ) m ' - r / i ' ] - T L A c ( d i m g ) } ,

(2.2)

where m and n are sets of/ integers. Throughout this paper, j is an index. When we need the square root of minus one, we shall write it explicitly in order to avoid confusions. %, is the inverse of the Cartan matrix of the group g. The series (2.1) is obviously such that C<1. It was first derived by Fateev and Zamolodchikov [15], for SU(3). Both eqs. (2.1) and (2.2) appear, but without complete proof, in an article by Bais, Bouwknegt, Surridge and Schoutens [ 16 ], who carry out the GKO [12] construction for general groups. A recent paper " of Fateev and Lykyanov treats the case of SU (N) [17]. The generalization of eq. (2.2) to other values of C may be written down by extending relation (2.1) to noninteger values of r. One may then determine r as a function of C, obtaining / • = j { - l ± v / [ C - / ( l + 2A c ) 2 ]/(C-/)}

(2.3)

and, substituting into (2.2), im..= [(C0-C)/ilhc(hc X%l(m'

+ l)][(m' + n-) +

+ n') + (m'-n')s/(C-C,)/(C-Co)]

(m,-nl)y/{C^C\)/(C-C0)] + [(C-C0)/24l][l/(hc

+ \)]dim&,

(2.4)

where C0 = l,

C,=l(\+2hc)2.

(2.5)

The plus or minus sign ambiguity in eq. (2.3) corresponds to exchanging m and «, and is thus irrelevant. As we shall see later on, eq. (2.4) is the correct continuation of (2.2). Eq. (2.4) is the generalization of (1.4). The value C0 corresponds obviously to / free bosonic fields. It separates the region where eq. (2.4) gives positive or negative values for m' and n' all positive. As section 3 will show, with this choice formula (2.4) gives the highest weights of the null states. Thus, in the same way as for the standard case, C0 separates the region where all positive highest weight representations are unitary (C> C 0 ), from the region where this is true only for partic­ ular values of C, that is, presumably, for the series (2.1). Clearly C0 is larger than one except for SU (2). There­ fore, some of the members of the series (2.1) share this property. They are mathematically similar to the theories of the FQS series however. Indeed, their spectrum of highest weights coincides with a set of null states and their Green functions satisfy differential equations of the hypergeometric type, as was shown for the case of SU (3) [15] and very recently *' for SU (N) [17]. The complementary region C>Ca will be most naturally described from the generalized Liouville theory, called Toda field theory, to be discussed in section 3. The discrete series (2.1) is a particular case of the ansatz C=t{\+hc(h,

+ \)(r+s)2/rs]

, r, .s integers ,

(2.6)

that generalizes formula (1.6). This again leads to rational highest weights since, in this case, (C-Cl)/(C-C0)

= [(r-s)/(r+s)]2.

(2.7)

We shall show that the generalization of (1.3) is indeed given by eq. (2.6) with r positive and s equal to one. This last series lies in the range C> C,. Finally, for C'0< C< C,, a characteristic feature of the three values (1.2) is that they are equally spaced between C0 and C,. (This is also true for the supersymmetric Liouville theory where C0 = \ and C, = ^ ) . Thus we believe that the generalization of (1.2) is C=/[l+Wi c (Ac + l ) ] , ,J

y=l,2,3.

This preprint was circulated while the present work was being completed.

414

(2.8)

92

Volume 206, number 3

PHYSICS LETTERS B

26 May 1988

By analogy with SU (2) (Liouville), the spectrum of highest weights should be given by tm^

= l(4-v)m'9,lm>+iiVthc(hc

+ i) = t(m).

(2.9)

For the series (1.2) of SU (2), we pointed out [ 7,8 ] that the partition function associated with the highest weights, that is to (1.4) with m = — n, has special properties that are crucial in the building of the associated string dynamics. For a general group g, one has a similar situation. The basic point is that the partition function may be written as a sum over all weight lattice vectors w (the common length squared of all roots is taken equal to 2): Zo^I<72'<-)-2,<,)=I<7(4-""1. m

(2.10)

w

After a Jacobi transformation \n{q) In(i7) = (2n)2t this becomes the following sum over the root lattice:

£f/«->.

(2.11)

a

It only involves integer powers of q for v=2 and 3. For the Liouville Neveu-Schwarz-Ramond models, the equivalents of the last two choices give the Liouville superstring theories that are space-time supersymmetric in three and five dimensions, respectively [ 7,8 ]. The authors of ref. [16] have argued that C crll =2/(1+2A C + 2AC2).

(2.12)

Thus eq. (1.8) holds for any group. The formulae we just wrote are all such that the expressions recalled in section 1 correspond to g=SU(2). Indeed this group has a rank equal to one and a Coxeter number equal to two.

3. The Todafieldtheories Let us now consider the Toda field theories [13]. The action is given by S-jiodxUdrf'K^-VW)],

K(0)=Xexp(K„
(3.1)

where K„ is the Cartan matrix of g. The equations of motion are 6 + a_0'=exp(AO,0')

wilhx± = {(a±r)

, 31=a„±3, .

(3.2)

These Toda field theories are conformally invariant. They are essentially the only interacting bosonic two-di­ mensional field theories having this property. Indeed, for a field theory (3.1) with a general potential V, the Noether energy-momentum tensor Tm has a trace equal to 2 V. The improved energy-momentum tensor 0„, = 7-„, + (6„6„-»;„,a 2 )/(0)

(3.3)

istraceless (using the equations of motion (3.2)) provided (1) / i s a linear function of $:f— Y.', (2)2K+c,(/ir- | ), / 8K/6>'=0.

(3.4)

Hence the theory is conformally invariant if and only if V is either zero or an exponential of the fields. If (in the latter case) we further demand that the theory be integrable, one ends up with the Toda field theories (3.1), i.e. the matrix K,t must be the Cartan matrix of some group g. The general solution of the Toda field theories as given in ref. [13] makes use of the underlying group struc415

93

Volume 206. number 3

PHYSICS LETTERS B

26 May 1988

ture encoded in the Cartan matrix. It relates the fields 0' to / arbitrary functions of x* and / others of x~, thus separating the left- and right-moving degrees of freedom. One has exp(-0') = ( < A , | [ A / + U + ) ] - | / n / / U + ) * ' ^ A / - ( j : - ) i / n / 7 ( x - ) » ^ | > l , > J .

(3.5)

The X, are the fundamental weights, defined as the basis which is dual to the one given by the simple roots. The M ± are matrices acting in the corresponding representation space. They are solutions of the differential equations Z.M*-=M*Ydf;-(z*)Etl,

(3.6)

y-i

where the Etl are the step operators acting along the directions of the simple roots. In the case of/= 1 (SU(2): Liouville) any two solutions are conformally equivalent, the arbitrariness of the two functions/ 1 being absorbed by appropriate conformal transformations of x ±. For higher / one thus expects that the arbitrariness in the choice of the 2/ functions/,1 corresponds to a bigger symmetry. We will indeed show that extended Virasoro symmetries appear. The general solution (3.S) is given by the scalar product of a left-moving and an analogous right-moving piece. It is sufficient to concentrate on one of them, say the right-moving one. We will simply write x instead of x~. To be definite we will choose in the following the case of the simply-laced algebras SU(Ar). Let us consider the construction of exp(—0'). X, is the heightest weight of the Af-dimensional defining representation. The right-moving piece of the solution is then an M-component vector depending on N— 1 arbitrary functions / - (x)=A'j(x) (primesdenote derivatives):

Y\A;(x)-9-M(x)U,y=n^M-91]

A,{x) /'dx,^U,Ml(x,) IH V> I/'(be, S''dx2As(x1)A-2(x2)A\(x,)

<3-7>

We now show that the y/, satisfy a linear differential equation of order N. First, the common factor UJA'J (x)_ '*"=Vi is precisely such that the wronskian of Vu ¥2 VN is equal to 1 as one may verify by explicit calculation: tf'i

¥2

•••

V'N

N-\

=i»f,-*=nKr,

0.8)

which coincides with the above value of i//\ since %=i(N-j)/N

fori^.

(3.9)

Next, each of the if/,,..., y/N is a solution of a linear differential equation of order N, obtained by writing that the wronskian of the v, and any linear combination i// of them is zero: Ci

-

v\N>

... i//tf'
VN

V

0. Developing with respect to v we obtain the differential equation 416

(3.10)

94

Volume 206, number 3

(-V»> + Ua)V»-2>+U{,)df»-:"

PHYSICS LETTERS B

26 May 1988

+ ...+ UwW=0.

(3.11)

It may be seen that, conversely, any set of N independent solutions of this equation can be brought under the form (3.7). For this, one combines them in such a way as to eliminate the potentials UU). It is remarkable that (3.7) was first derived purely from the Lie algebra so that there appears an interesting relationship between linear differential equations and group theory. The coefficients U(i) are only functionalsof A" /A] = -p'(x) and of its derivatives. For N=2 (Liouville) eq. (3.11) is nothing but the associated Schrodinger equation intro­ duced by GN [8]. For general N, computing the potentials U0) is tedious but straightforward. Only U(2) has a simple expression: Vw = { I %(p'p' + NSudp'/dx).

(3.12)

The fields p' are the fundamental dynamical variables that separate the degrees of freedom. Using the Poisson brackets (A is a parameter that plays the role of Planck's constant) {P*(x),pJ(X-)} = 47thK,J6'(x-x') ,

(3.13)

one shows that Ul2 > 12ft generates the Virasoro algebra with a non-vanishing central charge even at the classical level: IN Cci.„=^-I».. = *(*2-l)/2ft.

(3.14)

From our experience with the case N = 2 (Liouville) it is clear that the Poisson bracket (3.13) leads to canonical Poisson brackets between 0' and its conjugate momentum Kv$'. The next potential l/ l3) contains term cubic in the p\ terms quadratic with one derivative and terms linear with second derivatives. We will only write it here explicitly in the case N= 3: Uw = 2LA2pl+p2){2p2+p,)(p,-Pl)

+ \(2pl+pi)p\+\(2p';+p';).

(3.15)

Its Poisson bracket with Ul2), is such that it transforms as a conformal field of weight 3, up to central terms. The Poisson bracket of Uai with itself has an interesting structure. It does not close in the usual Lie algebra sense, since its expression involves Ul2h V{i), and (l/< 21 ) 2 : Denoting by L„ the Fourier modes of U(2) /2ft, and by T„ those of C, 3, / 2 , / ^ T ft one finds ^{T„,T„}

= -2(n2-m2)T^m-Un-myL„„

+ 2h(¥)Hn-m)^LkLf,.„_k

+

(4/3ft)niS„._„. (3.16)

It is also possible to express the generators U0) in terms of the fields 0', 0'. For example, l/ (2) is nothing but the (—component of the) improved energy-momentum tensor as computed from the action (3.1). It gener­ ates the conformal symmetry of this action. In the same way we can compute the transformations of the fields 0' as induced by Poisson brackets with U,,t (0, 0). It is clear that these should be symmetries of the action, too. Hence it should be possible to exponentiate them. Therefore the Poisson bracket of any two potentials U0) 2 for general N are quite complicated, it is easy to write down their expressions for constant p'=p'0- Indeed, in this case all U(,, are also constant. Integra­ tion of A" /A; = —p8 yields^; =exp(-p'0x), and by eq. (3.7) we find that the solutions v,(0) are Vl0]

~np(n\,)p'0x)

with 417

95

Volume 206, number 3

PHYSICS LETTERS B

tfi) = ».„ tfo = %-».->./

forl
26 May 1988

(3.17)

The differential equation they satisfy can thus be written in a factorized form: Y\(-d+)i'l,)Pi)v",)

= 0.

(3.18)

Comparing eqs. (3.11) and (3.18), one immediately finds (C/(„)o~ ( - ) ' " '

I

/<<,,,-Vo,, *>$'•••/>£•

(3.19)

Going back to the case of non-constant p', we remark that, for Liouville [ 18], the factorized form (3.18) remains correct, with an appropriate ordering. We have checked that this generalizes to yV= 3. It is a conjecture that this is true for all N. Consider next the classical conformal family. By integration of A" /A', = -p' we reconstruct /i;=exp(-Jd>cV(*'))=exp[-
(3.20)

and hence obtain the solutions C'= I~I-<; "'"'=e*P( *./?/).

V2 = Vi j
(3.21)

They can be expressed in terms of the basic fields 0,=exp($,n),

/=l

JV-l

(3.22)

as y,~(0,_i)"'0, (with e o = 0 * = l ) . The basic fields d, generate the conformal family by taking products of n' fields*?,. Such a product (classically) has a conformal weight —NJ.,%n'/2. Let us now turn to the quantum case. The fundamental Poisson bracket (3.13) gets replaced by a commutator v/'— 1{ , } - » [ , ]. For the Fourier modes we have [p'„,P'„]=2hnK„6n._„.

(3.23)

The classical expressions for (/„, now have to be normal ordered. The Virasoro operators are L„=£„ + CQlm„/24: L„=^l%(l;:p'„P'„-„:-J:r\N6IJnp^+[N{N2-\)/4it1]S„,0,

(3.24)

and the quantum central charge is C=(N-\)[\+N(N+l)/2fi]=l[l+hc(hc

+ \)/2fi] .

(3.25)

As usual the pL„ create harmonic excitations on top of ground states labelled by their p'0 eigenvalue. These ground states are heighest weight states of the extended Virasoro algebra. We have to determine the spectrum of all possible zero-modes p'0. TheSL(2,C) invariant vacuum, i.e. the state annihilated by £<, and L ± 1 , haspj> = -x/^Tforali;=l,...,Ar-l. The quantum version of eq. (3.20) now reads A',= :exp( -t\\

dx'p'(x') j:= : e x p [ - w , ( x ) ] : .

(3.26)

The value of 7 is determined in such a way that all 1//, are conformally covariant operators. By eq. (3.21) we see that this is only the case provided A) has conformal weight 1. This fixes r\ to be a solution of 418

96

Volume 206, number 3

PHYSICS LETTERS B

26 May 1988

2h)i2-ti+l=0^tit=(\/4fi)(\±s/i-ifi) .

(3.27)

There are two solutions i)± and the basic operators that generate the conformal family become $r= :exp(rit <3vq,y. , i= 1,..., N- 1 .

(3.28)

A general conformal operator of this family is obtained by taking products of n\ Sf and n'_ d,~ and removing the short-distance singularities. As can be explicitly seen from (3.23), (3.24) and (3.28) such an operator has a conformal weight A(»,M) = -h9uCi:j-lN9uC'.

f' = n' + >; + +'! , _»;_ ,

(3.29)

and, using (3.27), (1 denotes the vector (1, 1,..., 1)) J{H,m) = ((n + l,m+l),

(3.30)

where the right member coincides with (2.4). One may further write t(n,m)=(l/4fi)9„Pl0p>0

+ N(N2-\)/4&fi,

p'0(n,m) = -llJ^\[(n'+m')

+ (n<-m,)sJ\-M,),

(3.31)

showing what are the eigenvalues of p i in the quantum case. This determines the eigenvalues of Lo for all highest weight states, together with the eigenvalues of the zero modes of the higher spin operators. These are, in general, obtained by substituting (3.31) into (3.19). This general situation is very similar to the case of Liouville [8]. One thus expects that, here also, the classical differential equation (3.11) will carry over to the quantum case where it will lead to differential equations for correlators describing the decoupling of null vectors. Moreover, higher positive powers will also satisfy similar differential equations showing that formula (2.4) with m ' > 0 and n' > 0 does give the appropriate generalization of Kac's formula, as anticipated in section 2. This, of course, needs further study but the analogy with the Liouville case is compelling. Let us finally derive the generalization of the discrete series (1.3) of central charges, that is eq. (2.6) with s=l. This corresponds to the region C>C,, i.e. h< J andpo is purely imaginary. In this region which is con­ nected to the classical limit ft-»0 (where»;+ is singular) only */_ has to be used in constructing conformal fields. Hence only

p'0(l-l+m) = -yf^\[\ + \m>(\-J\-m]

(3.32)

are allowed. The SL(2,C) invariant vacuum p0 = —s/—l 1 corresponds to m=0. In analogy with the case N=2 (Liouville) it is clear that the local field exp( —0') shifts m' by two units. Starting from the SL(2,C) invariant vacuum, p0=-sp-\ 1, its conjugate Po = +\f—i 1 can only be reached if (r+l)(l-yi-8«) =2

(3.33)

for some positive integer r. This leads to the special values of ft: *=r/2(7-+l) 2 , r=integer>0,

(3.34)

which are the same for any group SU{N). In particular, (3.34) coincides with the series of the Liouville theory. If we use eq. (3.25) to reexpress ft in terms of the central charge C we get eq. (2.6) w i t h s = l as claimed in section 2. It is only for these special values of the central charge that we can consistently restrict ourselves to - 1 < >/^-T pi < 1 and hence to positive heighest weights.

Acknowledgement One of us (J.-L.G.) is grateful to Davd Olive for detailed discussions about the Toda field theories. 419

97

Volume 206. number 3

PHYSICS LETTERS B

26 May 1988

References [ I ] D. Friedan, Z. Qiu and S. Shenker, in: Vertex operators in mathematics and physics (Springer, Berlin, 1984). [ 2 ] J.-L. Gervais and A. Neveu. Nucl. Phys. B 238 (1984) 125. [ 31 J.-L. Gervais and A. Neveu, Phys. Lett. B 151 (1985)271. [4) J.-L. Gervaisand A. Neveu, Nucl. Phys. B 257 [FS14] (1985)59. [ 51 J.-L. Gervais, in: Unified string theories, 1985 Santa Barbara Workshop, cds. M. Green and D. Gross. | 6 ] A. Belavin, A. Polyakov and A. Zamolodchikov, Nucl. Phys. B 241 (1980) 333. (7] A. Bilal and J.-L. Gervais, Phys. Lett. B 187 (1987) 39; Nucl. Phys. B 284 (1987) 397; B 293 (1987) I; B 295 [FS21 ] (1988) 277. [ 8 ] J.-L. Gervais, Liouville superstrings, LPTENS preprint 87/39, note of lectures Poiania Brasov summer school (Romania). 19 ] V. Dotsenko and V. Fateev, Nucl. Phys. B 240 [ FS12 ] (1984) 312. (10) J.-L. Gervais and A. Neveu, Nucl. Phys. B 238 (1984) 396. 111 ] A Bilal and J.-L. Gervais, preprint LPTENS 87/33, submitted to Nucl. Phys. B (12] P. Goddard, A. Kent and D. Olive, Commun. Math. Phys 103 (1986) 105. 113 ] See e.g. P. Mansfield, Nucl. Phys. B 208 (1982) 277, and references therein. 114] A.B. Zamolodchikov, Teor. Mat. Fiz. 99 (1985) 108. 115 ] V Fateev and A.B. Zamolodchikov, Nucl. Phys. B 280 | FS 181 (1987) 644. 116 ] F. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, preprint ITFA 87-18, THU 87-21. 117 ] V. Fateev and S. Lykyanov, 1CTP preprint 6766/87. 118] J.-L. Gervaisand A. Neveu, Nucl. Phys. B 264 (1986) 557.

420

98

Reprinted with permission from Communications in Mathematical Physics Vol. 123, pp. 627-639,1989 © 1989 Springer- Verlag

Higher Spin Fields and the Gelfand-Dickey Algebra I. Bakas*** Center for Relativity, The University of Texas at Austin, Austin, Texas 78712, USA

Abstract. We show that in 2-dimensional field theory, higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by Gelfand and Dickey to describe integrable nonlinear differential equations in Lax form. The spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined. This provides a conformal field theory approach to the representation theory of Gelfand-Dickey algebras. 1. Introduction Recently Zamolodchikov investigated additional symmetries in 2-dimensional conformal field theory generated by higher spin local currents [1]. It is known that intwo dimensions the independent components of the stress energy tensor T(z), T(z), generate the (infinite) algebra of conformal transformations. The operator product expansion for the fields T(z) has the form TOW_

«

+"W+«W+....

2(z — w)* (z — wp

„,

z— w

where ••• denote all nonsingular terms. Introducing Fourier components L„{neZ), we obtain the Virasoro algebra [ A ^ J = (" - m)Ln+m + 23, (n3 - «)<Wo-

(2)

Primary conformal fields
* Supported in part by the NSF Grant PHY-84-04931 ** Address after 1 September 1988: Center for Theoretical Physics, The University of Maryland, College Park, MD 20742, USA

99

I. Bakas

628

The complete set of local fields, occurring in a conformal theory consists of conformal families [A[2]. In quantum field theory, the set of all conformal families forms a closed operator algebra. The spin of a primary field

(4)

where A and A are the conformal weights appearing in the operator product expansions T{z)
(5)

are conserved, i.e., diJf(z) = 0. The simplest example (s= 1) was considered in [3], where the additional infinite dimensional symmetry algebras were identified with Kac-Moody current algebras that arise in the conformal field theory of nonlinear
(6a)

\.L„Lmi = (n-m)Ln+m + ^(n3-n)6n+mt0, LWn,Wm} = ^n(n2-\)(n2-4)5n

+ m,0

(6b)

+ b2(n-m)An+m

+ (n - m)\^(n + m + 2)(n + m + 3) - i(n + 2)(m + 2)]LB+m.

(6c)

In Eq. (6), the following identifications have been made: /l„=

+

f :LkL._k: + lx.Lm

x2, = (1 + /)(1-/); x 2(+ , = (2 + 0(1-/).

(7a)

(7c)

The highest weight representations of the operator algebra (6) were studied in [4], and as a result an infinite series of new conformal models that possess a global Z3 symmetry were constructed in two dimensions. Furthermore, it was found that

100

Higher Spin Fields and the Gelfand-Dickey Algebra

629

these new (minimal) models with central charge c

,8) - 2 ( ' - p T ^ T ) ) ; '-*■'•*■" have a hidden relation with the SL(3, R) Kac-Moody algebra, thus making it possible to prove the positivity theorem that guarantees their unitarity. The hope of obtaining a systematic description of all types of criticality in two dimensions by considering all higher spin algebras with integer (or half-integer) values of s was also expressed in [4]. It is the purpose of this paper to provide a framework in which to study higher spin operator algebras in connection with the theory of integrable nonlinear differential equations and the hidden afline Kac-Moody symmetries they contain. In particular, we show that the spin 3 algebra (6) is contained in the Gelfand-Dickey algebra of formal pseudodifferential operators introduced to describe 1 + 1 nonlinear differential equations, such as KdV and KP, in Hamiltonian (Lax) form. We shall also comment on the generalization of this approach to include all higher spin operator algebras. Incidentally, we mention that the Gelfand-Dickey algebra of formal pseudodifferential operators has been used in [5] to construct sheaves of Lie algebras on algebraic curves which provide globalized generalizations of the Virasoro algebra (in the sense that a central charge is associated to each point of a Riemann surface and each closed oriented curved on it). This constuction has stimulated many attempts to develop the operator formalism for conformal field theories defined on higher genus Riemann surfaces (see for instance [6]), as well as clarify further the role that the diffeomorphism group of the circle plays in Polyakov's approach to string theory. The crucial ingredient is that the Virasoro algebra can be embedded in the Gelfand-Dickey algebra in a natural way (see also [7]). We shall return to this point later on. Here is an outline of what follows. In Sect. 2 we review the basic theory of formal pseudodifferential operators and define the Gelfand-Dickey algebras of the type GD(SL(n)). In Sects. 3 and 4 we study in detail the Virasoro and spin 3 operator algebras, respectively, using the Gelfand-Dickey bracket of the second kind. Finally (Sect. S), we discuss the generalization to all higher spin fields and indicate possible applications of this approach to 2-dim conformal field theory and statistical mechanics.

2. The Algebra of Formal Pseudodifferential Operators Let us now present some basic facts from the theory of formal pseudodifferential operators (see for instance [8]). First, consider the ring of all differential operators L = 11.(2)3" +11,-^)3"- ' + ••• + u,(z)0 + u0(z).

(9)

(Here d denotes the derivative with respect to z.) The multiplication law is provided by the Leibniz rule

(o,(z)a').(*i(z)ao = I (' )ai(z)dkbJ{z)di+J-\

(io)

101

630

I. Balcas

which, from the point of view of one-dimensional quantum mechanics, is equivalent to the normal ordering prescription.1 Even more important is the ring of formal pseudodifferential operators that consists of the formal series L = u1I(z)d" + - + nl(r)d + iio(z)+

~£ uk{z)#.

(11)

For notational purposes, it would be convenient to use the identifications L+=L;

L_=

£

«*(*)*: resL = «_1(r),

(12)

* - - o o

where "res" stands for residue. The multiplication law for formal pseudodifferential operators is also provided by the Leibniz rule; for completeness, we mention that for all k> 0 the following identity is true:

d-'°a(z) = | p ( - l)'y_~j, ) ! Mz)d-k-'.

(13)

At this point notice that primary conformal fields have a natural interpretation in the ring of formal pseudodifferential operators. In particular, by introducing the bracket [L ls L2]:= LJOLJ — L2°L„ we observe that [£l(z)<5,£2(z)d] = (£l(z)£2(Z) - e\(z)£l(z))8

(14)

and l£(z)d,(p(z)dkmoddk-l]

= (e(z)(p'(z)-ke'(z)
(15)

Equation (14) implies that the Lie algebra of vector fields on the circle is contained in the algebra of formal pseudodifferential operators, while (15) suggests that the operator (p(z)^ mod d*~1 represents a primary field with conformal weight A = — k (cf. Eq. (31)). The commutator [Llt L2] of two formal pseudodifferential operators Lj and L2 with degrees n t and n2, respectively, has degree n1 + n2 — 1. Therefore, all formal pseudodifferential operators of negative degree only form an algebra with respect to [, ] also known as the Volterra algebra. For convenience, we drop the multiplication symbol °. From now on, we let

X=td-%(z)

(16)

(=i

represent a generic element (symbol) of the Volterra algebra—usually, only a finite number of x.'s will be taken to be nonzero. Then we may define a pairing between the space of differential operators L and elements of the Volterra algebra by the formula [8]: :=Jdzres(L*).

(17)

This enables us to think of the space of differential operators L as being the 1

One may think of any such operator L as the normal ordered operator corresponding to the (classical) function u,[z)z" + u„_i(r)£"~ ' + ••• + Utiz'iz + u„(z) on the plane with canonical coordinates z and f

102

Higher Spin Fields and the Gelfand-Dickey Algebra

631

(smooth) dual of the Volterra algebra. Since the residue of the commutator of any two (formal) pseudodifferential operators is always a total derivative, we have that = J < izre S ([L,X 1 ] + X 2 ),

(18)

which defines the coadjoint representation of the Volterra algebra. For a given L = uH{z) 0 with un(z) = 1. i.e., L = V + u„_ ^2)3"-» + ••• + Ul(z)d + u0(z). For any functional f[u0,...,um-i], X^td-'xM,

(19)

let Xf be the formal sum mthXi = j ^ - .

(20)

We can use the coadjoint representation of the Volterra algebra in order to define the Poisson bracket {f,g}2h.= ins(lL,X^ + Xg) (21) for any two functional/, g of u 0 ,...,!<„_!. Note that in Eq. (20) we have not included the i = n+l term allowed by the residue duality (17). This is justified because, in any case, [L,X/] + is a differential operator of degree at most n — 1; and so in computing the residue of [L, Xf~\ + Xr only terms with 1 ^ i ^ n will contribute. The Poisson bracket (21) associated with the space of differential operators (19) is the Gelfand-Dickey bracket of thefirstkind introduced to describe integrable nonlinear differential equations in Lax form (see for instance [8]). However, it is known that integrable systems in 1 + 1 dimensions are bi-Hamiltonian in the sense that they can be equivalently described using two different kind of Poisson brackets. The Gelfand-Dickey bracket of the second kind is defined [7,8] as {f,g}^:=iTes(VXf(L)Xt\

(22a)

where VXf(L) = L(XfLU - (LX,)+L.

(22b)

We note that the Gelfand-Dickey bracket of the second kind is more general than the first one, since an arbitrary shift of the form L-*L + k (where A is a constant) yields VX/(L)->VXf(L)-KL,X,l

+

,

(23)

i.e., [L, X] + behaves like a "coboundary" of VX(L). For this reason, in what follows, we choose to work with the Gelfand-Dickey bracket of the second kind. At this

103

I. Bakas

632

point, we note that VX(L) may be viewed as the coadjoint action operator of the algebra of formal series (16) with respect to a new commutator [, ]. The latter is determined by the equation {L,lX1,X2}y.=

iKs(VXl(L)X2),

(24)

thus generalizing (18). The algebra constructed this way from the space of differential operators (19), with [, ] as its bracket, is called the Gelfand-Dickey algebra GD(GL(n)), where GL(n) denotes the general linear algebra in n dimensions.2 It will be shown later on that this provides a natural generalization of the Virasoro algebra, while the associated operator VX(L) generalizes the coadjoint action operator of the Virasoro algebra (in the sense that AdJ/(L - d") = VXf(L)). Next, we restrict ourselves to G = SL{n) that arises as a special case of the GD(GL(n)) algebra. The GD(SL(n)) algebra is constructed from the space of all differential operators (19) with u,,., =0, i.e., L = d* + um-2(z)d"-2 + - + u0(z),

(25)

and as we shall see, it is most suitable for describing higher spin operator algebras. In analogy with the GD(GL(n)) algebra, we define the bracket of the second kind by Eq. (22). But since u„_ 2 = 0, the i = n term in the formal sum (20) is meaningless unless it is taken to be zero. In such case, straightforward calculation shows that the operator VXj(L) involves a term of degree n — 1 and so the choice u„.l = 0 does not seem to be invariant. This problem is resolved if one considers * / = I

1

^ i # - + d"n*..

(26)

with x„ determined by the requirement res [L, X/\ — 0—as it turns out, res [L, Xf~] is the coefficient of the term with degree n - 1 in VX/(L). Therefore, the GD(SL(n)) algebra is well defined provided that x„ is chosen appropriately. In what follows we show that the Virasoro, as well as higher spin operator algebras, are described in terms of GD(SL(n)) for all n = 2,3, 3. L = d2 + u and Spin 2 Following the general construction described above, we consider Xf = 8-l^+8-2x2

(27)

with x2 determined by the requirement res [L, X{~\ - 0. Explicit calculation shows that res [d2 + u, X/] = 2x'2 - (df/du)" and so the GD(SL(2)) algebra is well-defined provided that , 1 *2 = 2 2

(28)

Gelfand-Dickey algebras labeled by more general simple Lie algebras G, GD(G), have been introduced in [9] in connection with generalized hierarchies of nonlinear differential equations in 1 + 1 dimensions

104

Higher Spin Fields and the Gelfand-Dickey Algebra

633

Furthermore, we find that in this case

ifsfy

V

*W — l\t)

, fdfX (bf

-2u[fu)-U\fu)-

(29)

Therefore, the Gelfand-Dickey bracket of the second kind takes the form {Mzngtom^-jdz-^G^-^,

(30a)

where Cu,f, = i ^ + « ( ^ + ^"(z).

(30b)

The Poisson bracket (30) has already been used in the Hamiltonian formulation of the KdV equation, as well as in 2-dim conformal field theory (see [10] and references therein). This is the bracket between any two functionals / , g defined on the dual of the Virasoro algebra. Moreover, the operator 6^ describes the conformal variation of the quadratic differentials u(£) and hence the coadjoint action of the Virasoro algebra. The commutation relations between the (coordinate) functionals u are easily found to be {u(z)Mz')f^„ = Hz) + " ( z W ( z - A +tf\%z~ z')-

(31)

This result is not surprising at all because the space of differential operators L = d2 + u is known to be isomorphic with the (smooth) dual of the Virasoro algebra (31). Recall that under arbitrary reparametnzations of the circle Z-KT(Z), the operators d2 + u transform as d1 + u^o,-it\d2

+ au)&-m,

(32a)

where 'u(z) = &McW) + \(^-\(£jy

(32b)

Also, it is known that the space of quadratic differentials is isomorphic with the (smooth) dual of the algebra of vector fields on the circle, and so the desired correspondence is established with the aid of densities of weight — \ [10]. The value of the central charge of the Virasoro algebra (31) is c = 6 (cf. Eq. (2)). This is because we chose to work with the differential operator L = d2 + u. Having chosen Xd2 + u instead (with constant X = c/6), the Gelfand-Dickey bracket of the second kind would have been described by (30) with (9U — c/1233 + ud + du. This way, we conclude that the GD(SL(2)) algebra provides a realization of the Virasoro algebra. We also remark that for d2 + u the Gelfand-Dickey bracket of the first kind is

{MzngW))}^

= idzJ-^l(-

23^-^1.

(33)

This gives only coboundary contributions (~ dzS(z — z')) to the commutation relations of the Virasoro algebra, as expected from Eq. (23) that relates the Gelfand-Dickey brackets of the first and second kind in the general case.

105

634

I Bakas

4. L = d3 + utd + «0 and Spin 3 For any functional/[u^uj, let us consider the formal sum (cf. Eq. (26))

x^d-^

+ e-^+d-W

04)

3u0 du, In this case we find that res \_d3 + «,d + u0, Xf~\ = 3x'3 + (Sf/Su0)'" - 3(<5//<5u,)" + (ui(<5//<5u0))' and so, according to the general theory, the GD(SL(3)) algebra is well defined provided that x3 satisfies the equation

^(0-(0'-(»# Furthermore, we find by a straightforward (but lengthy) calculation that Vx (L) is given by VXf(d3 + uld + u0) = u1d + u0,

(36a)

where

-<0-<0-(^)"-«1(0-«o(0 + «„l ^)" + «,(«,£)' + Mlxi - (u, j£)' - 2 U l ( 0 - ( u o 0 , (36c)

and x 3 satisfies the requirement (35). The equation above describes the coadjoint action of the GD(SL(3)) algebra in the sense that AdXf(L-d3):= VXf(d3 + utd + u0) = u1d + u0. Comparison with Eq. (26) suggests that (36) generalizes the coadjoint action of the Virasoro algebra in a very interesting way. To be more precise, let us examine first the transformation properties of the operators d3 + u1(z)d + u0(z) under arbitrary reparametrizations of the circle z -»a{z). Explicit computation yields 83 + Ul(z)d + u0(z)^a'-2(83

+ °Ui(z)d + 'U0(Z))G'~ \

(37a)

where »u1(r) = <7'2u1((T(Z)) + 2Sff(z), 'u0(z) =
(37b) (37c)

Here, Sc(z) = (o"'/o')-{3/2)(o"/o')2 denotes the Schwartzian derivative of the diffeomorphism a. Note that u^z) transforms (up to the Schwartzian term) as a quadratic differential, while (37c) suggests the presence of a spin 3 field in the formalism. In particular, using (37b), (37c), we have that

106

Higher Spin Fields and the Gelfand-Dickey Algebra

535

*u0(z) - iX(z) = a'3 fu0(
^FU'MZ))

\

(38)

and so the combination u0(z) — \u\(z) transforms as a conformal field of weight 3. In establishing these results, we found it most useful to think of the operators L = d3 + uxd + u0 as acting on densities of weight —1 rather than on scalar functions. In fact, as we shall demonstrate later on, there is a natural generalization of the transformations (32) and (37) to any differential operator L = d" + u„ _ 2d""2 + —h u0 viewed as acting on densities of weight — (n — l)/2. At the moment, we would like to investigate the GD(SL(3)) algebra in more detail, since Eq. (37b) indicates that the Virasoro algebra is contained as a subalgebra in GD(SL(3)). For this, we make use of the Gelfand-Dickey bracket of the second kind that is associated with the space of differential operators d3 + utd + u0. Recalling Eq. (36) for the coadjoiqt action, we obtain the following expression for the bracket: {/(u1(z),uo(z)),0(«1(z'),«o(2'))}al Ja3+u,d+»o

- «oW(

ITTSY

Su0(z)J

- («i«>X77*Y " 4u0(z) \ Su^z)/ V

+ S5uo0(z)\ ^LA&^J

"VM*))

»■" K^sss)'" {""mf

+

5f 5M

O(2)

3VI(Z1K(ZV

~ (u°m£(

(39)

Note that the bracket between the (coordinate) functionals u, is equal to {uMuiWYXu^ = («i(*) + UiWWA* - A + 2d38(z - z'), (40) which shows that the Virasoro algebra is, indeed, contained in GD(SL(3)). However, thanks to all the terms that appear in Eq. (39), this is not the whole story. Further calculation shows that for w{z)'= u0(z) — \u\(z) the following is true: {Ul(z), H
(41)

i.e., w(z) transforms as a local field of conformal weight 3. Moreover, we find that the Gelfand-Dickey bracket for the fields w is given by Mz), wM$ +He+Uo = - R5<5(z - z') - i(ufo) + u\{f))dAz - zO)

-A("i(z)+«i(zm3*z-z') + i(ul(z) + u'l(z'))dAz-z')(42) Consequently, u^z) and w(z) form an algebra with quadratic determining relations

107

636

I. Bakas

(40H42). Here, the central charge of the Virasoro algebra (40) is c = 24. Note, however, that all other values of c are easily obtained by considering L = kd3 + Uid + u0 with A = c/24. For completeness, we mention that in this case the Gelfand-Dickey bracket of the first kind is equal to

"••^-'Ks&C+i&C)}

<431

Shifting u0(z) (and hence w(z)) by a constant is equivalent to modifying the Gelfand-Dickey bracket of the second kind by the expression (43). It can be readily checked that such modifications do not alter the commutation relations (40)-{42) at all. On the other hand, we may shift ut(z) by a constant (which is equivalent to redefining the vacuum expectation value of the stress-energy tensor of the theory). For u^-^u^z) — 1, the commutation relations (40)-(42) become K(z), Ul(z')} - («,(z) + 11,(20)3,** - z') + 2d}S(z - z') - ZdA* ~ A {Ml(z), w(z')} = Wz) + 2w(z'))d,6(z - z'), 3 {H
(44a) (44b)

- $(uftz) + u\(z'))dAz - z') + §(Ul(z) + Ul(z'))dAz ~ *') - £( U l (z) + utfWldiz - z') + i K ( z ) + u'fr'WAz - z')- (44c) (Here, for convenience, we have dropped the labels of the Gelfand-Dickey bracket.) Although the algebra (44) is not a Lie algebra, it has a very natural interpretation in the context of 2-dim conformal field theory. Comparison with the commutation relations (6) shows that the spin 3 operator algebra is a particular representation of (44). To illustrate this result in more detail, we point out that the quadratic terms appearing on the right-hand side of Eq. (44c) need to be regularized upon quantization. An appropriate regularization (which is also consistent with the algebra commutation relations) is acquired by assigning (A(z) + A(z'))dz6(z - z') + ^

5

- Sd3 + Ad^z - z') - [i(M,(z) +

- i(u,(z) + Ul(z'))dl + A("i(z) + uUWW

Ul(z'))dz

- z')

to the classical quantity (u\(z) + u\(z'))dz6(z — z'). In the expression above, A(z) is essentially the normal ordered operator that represents u\{z) with Fourier modes given by Eq. (7). This way, the right-hand side of Eq. (44c) assumes the form - \(dl - 583 + 4dz)d(z - z') - i(/l(z) + A(z'))dxd(z - z') -M5("i(z)

+ u,(z'))a23 - 3«(z) + u';(z')& - 8(u,(z) + u ^ z ' ) ) ^ ] ^ - z'),

and so Zamolodchikov's spin 3 operator algebra is a well defined representation of (44) with W(z)*->(i/v/3)w(z), T(Z)<-»M,(Z), and c = 24. In general, if we consider L = (c/24)d3 + uid + u0, the operator algebra (6) will result for all values of the central charge c.

108

Higher Spin Fields and the Gelfand-Dickey Algebra

637

5. Higher Spin Fields and Conclusions Next, we discuss briefly the relation between higher spin n operator algebras and the Gelfand-Dickey algebras GD(SL(n)) constructed from the space of differential operators L = 5" + uH-2{z)dH~2 H 1- u0(z). It is relatively easy to show that in this case the Gelfand-Dickey bracket of the second kind for the (coordinate) functionals u„_2 is equal to {uB-2(z), un-2(z')}™ = (uK.2(z) + uH.2(z'))dAz - z') + ^8(z

- z')

(45)

with cK = n(n— 1)(« + 1), which in turn implies that the Virasoro algebra is a subalgebra of GD(SL(n)) (see also ref. [5,7]). Alternatively, this result is established using the transformation properties of the operators d" + u,,_2(z)3""2 H h u0(z) under arbitrary reparametrizations z -*a{z). Explicit calculation shows that a" + «„.2(z)5"-2 + -+U 0 (z)^(7'- ( " +1,/:2 (5- + X - 2 ( ^ " 2 + -+'u 0 (z)K" < "" 1 ) / 2

(46a)

with X-2(Z) = °'2Un-2(°(z)) + T ^

:

~ ^ S M

(46b)

which generalizes the transformations (32) and (37) for all values of n. For clarity we note that under arbitrary reparametrization z -♦ a{z), the form of the operators L = dn + u„-2dn~2 H hu0 m a y n o t be preserved due to the occurrence of the term - ( « ( " - l)/2){ff"/a"'*1)ff'~i of degree n- 1. This will definitely be the case if we assume that the operators L act on scalar functions. However, if we think of them as acting on densities of weight — (n — l)/2 (as illustrated by (46a)), all terms of degree n — 1 cancel, which makes the choice u„_ t = 0 consistent. We point out that this is equivalent to the requirement res [Xf, L] = 0 imposed for well-definiteness of the GD(SL(n)) algebras earlier on. Further study of the transformation (46a) shows that up to Schwartzian terms, uH.2{z), u»_3(z),...,u0(z) (or appropriate combinations of them) behave as conformalfieldsof weight 2,3,..., n, respectively. For instance, forn = 4 wefindthat "Ml(z) = ff'^.Wz)) + 2&a"u2{o{z)) + SS'M X(z) =
(47a)

(47b)

In analogy with the n = 2,3 cases that we have already investigated, the GelfandDickey bracket for these fields would provide us with an algebra equivalent to the spin n operator algebra of 2-dim conformal field theory. This way, we identify the space of differential operators L = 5" + u„_25*"2H hu 0 with the (smooth) dual of the spin n operator algebra. Using the coadjoint action operator (22b), we may extend the methods of ref. [10] to all higher spin operator algebras and study the geometry of the resulting coadjoint orbits. Such orbits might be found useful

109

638

I. Baku

for investigating the geometric role of higher spin fields. We intend to study this problem in more detail elsewhere. Moreover, highest weight representations of all spin n operator algebras are expected to be of paramount importance in quantum conformal field theory, as well as in statistical mechanics in 2 dimensions. According to the general philosophy of 2-dim critical phenomena, the investigation of critical singularities reduces to the problem of finding appropriate conformally invariant quantum field theory solutions (see [2,4] and references therein). Such conformal (minimal) models correspond to degenerate Verma module representations of the underlying sym­ metry algebra for discrete values of the central charge c. For the spin 2 algebra, the corresponding discrete values of c are given by 1 — (6/p(p +1)) w ' t n p = 3,4,5,... [11], while for the spin 3 operator algebra by Eq. (8). In generalizing these results to higher spin operator algebras, we realize that unitary representations of the GD(SL(n)) algebras play an important role. Taking into account the embedding of the Virasoro algebra in GD(SL(n)) described by Eqs. (45) and/or (46), wefind(as a result of some preliminary computations) that the following set of discrete values c = (n-l)(l-,^-^-\

p = n+l,n + 2,...

(48)

is associated with (minimal) models of spin n operator algebras. Details of the underlying calculations will be presented in [12]. However, it is worth mentioning here that the representations of affine Kac-Moody algebras are closely related with the theory of integrable nonlinear diferential equations and hence with the Gelfand-Dickey algebras (see, for instance, [13] and references therein). In particular, the GD(SL(n)) algebra may be viewed as a reduction of the SL(n) Kac-Moody algebra. This seems to provide the basic ingredient for understanding hidden relations between (minimal) conformal models of spin n operator algebras and highest weight representations of the SL(n) Kac-Moody algebras in favor of showing the positivity theorem that guarantees the unitarity of these models. Along these lines, it would be most interesting to generalize the Goddard-Kent-Olive construction [11] to all values of n in a well prescribed way. Finally, we note that it is possible to extend our results to superconformal (and more generally to higher half-integer spin) operator algebras by using the supersymmetric generalization of the Gelfand-Dickey algebra of formal pseudodifferential operators introduced in [14] to study the super KP hierarchy of nonlinear differential equations. This will be the subject of future publications [12]. For completeness we mention that certain results somewhat related to ours have also been discussed by others [15]. Acknowledgements. I am grateful to Prof. Y.-S. Wu for a stimulating conversation during the course of this work. I also thank the theorists at Syracuse University for their encouragement to present these results.

References 1. Zamolodchikov, A.B.: Theor. Math. Phys. 65, 1205 (1985) 2. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Nucl. Phys. B241, 333 (1984)

110

Higher Spin Fields and the Gelfand-Dickey Algebra

639

3. Knizhnik, V., Zamolodchikov, A. B.: NucL Phys. B247, 83; (1984); Gepner, D., Witten, E.: Nucl. Phys. B278, 493 (1986) 4. Zamolodchikov, A. B., Fateev, V. A.: Nucl. Phys. B280, [FS18], 644 (1987) 5. Beilinson, A. A., Manin, Yu. I., Schechtman, V. V.: preprint (1986) 6. Alvarez-Gaume, L., Gomez, G, Reina, G: Phys. Lett. B190, 55; (1987); Vafa, C: Phys. Lett. B190, 47 (1987); Witten, E.: Commun. Math. Phys. 113, 529 (1988) 7. Khovanova, T. G.: Funct. Anal. AppL 20, 332 (1986) 8. Gelfand, I. M., Dorfman, I.: Funct. Anal. Appl. 15,173 (1981); Lebedev, D. R., Manin, Yu. I.: Funct. Anal. Appl. 13, 268 (1979); Adler, M.: Invent. Math. 50, 219 (1979); Dickey, L. A.: Commun. Math. Phys. 87,127(1982); Manin, Yu. I.: J. Sov. Math. 11,1 (1979); Guillemin, V., Steinberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press 1984 9. Drinfeld, V. G., Sokolov, V. V.: Sov. Math. Dokl. 23, 457 (1981) 10. Kirillov, A. A.: Funct. Anal. AppL 15, 301 (1981); Bakas, I.: NucL Phys. B302, 189 (1988); Witten, E.: Commun. Math. Phys. 144, 1 (1988) 11. Friedan, D., Qiu, Z, Shenker, S.: Phys. Rev. Lett. 52,1575 (1984); Goddard, P., Kent, A., Olive, D.: Phys. Lett. B152, 88 (1985) 12. Bakas, I.: in preparation 13. Drinfeld, V. G„ Sokolov, V. V.: J. Sov. Math. 30, 1975 (1985); Frenkel, I. B.: In: Lie algebras and related topics. Proceedings Lecture Notes in Mathematics vol. 933, p. 71 Berlin, Heidelberg, New York: Springer 1982; Jimbo, M., Miwa, T.: In: Integrable systems in statistical mechanics. D' Ariano, G. M. (ed.) et al.; Singapore: World Scientific, 1985 14. Manin, Yu. I., Radul, A. O.: Commun. Math. Phys. 98, 65 (1985) 15. Yamagishi, K.: Phys. Lett B205, 466 (1988); Bilal, A., Gervais, J.-.L.: Phys. Lett. B206, 412 (1988) ThierTy-Mieg, J.: DAMTP preprint (1987) Communicated by L. Alvarez-Gaume Received November 5, 1988 Note added in proof. Recently I became aware of reference [16] where Gelfand-Dickey algebras have also been used in the study of higher spin operator algebras. 16. Mathieu, P.: Phys. Lett. B208, 101 (1988)

Ill

Reprinted with permission from Annals of Physics Vol. 203, No. 1, October 1990 © 1990 Academic Press, Inc. (New York and London)

Toda Theory and ^-Algebra from a Gauged WZNW Point of View J. BALOG,* L. FEHfcR,1 AND L. O'RAIFEARTAIGH Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland P. FORGACS* Max-Planck-Institut fur Physik und Astrophysik (Wemer-Heisenberg-Institut fur Physik), P.O. Box 401212, Munich, Federal Republic of Germany AND A. WIPF Institutfur Theoretische Physik, ETH-Honggerberg, CH-8093 Zurich, Switzerland Received December 14, 1989

A new formulation of Toda theories is proposed by showing that they can be regarded as certain gauged Wess-Zumino-Novikov-Witten (WZNW) models. It is argued that the WZNW variables are the proper ones for Toda theory, since all the physically permitted Toda solutions are regular when expressed in these variables. A detailed study of classical Toda theories and their Tlf-algebras is carried out from this unified WZNW point of view. We con­ struct a primary field basis for the ♦'-algebra for any group, we obtain a new method for calculating the iV-algebra and its action on the Toda fields by constructing its Kac-Moody implementation, and we analyse the relationship between Tt^-algebras and Casimir algebras. The ^-algebra of G2 and the Casimir algebras for the classical groups are exhibited explicitly.

© 1990 Academic Pren, Inc.

I. INTRODUCTION

Two dimensional conformally invariant soluble field theories are based on various extensions of the chiral Virasoro algebras. The best known extension is the Kac-Moody (KM) extension [ 1 ] , whose most prominent Lagrangean realization is the Wess-Zumino-Novikov-Witten (WZNW) model [2]. There are various indications that the KM algebra may even underlie all the rational conformal field theories. For example, the Goddard-Kent-Olive (GKO) construction [3] * On leave from Central Research Institute for Physics, Budapest, Hungary. * On leave from Bolyai Institute, Szeged, Hungary.

112 TODA THEORY AND T^-ALGEBRA

77

generates a huge class of rational conformal field theories. Another extension is the so-called iF-extension, which is a polynomial extension of the Virasoro algebra by higher spin fields. The study of polynomial extensions of the Virasoro algebra was initiated by Zamolodchikov [4]. Later it was realized [5, 6] that a large class of polynomial extensions of the Virasoro algebra can be constructed by quantizing the second Gelfand-Dickey Poisson bracket structure of Lax operators, used in the theory of integrable systems. These ^-algebras proved very fruitful in analysing conformal field theories and they have become the subject of intense study [5-8]. Recently it has been found by Gervais and Bilal that Toda theories provide a realization of 1^-algebras [8,9]. Toda theories are important in the theory of integrable systems and include the ubiquitous Liouville theory, which, among other things, describes two dimensional induced gravity in the conformal gauge. There are a number of results suggesting that Toda theories must be closely related to WZNW models. First, in both cases the fields can be recovered from the generators of the respective extended Virasoro algebras (KM and iST-algebras) by means of linear differential equations [8]. Second, the Gelfand-Dickey Poisson bracket structure can be obtained by a Hamiltonian reduction from a KM phase space [10]. Finally, it has been shown by Polyakov [11] that two dimensional induced gravity (in the light cone gauge) exhibits (left-moving) SL(2, R) KM symmetry. In a recent letter [12] we have shown that the exact relationship is that Toda theories may be regarded as WZNW models (based on maximally non-compact, simple real Lie groups) reduced by certain conformally invariant constraints. To be more precise, Toda theory can be identified as the constrained WZNW model, modulo the left-moving upper triangular and right-moving lower triangular KM transformations, which are gauge transformations generated by the constraints. The advantages of treating Toda theory as a gauge theory embedded into a WZNW model are the following: First, the coordinate singularities of Toda theory disap­ pear by using the WZNW variables. Second, the ^-algebra of Toda theory arises immediately as the algebra formed by the gauge invariant polynomials of the con­ strained KM currents and their derivatives. Third, the general solution of the Toda field equations is easily obtained from the very simple WZNW solution. Finally, there are natural gauges which facilitate the analysis of the theory. In this paper we exploit the embedding of Toda theory into the WZNW model to obtain a number of new insights and results about the structure of Toda theory and iF-algebra. All our considerations are classical. We hope that their quantum generalizations will provide new constructions of quantum Toda theories [13] and ^-algebras. We first set up a Lagrangean framework for the WZNW-Toda reduction, namely we establish that Toda theories can be identified as the gauge invariant content of certain gauged WZNW models. Our gauged WZNW models differ from the usual gauged WZNW models [14] used in the path integral realization of the GKO con­ struction not only in the non-compactness of our groups, but also in that instead of a single diagonal subgroup we gauge two subgroups of the left x right WZNW group, the upper triangular maximal nilpotent subgroup on the left and the lower

113 78

BALOG ET AL.

triangular one on the right hand side. The nilpotency of the triangular subgroups is crucial to this ambidextrous generalization of the usual vector gauged WZNW models, and in fact the nilpotency of the gauge group is the reason for the appearance of the simple polynomial structures in Toda theory. The constrained WZNW model is recovered in this framework by an appropriate partial gauge fixing which leaves the left- and right-moving triangular gauge transformations mentioned earlier as a residual gauge symmetry. In most of our considerations we rely heavily on the use of a class of natural gauges used in studying the gauge invariant differential polynomials in the review paper [10] by Drinfeld and Sokolov. The basic property which makes the DS gauges convenient is that in each DS gauge the surviving components of the KM current serve as a basis for the #"-algebra. Working in the DS gauges, we give a simple algorithm to find the KM transfor­ mations which implement the canonical transformations generated by the i^-algebra. This provides us with a new method both for computing the iS^-algebra and for determining the action of the iF-algebra on the Toda fields. Our method crucially depends on using the embedding WZNW theory and its full, non-con­ strained KM algebra. We illustrate the method on the examples of A2 and B2 and demonstrate its power by computing the complete iF-algebra relations for the rather non-trivial example of G2. We find a DS gauge which enables us to construct a primary field basis of the iST-algebra. As far as we know a general algorithm for constructing primary ^-generators has not been known before, although such generators have been found in low dimensional examples [6]. We note that even the existence of a primary field basis is not completely trivial, since such a basis is constructed by a non-linear transformation [6] even if one starts from "^-generators transforming in a linear (inhomogeneous) manner under the Virasoro algebra. Our construction of the primary TiT-generators is based on a special SL(2, R) subgroup of the WZNW group, which plays an important role throughout the theory. The primary ^"-generators are associated in a natural way to the highest weight states of this SL(2, R) in the adjoint representation of the WZNW group. There have been attempts [15] at constructing polynomial extensions of the Virasoro algebra from a KM algebra by using the higher Casimirs of the underlying Lie algebra similar to the manner in which the second order Casimir is used in the Sugawara construction. On the quantum level these Casimir algebras close only under very restrictive conditions on the KM representation. We show that the leading terms (i.e., terms without derivatives) of the i^-generators are always Casimirs, and that the Poisson bracket version of the Casimir algebras always close. In fact, we prove that these classical Casimir algebras are obtained from the corresponding iS^-algebras by a certain truncation, and thus the Casimir algebras can be used to investigate the leading terms of the i^-algebras. For the classical Lie algebras A,, Bh and C, we give the explicit form of the Casimir algebra. We also consider the existence of quadratic relations for the Ti^-algebras. In the case of the Ah B,, and C, Lie algebras it is easy to display ^"-generators with quad-

114

TODA THEORY AND iF-ALGEBRA

79

ratic relations. The above mentioned relation between the Casimir and i^-algebras shows that for the other Lie algebras the ^-relations are necessarily of higher order. Finally, we investigate how the Toda fields can be reconstructed from the ^-generators. This reconstruction is a reduced version of the reconstruction of the group valued WZNW field from the KM currents, and this tells us that every Toda solution with regular iF-generators can be represented by a regular WZNW solu­ tion. The reconstruction problem leads us to studying the differential equations satisfied by the gauge invariant components of the constrained WZNW field. This way we recover the Lax operators studied in [10], which also appear in the generalized Schrodinger equations of Ref. [ 8 ] . For A,, Bh Ch and G2 the reconstruction problem can be reduced to solving a single ordinary differential equation of the order of the defining representation of the corresponding algebra, in all other cases one inevitably has a pseudo-differential equation. We will see that one has a single ordinary differential equation exactly when the representation in which the group valued WZNW field is taken is irreducible under the SL(2, R) sub­ group mentioned earlier, and that in general the structure of the pseudo-differential operator depends on the decomposition of this representation under the SL(2, R) subgroup. The plan of the paper is the following: In Section II we present a short review of the reduction of the WZNW model to Toda theory and describe the gauged WZNW framework. We elaborate on the role of the residual gauge invariance and on the gauge invariant quantities in Subsection II.2. The longest and most impor­ tant section is III. We start it with the definition of the liT-algebras. In Subsec­ tion III. 1 we present the construction of the Drinfeld-Sokolov gauges and observe that in these gauges the ^"-algebra reduces to the Dirac bracket algebra of the sur­ viving KM current components. In Subsection III.2 we exhibit a primary field basis of the "^-algebra and illusrate it with B2. In Subsection III.3 we give an algorithm to implement the action of the ^-algebra by means of KM transformations and illustrate the procedure with A2 and B2. In Subsection III.4 we first display a sub­ class of Drinfeld-Sokolov gauges where the ^-algebra relations are quadratic for A i, Bh and C,. Then we introduce the "diagonal" gauge, which is frequently used in Section IV, and briefly discuss the related Miura-transformation. Subsection IV.l contains a detailed analysis of the relation between the Casimir Poisson bracket algebras and the Ti^-algebras. In Subsection IV.2 we present the explicit Poisson bracket algebra of the Casimir operators of the classical Lie algebras Ah B,, and C,. In the last section, V, we study the differential and pseudo-differential operators which appear when the Toda-fields are reconstructed from the iS^-generators (or the constrained WZNW fields are reconstructed from the KM currents). There are three appendixes; Appendix A contains our conventions and some important group theoretical results, Appendix B contains the complete if(G2) algebra and Appendix C contains the details of the calculations of the Casimir algebras. We end the paper by summarizing the main results and giving some conclusions.

115 80

BALOG ET AL.

II. TODA FIELD THEORY AS A GAUGE THEORY

In this section we first summarize the main points of the redution of WZNW models to Toda theories. Then we show how to set up a Lagrangean framework for the reduction, using an ambidextrous generalization of the usual vector gauged WZNW models. Then we elaborate on the concept of residual gauge transforma­ tions and on the corresponding gauge invariant quantities. In particular, we point out that in the WZNW framework 1^-algebras appear naturally as symmetry algebras of Toda theory. II. 1. Toda Theory as a Gauged WZNW Model The so called Toda field equations constitute a rather interesting set of integrable (soluble) equations. These equations appear naturally in various problems (cylindrically symmetric instantons [16], etc.) and they can also be thought of as a generalization of the ubiquitous Liouville equation: d+ d_ + Me' = 0,

where M = const..

(2.1)

Now the Toda equations are given as a + <5_^ + ^|a| 2 A/"exp j i £ K ^

= 0,

(2.2)

where Krf is the Cartan matrix1 of a simple Lie algebra, A denotes the set of simple roots and the A/°"s are (positive) constants. The corresponding Lagrangean is

where K is the coupling constant of the theory. Clearly (2.2) reduces to the Liouville equation (2.1) by making the simplest choice for Kafi, namely the choice when K„f is just a number (corresponding to a rank one algebra). In fact Toda field theories are also distinguished by being the only two dimensional, nontrivial conformally invariant models which are soluble [8, 16] in the class of scalar theories without derivative couplings. These theories possess an imporved energy-momentum tensor

with vanishing trace, 8 + _ =0, on shell. Interestingly, the general solution of (2.2) can be written in closed form [16]. 1

Our conventions are collected in Appendix A.

116 TODA THEORY AND iST-ALGEBRA

81

Let us recall first, how Toda theories can be regarded as constrained WZNW models. We start with the WZNW action based on a connected real Lie group G (with maximally non-compact simple real Lie algebra (S)

+

isJ* Tr «'- , * ), >-

(15)

where g is a group-valued field and B3 is a three dimensional manifold whose boundary is Minkowski space-time. We choose the coupling constants K and k to be related by the equality k= —ATIK. This action possesses left and right KM symmetries. Their Noether currents associated to some Lie algebra element, A, are given as J=K(d+g)g~l

JW = Tr(lJ),

J=-Kgl(d_g).

7(A) = Tr(A.J),

Thefieldequations are equivalent to the conservation of the left and right currents: d_J = 0,

d + J=0.

(2.7)

Let now n" and v" (a e A) be arbitrary positive numbers and let us denote the set of positive roots by
J(£_J=-KV",

J(Ev) = 0,

7(£_„) = 0,

zeA


(2.8)

the equations of motion of the WZNW theory (2.7) reduce to the Toda field equa­ tions (2.2). To prove this result we start with the (local) Gauss decomposition g = ABC

(2.9a)

of the group-valued field g, where A = exp\ £ U *+

O'EX J

C = exp( £ U *+

c*E_\ J

(2.9b)

This group-valued Gauss decomposition is locally unique for Lie groups G with maximally non-compact Lie algebras.

117 82

BALOG ET AL.

Now exploiting the fact that n* and v" are zero for all but the simple roots, the constraints (2.8) can be rewritten as /r'<5_/l=xi|a|2v"£.exp{ix 1 P J * 2 ( d + C ) C - ' = £ i|a| Ai"£_ a exp{i £

K^A J

(2.10)

K^A

Substituting (2.10) into thefieldequations (2.7) one indeed recovers the Toda equa­ tions (2.2) (with M" = \a\2 ft'v"). It can be shown that this reduction is canonical in the sense that the Poisson brackets of the Toda variables ^ and ^ can be calculated either from the Toda or from the WZNW action (as a requirement, this fixes the relationship between the coupling constants). We remark that the famous Leznov-Savaliev general solution of the Toda field equations [16] can be derived effortlessly from the general solution of the WZNW field equations (2.7), g(x\x-) = gL{x + ) . g R ( x

),

(2.11)

where gL and gR are arbitrary group-valued functions constrained only by the boundary conditions and there is an obvious constant-matrix ambiguity in the definition of gh and gR. The general solution of the Toda equations can be obtained from (2.11) by first imposing the constraints (2.8) and then decomposing the constrained WZNW solution according to (2.9) [12]. In Section V we shall show that it is equally easy to recover the solution of the Toda field equations in the form recently found by Gervais and Bilal [8] from (2.11). As the quantization of Liouville and Toda theories is expected to be simpler in the WZNW formulation, it is worthwhile to find a Lagrangean realization of the reduction of the WZNW model to the Toda theory. In the following we show that an ambidextrous generalization of gauged WZNW models [14] provides a natural framework to carry out this reduction. For example, gauged WZNW models turned out to be useful in the Lagrangean description of the Goddard-Kent-Olive coset construction (GK.O) [3]. We shall need the Polyakov-Wiegmann identity [17] expressing the WZNW action for the product of three matrices A, B, C as the sum of the respective actions for A, B, and C, modulo local terms: S(ABC) = S(A) + S(B) + S{C) + K | d2xTr{(A

' d _ A)(d +

+ (B~l d_B)(d+C)C~i

B)Bl

+ (A-1 d_A)B(8+C)C-lB~i}.

(2.12)

118 TODA THEORY AND 1^-ALGEBRA

83

Next we want to consider the gauged version of the WZNW theory, i.e., we are looking for an action invariant under the transformations, g^agp-1,

*eH,peft,

(2.13)

where a, fi are functions of both x+ and x~, and H, ft are two isomorphic subgroups of G. Let us first recall the "usual" gauged WZNW models [14]. In the standard case one gauges a diagonal (vector) subgroup, H, of the Kac-Moody group G L xG R . Now the transformation of g under the vector subgroup is given as

g^ygy-1,

y(x+,x)eH.

(2.14a)

It is easy to see that the action functional I(g,h,Ji) = S(hg7i-1)-S{h}i-1),

h,%eH

is gauge invariant, provided (2.14a) is supplemented with 7i->7iy-1.

h-+hy-\

(2.14b)

Using (2.12), I{g, h, %) can be rewritten as I(g,A_,A + ) = S(g) + K\d2xTT{A_(d+g)g-1 + A_gA+g~l-A_A

+

},

+ (g-id_g)A

+

(2.15)

where 5(g) is the WZNW action (2.5) and A_=hld_k,

A+=(d+%-x)%.

(2.16)

In the action functional (2.15) A_, A+ are regarded as the light-cone components of some "gauge field" belonging to the adjoint representation of H, transforming according to (2.16), and its gauge invariance is obvious from the above construc­ tion. The variation of this action with respect to the non-propagating gauge fields A ± provides constraints which classically set the currents of H to zero. It has been demonstrated [14] that a careful quantization of (2.15) yields the GKO coset construction. At first sight it seems impossible to generalize (2.15) to be invariant under the more general transformations (2.13), since now the only obvious candidate for an invariant action is just S(hgh ~') which is non-local in the gauge fields. However, in the rather degenerate case when H and ft are the subgroups of G generated by the step operators associated to the positive and negative roots, and denoted by A^

119 84

BALOG ET AL.

and N, respectively, their Lie algebras are nilpotent, and hence one has the crucial property that S(h) = S(h~) = 0.

(2.17)

So S(hgh~1) — S{g) is local, therefore the gauge fields A_, A+ in Eq. (2.16) (where now heN and HeN) can be used in this case in the same way as for the case of a diagonal subgroup to set the corresponding N and N currents to zero. Since the constraints we want to implement set certain currents to constants rather than to zero, we consider the action I(g,A_,A

+)

= S(g) + K\d2xTr{A_(d+g)g-i + A_gA+g-l-A_n-A

+ (g-id-g)A +

v},

+

(2.18)

where n, v are special (constant) matrices, given by v = Z ^|a| 2 v«£ a ,

/;=£

\\*\2n°E_a.

A_, A+ are now independent gauge fields in the adjoint representation of the sub­ groups N and N so they are nilpotent matrices. The invariance of the action (2.18) under the gauge transformations, g-ag/T1,

/f.^d.a-'+aa.a"1,

A + ^/J/l + / r ' + (d + P)p~l, (2.19a)

where <x = x(x+,x-)eN

and

p = p(x + ,x~)eN,

(2.19b)

is now not completely obvious because of the non-gauge-invariant looking terms, TT(A + v + A _ /i). However, these terms change by a total derivative under gauge transformations because of the special form of A +, A_ and because the matrix v (resp. n) contains only step operators corresponding to simple positive (resp. negative) roots. For example, under the transformation (2.19) with

/?(*+,*-) = exPr X * , £ _ , ] we have Tr{v(8

+

/?)/?-'}= I
v'a + ^ ,

120

TODA THEORY AND Ti^-ALGEBRA

85

and the term Tr A + v indeed changes only by a total derivative. The equations of motion following from (2.18) are (with
+ g-lA_g)-lA

+

,g-l8_g

+ g-,A_g^

+ gA+g-l)+\_A_,d+gg-,

+ d_A

+

=0

+ gA + g-i] + d + A_=0 l

l

TrlE_v(g- d_g + g- A_g-vn = 0 l

1

TT\_E
= 0.

(2.20a) (2.20b) (2.20c) (2.20d)

Now making use of gauge invariance, A+ and A_ can be set equal to zero simultaneously and then we recover from (2.20) the equations of motion of the WZNW model (2.7) together with the constraints (2.8). Note, however, that setting A +, A _ to zero is not a complete gauge fixing. Indeed, it is clear that the condition A± =0 is invariant under chiral gauge transformations <x = a.(x + ) and f} = f}{x_) which are in the intersection of the gauge group and the KM symmetry group of the theory. Since in the A± = 0 gauge (2.20) reduces to (2.7) and (2.8), it follows that the residual gauge transformations g^xxgp-i,

where a =
(2.21)

must leave (2.8) invariant. This can also be verified by using the standard transfor­ mation property of the currents J and J under KM transformations: J-+oJz-l + K(d + a)(x-1

and

J-* pJp'1 + K(d_p)p~l.

(2.22)

Note that these chiral gauge transformations (2.21) form the complete residual gauge group of the gauge A ± = 0. From now on we stay in this gauge. Here we point out how the residual gauge transformations (2.21) arise from the Hamiltonian point of view. For this, as well as in the rest of the paper, we take the space of solutions, given by (2.11), of the WZNW theory as our phase space. This is convenient here because of the left-right factorized form of the general solution. The translation to the equivalent equal time canonical formalism could be made by parametrizing the solutions by their initial data and expressing the initial data in terms of the canonical variables. To make this translation as easy as possible, in this paper we use equal time Poisson brackets on the space of solutions. After these remarks, let us observe that the KM Poisson brackets of those current components which are to be constrained according to (2.8) vanish on the submanifold of the phase space defined by (2.8) (constraintsurface), i.e., we are dealing with first class constraints. Now first class constraints always generate such canonical transformations which leave the constraint-surface invariant, and it is easy to see that in our case these are naturally identified with the residual gauge transformations. II.2. Gauge-Invariant Quantities Clearly the Toda fields,
121

86

BALOG ET AL.

fields constitute a complete system of independent invariants with respect to these transformations on the "constraint-surface." In other words, Toda theory can be identified, at least locally, with the constrained WZNW model modulo residual gauge transformations. From now on we shall refer to the residual gauge transfor­ mations (2.21-2.22) simply as gauge transformations. It is important to note, that (2.9) is valid only in a neighbourhood of the identity of G. As a consequence of this non-global nature of the Gauss decomposition, our reduction can generate singular Toda solutions from perfectly regular WZNW solutions. This is the basis of one of the most important properties of the WZNW setting of Toda theory, namely, that the physically allowed singularities of the Toda solutions are precisely those which disappear by using the WZNW variables. We have shown this in Ref. [12] in the special case of 5L(2, R) by proving that the requirement that a Liouville solution be obtained from a regular solution of the WZNW theory is equivalent to demanding that the associated energy-momentum tensor (2.4) be regular. In Section V we shall show that this generalizes for a rank / algebra where besides the energy-momentum tensor there are / — 1 additional ""^-densities." In that case the Toda solutions with regular iST-densities can be represented by regular WZNW solutions, even if they appear singular in terms of the original local Toda variables * are invariant, but they are only well defined for WZNW solu­ tions in that neighbourhood of the identity where the Gauss-decomposition (2.9) is valid. Of course one could cover G with a finite number of patches and introduce locally regular Toda fields on them. These local fields would be related by some group transformations on the intersections of these patches and together they would define a global Toda field. Fortunately there is a simpler and more direct way to find globally well defined quantities which reduce to the local Toda fields in the neighbourhood of the iden­ tity. Consider some (d-dimensional) representation of G and choose a basis such that the Cartan subalgebra is represented by diagonal matrices, and the Lie algebras of N and N are represented by upper and lower triangular matrices, respectively. Then, because a. and /? in (2.21) are upper and lower triangular matrices, respectively, with l's in their diagonals, it follows that the lower-right sub-determinants

(

gu ■ ■ ■

: gdi--gdd)

Su\

(2-23)

122

TODA THEORY AND T^-ALGEBRA

87

of the matrix (g„) are all gauge-invariant quantities. It is also easy to see that in the Gauss decomposable case the Toda fields ^" can be recovered as linear com­ binations of logarithms of the &,. For example, let us consider A, and take G — SL(I+1, R) in the defining representation. Using the standard convention, in which Hai has 1 in its //-slot, — 1 in its (/+1)(/ + l)-slot, and O's elsewhere, for a Gauss decomposable g one obtains the simple formula, %=e-*>-'n,

where k = "k.

(2.24)

The local Toda field ^ indeed becomes singular where the Gauss-decomposition ceases to be valid, that is where one of the sub-determinants % changes sign. In general the globally well-defined sub-determinants (2.23) yield an overcomplete system of invariants, but in each concrete case one can single out / independent ones. For example, for the defining representations of the classical groups, the last / sub-determinants starting from gM suffice. They can be used as global variables for the Toda theory, after imposing the constraints (2.8). Since these sub-determinants are polynomial in the components of the basic WZNW field, g, they appear better suited for quantizing Toda theories than the original Toda fields themselves. For later use we note that beside the sub-determinants, which are fully gaugeinvariant polynomial quantities, there are other important quantities, which are linear in g, but invariant under left (or right) gauge transformations only. These are simply the elements of the last row (column) of g (and of gL and gK in (2.11), respectively). As the KM algebra plays a central role in WZNW theories, it is clear that gaugeinvariant quantities formed out of the KM current / (and J) will also be important in the Toda theories. To illustrate this, we recall how the conformal invariance of the Toda theory appears in the WZNW framework. Here we restrict ourselves to the left-moving sector. It can be shown that there is a unique Virasoro algebra in the semidirect product formed by the KM algebra and its associated Sugawara Virasoro algebra, weakly commuting with the constraints (2.8). Since the residual gauge transformations are generated by these constraints, the energy-momentum density L = L'-Tr{J'p),

where U = ±-Tr{J2),

p=\

£

Ha

(2.25)

giving rise to this Virasoro algebra, becomes gauge invariant on the constraint-sur­ face. It follows that L must generate the conformal symmetry of the constrained WZNW, i.e., of Toda theory. (One can verify that, after imposing (2.8) and using the local coordinates defined by the Gauss decomposition (2.9), L indeed reduces to the improved energy-momentum tensor 0 + + (2.4).) Note that p in (2.25) has the property [p, £,] = £„,

when

aeJ,

(2.26)

123 88

BALOG ET AL.

and that the classical centre of the (Toda) Virasoro algebra is c=-12fcTr(p 2 ),

(2.27)

where k is the level of the underlying KM algebra. We will see in Section III that, besides L, there are other gauge-invariant polyno­ mial quantities formed out of the constrained KM current and its derivatives. These objects will be referred to as gauge-invariant differential polynomials. A crucial property (which we elaborate on in Section III) of the gauge-invariant differential polynomials is that they form a closed algebra under the KM Poisson bracket operation. That is, the Poisson bracket of two gauge-invariant differential polynomials is again expressible in terms of gauge-invariant differential polynomials and ^-distributions. This means that if the quantities W' form a basis in the ring of gauge-invariant differential polynomials then we have \W\x\ W^y)}=ldP'i(W)6{k)(xl-yl),

(2.28)

where the PiJk are polynomials of the W's and their derivatives. These Poisson bracket relations generate a non-linear algebra, reminiscent of a universal envelop­ ing algebra. This non-linear algebra of the gauge-invariant differential polynomials always contains the Virasoro algebra, hence it is a polynomial extension of it. This way one associates an extended conformal algebra to every Kac-Moody algebra based on maximally non-compact simple real Lie algebras, for any level k. It turns out that this polynomial algebra is always finitely generated, by / = rank (3?) elements. In the literature these algebras are referred to as classical 1^-algebras. The quantum analogues of these Poisson bracket algebras play an important role in conformal field theory [4-9]. It has recently been realized [5-6] that quantum T^-algebras can be constructed by quantizing the so-called second Gelfand-Dickey Poisson bracket algebra of pseudo-differential operators, which has been studied earlier in the theory of integrable systems and is known to be isomorphic to the algebra of gauge-invariant differential polynomials [10] mentioned above. It is worth noting that the differential operators which provide the bridge between the original Gelfand-Dickey construction and the KM approach to T^-algebras [10] (also constructed by an independent reasoning in [8]) appear naturally in our framework. They are nothing but the operators defining the differential equations satisfied by those (last row) components of gL which are invariant under left gauge transformations. These differential equations can be obtained as a consequence of the obvious relation (Kd+-J)gL

= 0,

(2.29)

where (2.29) is taken in the defining representation of the corresponding maximally non-compact real Lie algebra & (see Section V for more details).

124

TODA THEORY AND T^-ALGEBRA

89

In their review paper [10] Drinfeld and Sokolov studied the algebra of gaugeinvariant differential polynomials by making use of the constrained KM algebra. We shall see that exploiting the full (unconstrained) embedding KM algebra yields further insight into the structure of classical ^-algebras and leads to new results.

III. THE T^-ALGEBRA

In this section we undertake a detailed analysis of the ^-algebra introduced in Section II. We first make the definition of the 1^-algebra more explicit. The basic objects we are dealing with are gauge-invariant differential polynomials, W\ defined on the space P of the constrained KM currents (i.e., currents J satisfying (2.8)). The Poisson brackets of the Wl are obtained by first extending their domain to the whole KM phase space, K, computing the Poisson brackets on K and then restricting to P. The Poisson brackets on K depend on the chosen extension of the W"s (denoted by W), but their restrictions to P, which are again gauge invariant, do not. This follows by using the standard properties of the Poisson bracket from the first class nature of the constraints, and from the fact that the W"s are invariant under the gauge transformations generated by the constraints (and from the assumption that the W are real analytic in a neighbourhood of P). There is no reason to expect that a generic extension of the W's closes under the Poisson bracket on K, but there is a trivial extension, which has the property that the Poisson brackets of the Wl,s not only close but have the same formal structure on K as on P, i.e., {W\x), W(y)} = £ P i ( W ) S ^ i x 1

-yl),

k

where the P'fs are the "structure differential polynomials" (2.28) of the 1^-algebra. This particular extension is constructed as follows. First one expands the general KM current JeK in the Cartan-Weyl basis and notes that in P the upper trian­ gular and Cartan components vary freely, while the lower triangular components are completely fixed by (2.8). The trivial extension W of W is then defined to be the one which simply does not depend on the lower triangular current components. Every element W of the ^-algebra generates canonical transformations on the KM phase space by the formula J-+J + d*J,

S*J= -C" dxl

a(x){W(x),J},

where ty(x) is any extension and a(x) is an arbitrary test function. (Note that our equal-time Poisson brackets and spatial S's are in fact equivalent to light-cone brackets and £'s. Prime everywhere means, even on <5's, "twice spatial-derivative"

125

90

BALOG ET AL.

and this reduces to d+ on quantities, J(x), W'(x), and our test functions, which depend on x = (x°, xl) through x + only.) Since the transformation S&- is canonical (preserves the KM-structure and hence the co-adjoint orbits in K), it follows that it can be represented as a field dependent KM transformation, i.e., Sn,J = dRJ=[R,J~\

+ KR\

where R(J) is some (/-dependent) element of the KM algebra. The transformation da- transforms P into itself, and in fact induces a transformation 8%, on the space M of the gauge-orbits in P. The transformations 3$, corresponding to different extensions ft'of W^ differ on P only by (field dependent) gauge transformations, and thus the induced transformation 5^ does not depend on the extension (only on W). Of course, the reduced phase space M carries its own Poisson bracket structure which is inherited from the Poisson bracket structure of K, and is described by the standard Dirac bracket formula if one parametrizes M with some section of the gauge orbits in P (gauge choice). The induced ^-transformations 5%, are canonical transformations on M with respect to this induced (Dirac) Poisson bracket. In Subsection III. 1 we introduce some convenient gauges (called DS gauges), which will be used to show that the i^-algebra has a finite (/-dimensional) basis and to exhibit some particular bases W' (/= 1 •••/). The particular iF-generators W will be the gauge-invariant extensions (from the gauge section to P) of those current components (called DS currents) which survive the gauge fixing. Thus, in these gauges the ^-algebra appears as the Dirac bracket algebra of the DS currents. This is the basic fact on which most of our results are based. In Subsection III.2 we exhibit a conformal field basis of the ^"-algebra. In Subsection III.3, working in a DS gauge, we shall present an algorithm for finding the field dependent KM transformations which implement the induced iT-transformations <5£,. This algorithm is our main result since it enables us to calculate the action of the iST-algebra on any gauge-invariant quantity. In the last section we deal with some particular gauges which facilitate the study of some properties of the i^-algebra. III. 1. Drinfeld-Sokolov Gauges In this section we recall the construction of a class of particularly convenient gauges in which the properties of the ^-algebra become apparent. This class of gauges has been introduced first by Drinfeld and Sokolov [10], so we call them DS gauges. First we consider a special 5/(2, R) subalgebra of &, y, which will play an impor­ tant role in what follows. This subalgebra is spanned by the Cartan element p in (2.25) and nilpotent generators I± such that [ / + , / - ] = 2p,

[p,/±] = ±/±.

(3.1)

126 TODA THEORY AND iF-ALGEBRA

91

The step operators are explicitly given by ' - = I *.£-,.

I+=l^Eai,

(3.2a)

where X, = \KH'\OL,\\

«, = 2 £ ( * " ' ) , .

(3.2b)

Note that since Tr(/_ Eai) = Kfi' any element of P, i.e., any current fulfilling the con­ straints (2.8) (with nai = ii'), has the form /(*) = / _ + £ 0"(*)tf„+

I

CWiv

(3.3)

The adjoint representation of # decomposes into if multiplets. Since p is an element of the Cartan subalgebra of If the step operators are p-eigenstates, rj5,£J = />(<*>)£*.

(3.4a)

where A(
>f <*>= £"»/<*/•

j-i

(3.4b)

/-i

Let *» be the eigensubspace of p of eigenvalue A. If A # 0, then dim 95k = number of roots of height A.

(3.5)

It can be shown [18, 10] that, if for 1
«A = dim^-dim^ + 1

(ln* = A

(3.6)

is not zero, then A is an exponent of <§ with multiplicity nh. We recall the meaning of the exponents and their multiplicities [18]: The ring of group-invariant polynomial functions on # is generated by / homogeneous elements whose degrees are determined by the exponents, A. More precisely, there are exactly nh independent generators of order A + 1. In other words, these generators define a complete set of independent Casimir operators. We note that A = 1 and A = h^ are always exponents. The multiplicity of the exponents is always 1, except for Dv, where there are two independent Casimirs of order 21. Note that for (A > — 1) /_ maps 9h + , into \ injectively, that is dim/_(9 ! , + 1) = dim9!A+1,

(3.7)

127

92

BALOG ET AL.

where I_(% + 1)= [/_,%+\\ For any exponent, h, let Vh be a linear complement of I_(% + 1) in 9h (dim KA = /i/,) and let us also introduce the direct sum V=®

Vh

(dim^=/).

(3.8)

We choose a basis F( (i = 1,..., /) in V in such a way that [p,/-,]=/«,./•,

(3.9a)

l=/»is$/i 2 < ••• *SA, = /J*

(3.9b)

holds, where

is the list of the exponents with possible multiplicities included (see Appendix A). The basic fact we need is that any constrained current of the form (3.3) can be uniquely gauge transformed into a current J(x) of the form + KA'(X) A~l{x) = /_ + £ Wl{x)Flt

J(x) = A{x) J{x) A-\x)

(3.10)

I— 1

and that the W(x) and the parameters av(x) of the gauge transformation ,!(*) = expT £

a*(x)E9]

are differential polynomials in the components of J(x). The proof of this statement [10] is actually easy. Using the fact that the gauge transformations are generated by upper triangular matrices, the inspection of (3.10) reveals that it is uniquely soluble in purely algebraic steps for both W(x) and av(x) in terms of J(x). Denote now by Mv the space, whose "points" are currents of the form (3.10). The previous statement tells us that Mv defines a complete gauge fixing. Moreover, it also follows immediately that the components, W'(x), of the unique intersection point of My with the gauge orbit passing through JeP define gauge-invariant dif­ ferential polynomials on P, which freely generate the "^-algebra. In other words, the ff"s form a basis in the algebra of gauge-invariant differential polynomials. On the other hand, a completely general element of the KM algebra K can be expanded as / ( * ) = £ (/'(x)f,+

£

r9(*)E-.+

I

^(x)U_,Ev-\

(3.11)

and My is obtained by first constraining the £~v(x) by imposing (2.8) and then also fixing the residual gauge freedom by setting the fix) to zero. The current components, U'(x), which are not affected by this two step restriction and the corresponding gauge-invariant differential polynomials, W(x), are related by U'(x\Mv = W'(x)lMy.

(3.12)

128

TODA THEORY AND ^"-ALGEBRA

93

However, it should be stressed that conceptually the U'(x) (linear functions on K) and the W'(x) (gauge-invariant differential polynomials on P) are very different objects and must be carefully distinguished. To make this distinction even clearer we introduce a separate name for the £/'. From now on we shall refer to them as DS currents. It will turn out that most of our results are a consequence of (3.12). For example, this relation immediately implies that each differential polynomial W(x) contains a leading term, i.e., a term without derivatives. In Subsection IV. 1 we shall prove that the leading terms of any i^-basis are obtained by restricting Casimirs from K to P. Now we discuss how the ^-algebra appears in a DS gauge. Clearly Mv inherits a Poisson bracket structure from the embedding KM algebra. This induced Poisson bracket structure is given by the familiar Dirac bracket formula [19]

{/, g}* = {/. g} ~ Z

f2" f d*1 dyl{f, i'(x)}

xDafi(x,y){t<>(y),g},

(3.13)

which is valid for two arbitrary phase space functions (/and g are functions on the KM phase space but only their restriction to Mv really matters). In this formula the £" are the current components to be constrained (cf. (3.11)), and D„p(x, y) is the inverse of C*{x,y)={Z'{x),t>{y)},

(3.14)

which satisfies £

P dylC*(x,y)D»(y,z)

= 8;5(xl-zl),

(3.15)

for arbitrary a, ye + ) is regular on Mv. Hence C"p is also regular on Mv.) Now the DS currents, U'(x), which survive the complete gauge fixing provide us with coordinates for the phase space M v. Thus the induced Poisson structure of My can be described by specifying the Dirac brackets of the U'(x). The crucial point is that the Dirac brackets of the DS currents satisfy {U\x),Ui(y)}*

= {W\x),WJ(y)}

on Mv,

(3.16)

as a consequence of (3.12). As discussed earlier the Poisson brackets of the W"s are in principle calculated by first extending them to K and then restricting the Poisson brackets calculated on K to P. Because of the gauge in variance of the W"s, this is equivalent to calculating the Dirac brackets of the DS currents. To summarize, we see that if the space of gauge orbits M is parametrized by the

129

94

BALOG ET AL.

gauge section My, then its Poisson bracket structure is naturally described by means of the Dirac brackets of the DS currents, and that the i^-algebra can in fact be regarded as the Dirac bracket algebra of the DS currents. It will be demonstrated in the rest of this section that the properties of the T^-algebra are most effectively studied by making use of the DS gauges. The family of DS gauges is parametrized by the possible choices of the linear space V in (3.8). It is easy to see that Ui()x)~ L(x) on Mv and therefore Wl(x)~ L(x) on the constraint-surface P, for any DS gauge. III.2. Conformal if-Generators The energy-momentum density of the Toda theory, L in (2.25), generates the action of the conformal group on the KM phase space. This conformal action operates as J->J + SLJ f2*

SLJ= -

J

(3-17) dxla(x){L(x),J}

= (aJ)' + Ka"p + a'\_p,J],

o

where JeK and a(x) is any test function. The main point of this section is the observation that the ^-generators associated to a certain DS gauge (highest weight gauge) are primary fields with respect to this conformal action. To demonstrate this it will be useful to describe the conformal action in terms of field dependent KM transformations. Let R(J) be a KM algebra valued function defined on the KM phase space. Then it generates an infinitesimal (field dependent) KM transformation: J^J

+ 5„J,

dRJ=s[R,J~\ + KR'.

(3.18)

Now it is not difficult to verify that the conformal action SL is implemented by the field dependent KM transformation generated by the particular KM valued function R0(a,J) = -aJ + a'p,

(3.19a)

that is one has SLJ^S^J

for any J.

(3.19b)

The conformal action (3.17) transforms the set of constrained KM currents, P, into itself. Another crucial property of 5 L is that on P it commutes (modulo gauge transformations) with the action of the gauge transformations (2.22). Therefore (3.17) induces a conformal action on the gauge equivalence classes of the con­ strained currents, which amounts to an action on the set of gauge fixed currents,

130

TODA THEORY AND T^-ALGEBRA

95

My, representing those equivalence classes, for any choice of V. Our purpose below is to describe this induced conformal action J-+J + 6V

(JeMy)

(3.20)

operating on M v. In general J + SLJ$My, and therefore to determine 8fJ we must find the compensating (unique) gauge transformation, r = r(a, J), such that J + 5LJ + 8rJeMv,

for any JsMv,

(3.21a)

and then we have dtJ = SLJ + d,J = 5RJ

with R = R(a,J) = R0(a,J) + r(a,J).

(3.21b)

Before trying to determine r(a, J) let us recall that Sf is a canonical transforma­ tion on the reduced phase space My, generated by L by means of the Dirac bracket, &*LJ= - C" dx1 a(x){L(x), J}* = - C" dxla(x){U1(x),J}*

(3.21c)

on My. Here the second equality holds provided we normalize the DS current Ul according to Ul(x) = L(x)

on My,

(3.22a)

which corresponds to the normalization of the basis vector Fy, TTF1I_=K.

(3.22b)

With this normalization, as an obvious consequence of (3.16) and (3.21c), we have SfU1 =a(U1)' + 2a'Ul -KTr{p2)a'".

(3.23)

Next we want to determine the induced conformal transformation of the U' for / > 2. First, for an arbitrary gauge fixed current /(*) = / _ + £ V\x)Ft

(3.24)

i— i

one easily sees that 3 « / = I [a(t/')'+ (/>, + !)*'C/']F,. + Ka"p,

(3.25)

i- 1

where A, is the height of the Lie algebra element F, according to (3.9a). The last term is "out of gauge" so one indeed needs a "compensating" gauge transformation.

131

96

BALOG ET AL.

In principle it is a purely algebraic problem to find r(a, J), but in practice it is quite hard to produce an explicit formula for the solution in an arbitrary DS gauge for an arbitrary Lie algebra. However, one can find a special gauge in which the form of r(a,J) is particularly simple and the DS currents are primary with respect to the induced conformal action (3.21). The construction is based on the sl(2, R) subalgebra if introduced in the previous section. Since the adjoint representation of ^ decomposes into if mul­ tiples, it is natural to consider the corresponding highest weight states, i.e., those Lie algebra elements which commute with / + . It is easy to see that the highest weight states in % span a natural complement of / _ ( ^ + ,). Choosing this par­ ticular complement in the construction presented in Subsection III.l we obtain a particular DS gauge, which we call the highest weight gauge. By using the fact that the basis vectors F, of V in (3.8) now satisfy the condition F,~/+,

[/+,F,]=0,

i=2,...,/,

(3.26a)

one easily proves that in the case of the highest weight gauge the compensating gauge transformation r(a, J) is given by the simple formula r(a,7)=-|icfl*/+.

(3.26b)

The corresponding conformal variation of the DS currents U' then turns out to be <5J£/' = a(t/')' + (/>, + l ) a ' ^ '

for

i = 2,...,/,

(3.27)

i.e., they are indeed primary with respect to the induced conformal action (3.21). Equivalently, one can say that the corresponding gauge-invariant differential poly­ nomials, W\ are primary with respect to the original conformal action (3.17) (restricted to P). The conformal weights of the W's (t/"s) are (/i, + 1), i.e., they are in one-to-one correspondence with the orders of the independent Casimirs of 'S. To summarize, we have proven that the generators W (i = 2,..., /) defined by the highest weight gauge, together with L=Wl, constitute a natural, conformal field basis of the ^-algebra. This is one of our main results. As far as we know, an algo­ rithm to find a conformal T^-basis has not been known before in the general case, although conformal ^-generators were explicitly exhibited for some particular low dimensional examples [6]. We now illustrate the idea of both the DS and the highest weight gauges on the example of B2 = o(3, 2). We use the convention [20] in which this Lie algebra con­ sists of (5 x 5) matrices which are antisymmetric under reflection with respect to the "second diagonal." The Cartan subalgebra is spanned by the diagonal matrices in B2- In this convention the Lie algebras of N and N are represented by upper and lower triangular matrices, respectively. In particular, the E% for a e A have non-zero entries only in the first slanted row above the diagonal. The Cartan element p in (2.25) is then easily found to be p = diag(2, 1,0, - 1 , - 2 ) .

(3.28a)

132

TODA THEORY AND Ti^-ALGEBRA

97

By a convenient choice of the parameters T, in (3.2) we can choose the step operators of y as '0 4 0 0 0^ 0 0 6 0 0 /+ = | 0 0 0 - 6 0 |, 0 0 0 0-4 ^0 0 0 0 0 /

/O 0

0 0 0 0 /_ -= |0 01 1 0 0 loo - 1 0 \0 0 0 -1

io

0 0 0 |. 0 0

(3.28b)

(Note that the value of the parameters T, is irrelevant since they can be redefined by rescaling the simple step-operators.) The elements of % are now those matrices in B2 that have non-zero entries h steps above the diagonal only. Before describing the general DS gauge, we need to know the image /_(#2)- In fact, an easy calcula­ tion yields that /_(#j) is the set of matrices of the form 0 x 0 0 0 -x 0 0 0 0 0 0 ^ 0 0 0

0 0 x 0 0

0 0 0

(3.29)

—x

0

Since dim 9j = 2, there is now a one parameter family of (one dimensional) linear sub-spaces Vx of #, which are complementary to /_(# 2 ) m ^i- These are nothing but the "lines" spanned by the vectors of the form /0 0 Ft = Fl(p) = 0 0 \ 0

p 0 0 0 K-p 0 0 0 p-K 0 0 0 0 0 0

0 0 0 |, -p 0

(3.30)

for any real p. Note that F, has been normalized according to (3.22b). (For the B, algebras Tr means half of ordinary matrix trace in the defining representation.) The general current in the "DS gauge of parameter p" is written as

J(x) = /_ + {/'(x)Fl + U2(x)F2=\

0 1 0 0 0

pUl 0 u2 l 0 qU 0 1 0 -qU> -qU1 0 -1 0 0 0 -1

0

-u2 0 l

-Pu

0

,

(3.31)

133

98

BALOG ET AL.

where q = K — p. We designate this set of gauge fixed currents as Mp. Observe that for Sp = 2K the matrix F, is proportional to / + , so that this value of p corresponds to the highest weight gauge. It is not hard to calculate the compensating gauge transformation r(a, J) in (3.21) which cancels the last term in (3.25). The reader can check that the result is

(

0 0 y2 y,

0 \

0 0 0 0 -yA 0 0 0 0 -y2 , 0 0 0 0 0 / 0 0 0 0 0 /

(3.32a)

which reduces to (3.26b) in the case of the highest weight gauge, as it should. The with m corresponding yconformal variation of U2 onilMp is given y3 = X[ica""-a"U A = by K(2K - 5/>), (3.32b) 2 = Xa , which reduces to (3.26b) in the case of the highest weight gauge, as it should. The corresponding conformal variation of U2 on Mp is given by b*LU2 = 4a'U2 + a(U2)' - A((K - p) a'"Ul + K(a"U1)' - K2a'""). (3.33) Since U' generates the induced conformal action on Mp through the Dirac bracket, from (3.33), taking (3.16) also into account, we can read off the Poisson bracket of Wl with W2 (restricted to P), which is now given as { W\x), W2(y)} = iW2(x)' S(xl - yl) + AW\x)

S'(xl - yl)

+ X((p- K)( Wl5)'" - K( WlS')" + K2b'"").

(3.34)

For A = 0, that is for the highest weight gauge, the corresponding ^-generator, W2, is a conformal primary field of weight 4. The generator W2 = W2p) associated to any other DS gauge (of parameter p) transforms in a complicated, inhomogeneous manner under the conformal action. III.3. KM Implementation of

if-Transformations

Here our purpose is to study the canonical transformations defined (as discussed at the beginning of the section) by the liT-algebra on the space of gauge orbits M. For this we consider the transformation 8%, induced on M by the following ^-transformation d# (acting originally on K), J^J+5it,J,

5*J= - £ f " dxl aMifr'lx), J},

(3.35)

where the W(x) are some arbitrary extensions from P to K of the "^-generators W(x) associated to some DS gauge with gauge section My, and the at{x) are

134 TODA THEORY AND Ti^-ALGEBRA

99

arbitrary test functions. We parametrize M by Mv and in this parametnzation the transformation 5$, is generated by means of the Dirac bracket according to J^J+5*WJ,

6U=-{Q(a,J),J}*,

(3.36a)

with Q(a, J) = £ C" dx1 a,(x) U\x),

(3.36b)

where the U'(x) are the corresponding DS currents. Similar to the special case of the induced conformal transformation <5* discussed in the preceeding section, the induced ^-transformation Sfy can be implemented by some field dependent KM transformation R(a, J). Of course, this KM implementation is in principle possible in any gauge, but here we show that in the DS gauges there exists a simple, effective algorithm for actually computing the KM valued function R(a, J) which implements <$&< '-e-> which satisfies 8%.J=SKJ

for any JeMy.

(3.37)

This is immediately translated into the action of the i^-algebra on itself, since in the DS gauge the W(x) reduce to the current components U'(x). An extra bonus of the KM implementation is that the KM algebra acts also on the G-valued WZNW field g(x+, x~) and from that action we get S*wg = S*g,

that is,

{Q(a,J),g}*=-R(a,J)g,

(3.38)

where "dot" means ordinary matrix product. From this equation we can read off the action of the ^-algebra on the Toda fields, which are the sub-determinants off. In order to make the presentation more concrete, we consider as examples the ^-algebras of the rank 2 Lie algebras A2, B2, and G2. The A2 example, which is the simplest non-trivial case, is included for the purpose of illustration. The B2 example has some non-trivial features which will motivate some developments in subsequent subsections. Finally, G2 (in Appendix B) illustrates the power of the method, since it enables us to compute the very non-trivial structure polynomials of this iT-algebra. We start by presenting a general characterization of the tangential (gauge pre­ serving) KM transformations for an arbitrary DS gauge. First we pick a point, J0eMy, and consider the tangential KM transformations at JQ. In other words, we want to describe all elements R{J0) of the KM algebra, which map J0eMy into My, i.e., we want to solve the condition that JO + SKJ0 = J0+\.^JO1

+ KR'

is in My.

(3.39)

135 100

BALOG ET AL.

To give the general solution of this condition, it turns out to be useful to supple­ ment the decompositions introduced in Subsection III. 1, $=r*©/-(9U,),

h>l

(3.40)

by similar ones for the subspaces <S_h of $ corresponding to the negative roots (cf. (3.4)). Indeed, the decomposition we consider is induced by (3.40) as ^_A=F_„

©[/_„,

for h>\,

(3.41)

where V_h is the transpose of Vh, V.h={v'\v6V„},

(3.42a)

and U_h is the annihilator of Vh in cS_h with respect to the scalar product Tr: U_„={ue^_h\Tvuv

= 0, VveV„}.

(3.42b)

(The transpose in (3.42a) can be defined abstractly by means of the Cartan-Weyl basis as E'v = E_v, H'v = Hv, but in convenient conventions [20] it is the ordinary matrix transpose.) Having introduced the necessary definitions we now return to the study of (3.39) and decompose the quantities entering this condition as R(x)= I

(u_„(x) + v_h(x)) + £ yh(x),

(3.43a)

where u_h, v_h, and yh are in the subspaces U_h, V_h, and %, respectively, and Jo(x) = I-+Z

v°{x),

5RJ0(x)= £ vh(x),

(3.43b)

where both v% and vh must be in Vh. By analysing Eq. (3.39), one finds that if / 0 and all the v_h(x) are given, then the remaining components of R are uniquely determined differential polynomials in terms of these. Furthermore, it follows that the components of SRJ0 are differential polynomials of J0 and v_h as well. In fact, the differential polynomials, R and SRJ0, are linear in v_h(x), but in general non­ linear in J0. The above result provides us with a complete characterization of the tangential KM transformations at the arbitrarily chosen gauge-fixed current J0. To actually prove this, one has to consider Eq. (3.39) height by height, starting from below, and use the following two properties of our Lie algebra decomposition: First, for A>0, /_ maps 9h into ^,_, in a one-to-one manner and this map is in fact onto for those h which are not exponents. Second, for 1 ^h^(h^, — 1), /_ maps U_h onto #_*_ i also in a one-to-one manner. By using these properties of /_, it is not difficult to verify that condition (3.39)

136 TODA THEORY AND TiT-ALGEBRA

101

is indeed uniquely soluble for R(J0) and 8RJ0 by purely algebraic means at every height, once / 0 and the v_h are given, and that the solution is linear in v_h. Let us choose a basis {F~'} in 0 A V_h dual to the basis {Ft} in 0 * Vh F-^-^—r

(3.44a)

Tr F,F't Since we assume that F, e Vhj, the duality property (which we shall need later on) Since we assume that F, e Vhj, the duality property (which we shall need later on) Tr F,F-J=>6J„ (3.44b) is automatic in almost all cases, i.e., for those basis vectors which correspond to exponents A, #/», of multiplicity 1, and we can also ensure this by a choice in the case of those two basis vectors which correspond to that exceptional exponent h = (21 — 1) of D2i whose multiplicity is 2. Using this basis, we can now write R in (3.43) as R = R(a,J0)= £ a,(*)F-'+ £ u_h(x)+ £ yh(x), i-1

»>l

(3.45)

»J0

where the a,(;c) are arbitrary functions and the u_h and yh are differential polyno­ mials linear in the a„ but not necessarily in J0. It is important to emphasize that, since J0 was arbitrary in the construction, this equation defines an element R(a, J) of the KM algebra for any J and a,. According to its construction, at any fixed JeMy this KM valued function R(a, J) provides a parametrization of the set of tangential KM transformations at J, by the / arbitrary real functions a,(x). Hence it is clear that by varying J and at the same time promoting the parameters a, to functional of J one can write in the form R(a(J), J) the most general field dependent, gauge-preserving KM transformation on Mv. So, in particular, the field dependent KM transformation implementing the induced ^-transformation <5£, (3.37) can also be written in this form with some functionals a,(J). The result we prove is that the above constructed KM valued function R(a, J) when considered for fixed (/-independent) a, and varying J is the one which implements the induced lif-transformation 5%, according to (3.37) and (3.38). This result means that we in effect replaced the task of finding the inverse of the matrix Cap(x, y) (3.14) which enters the standard formula (3.13) of the Dirac bracket, by the much easier (as will be clear from the examples) task of solving Eq. (3.39). To justify our claim we now show that (S*f)(Jo)={f,Q}*(Jo)

(3.46)

holds for an arbitrary real function f(J), where R = R(a, J) is given by the above construction, Q = Q(a, J) is the moment of the DS currents defined in (3.36b) and

137

102

BALOG ET AL.

J0eMy is arbitrary. To accomplish this we first recall that any element R0 of the KM algebra defines a particular (field independent) KM transformation S^ on the full KM phase space K, which is an (infinitesimal) canonical transformation generated by means of the KM Poisson bracket by the function Q0(J) = C" dxl Tr R0(x) J(x). J o

(3.47)

This means that the relation SRoF(J)={F,Q0}(J)

(3.48)

is satisfied on the full KM phase space K, for any real function F(J). The trick is that now we take R0 to be R(a, J0) in (3.45) for fixed J0 and a. In this case we know that at J0 the variation 5,^ respects the constraints defining Mv (R(a,J0) was constructed by requiring this) and therefore at J0 the constraint-contributions drop out from the Dirac-bracket of Q0 (3.47) with any quantity. This way we derive {F,QoWo)={F,Qo}*(Jo).

(3.49)

On the other hand, it is easy to see that for R0 = R(a, J0) the functions Q0(J) (3.47) and Q(a, J) (3.36b) differ on Mv only by a constant. This implies that they can be interchanged on Mv under Dirac-bracket. Taking this into account we immediately obtain (3.46) by combining (3.48) and (3.49) and by taking FiMy=f^Mv. This finishes the proof. We now illustrate on the simplest non-trivial example, A2 = sl(3, R), how to calculate the iT-algebra by our algorithm. The TT"(/42)-algebra is well known but it is worth reconsidering it in the present framework as an illustration. We use again the conventions of [20]. The Cartan element of the speical 5/(2, R) is represented by p = diag( 1 , 0 , - 1 ) .

(3.50a)

Choosing T, = T2 = 1 in (3.2) the remaining generators of Zf are given by

/+ =

/0 2 0\ 0 0 2 \0 0 0/

and

/_ =

/0 0 0\ 10 0 . \0 1 0/

(3.50b)

As in the B2 example, there is a one parameter family of DS gauges, and the gauge fixed current in the "DS gauge of parameter p" is written as J(x) = I_ + Ul(x)Fl + U2(x)F2,

(3.51)

138 TODA THEORY AND "W-ALGEBRA

103

where we can take (Op 0 Fl = Fi(p) = 0 0 K-p | \0 0 0

and

F2 = | 0 0 0 ) .

(3.52)

Here F, is normalized according to (3.22b). The highest weight gauge corresponds to 2p = K, but here we choose to work in the "Wronskian gauge" p = K, which is the gauge usually considered in the literature [5-9] (the origin of the name "Wronskian gauge" will become clear in Section V). Our aim is to find the explicit form of R(a, J) in (3.45) in the Wronskian gauge. In this case 0 001 F-' = (1/K00|

F-2 = | 0 0 0 | ,

and

(3.53)

0 0 0, and [/_, in (3.42b) now consists of matrices for which only a32 is non-zero, while U_2 is trivial. The explicit form of R in (3.45) reads then as yo

^2 N

yi

R = R(at, a2, J) = | a,/ie (y0 - y0)

Pi |,

(3.54)

where the at are arbitrary functions and the other entries are to be determined by the condition that the variation 5RJ must leave J "form invariant." In our case this means that SRJ must be of the form SRJ=[R,J]

'0 K5U1 6U2S + KR' = \ 0 0 01, ,0

0

(3.55)

0

since in the Wronskian gauge

(3.56)

As is follows from our general result, substituting (3.54) and (3.56) into (3.55) one obtains a system of equations which is uniquely soluble in purely algebraic steps for both the component functions u_t ■■ -y2 of R and for the corresponding variation

139

104

BALOG ET AL.

of J. One has to consider (3.55) height by height, starting from below, and easily obtains the following formulae for the components of R(a, J): a, «_,=

, , yi=aiUl+a2Uz-Ky'0,

Ka'2,

yo = a\+j\_a2Ul-Kan,

y{ = a2U2-Ky'0,

y0 = 2y0-a\,

y2 = — U2 + Ky\.

(3.57)

K

Before proceeding let us note that R (a2 = 0) implements the induced conformal action in the Wronskian gauge, and in fact one can rewrite the above formula as J?(fl„a2 = 0,y) = r i a 1 y +

fli^l-icriflr/++KflrF2l,

(3.58)

which is consistent with (3.21) and (3.19a) describing the conformal action in general. The variation of J under the KM transformation 8R is found to be SUl = la^U1)' + la\ Ul - 2Ka?~\ + l2a2{U2)' + 3o2 U2 - K2(a2 Ul)" + K3a'2"'J

(3.59)

and 5U2 = [a,(f/ 2 )' + 3a\ U2 + K2a'[Ux -K3a'{"~\ + a2\_K2(U2)" + ]K3U1(U1)'

- §K4(£/')"']

+ a i C f ^ t / 1 ) 2 + 2K2(U2)' - 2K\U1)"~\ -2K4a'2,(Uly-^KAa'2"Ul + lKsa2"".

(3.60)

Now by combining Eqs. (3.36) and (3.37), it follows that SU'(x)= S

F dylaj(y){U'ix\

U\y)}*

(3.61)

>-"T,2 •*<>

holds, so from (3.59) and (3.60) one can read off the Dirac brackets of the DS currents, yielding immediately the Poisson brackets of Wl and W2 according to (3.16). (See Subsection IV. 1.) Observe that the W2 generator associated to the Wronskian gauge is not a primary field with respect to Wx = L. However, it is easy to see that the combina­ tion fV2-j(W1)'

(3.62)

140

TODA THEORY AND "^-ALGEBRA

105

defines a primary field of weight 3. By investigating the transformation rules between the 1^-bases corresponding to different DS gauges one can prove that (3.62) is precisely the 1^-generator associated to the highest weight gauge. Note also that in this example the components of R{a, J) in (3.57) are only linear functions of the current components, and as a consequence 5RJ is at most quad­ ratic in J, which implies that the Poisson brackets of the "^-generators are also (at most) quadratic polynomials. This is not always the case, as can be seen, e.g., in the example of B2. We now illustrate the action of the T^-generators on the components of the matrix-valued field g(x+, x ~) on this example. All we have to do is to use the results (3.57) for (3.54) and substitute this R{a, J) into (3.38). Let us discuss the conformal transformations first. For this case, we find *i S i / - * ! S u + («i U'-Ka'i)g2i

+ (^

U2-K2a'A g3l

(3.63a)

&ig2i = — gu-Ka"gji

(3.63b)

8ig3i = — g2t-a\gu-

(3.63c)

K

To simplify (3.63) we can make use of the relation between the currents and the matrix-valued fields, (2.6). In this example this gives g2t = Ktyi,

gu = K2d2il/i,

(3.64)

where ^, = £ 3 , and d = d/dx + . Equation (2.6) also gives a differential equation satisfied by i^, (see Section V), which we will not explicitly use here. Using (3.64), (3.63c) simplifies to ^ , = 0,3*,-a',*,,

(3.65)

which tells us that i/f, is a primary field with conformal spin — 1 , whereas the remaining equations in (3.63) describe the conformal transformation properties of the secondary fields (3.64). We now turn to the genuine "^-transformation generated by U2. Using (3.64) again, we find d2il,i = a2^K282-^Ui^i-K2a'2dipi

+^ a ^

i

.

(3.66)

Equation (3.66) can be thought of as the transformation rule for a "TiT-primary" field under the W^transformation (for the A2 "^-algebra). For the algebra B2 we have derived the conformal action in Subsection III.2. Thus it only remains to determine the canonical transformation generated by W2

141 106

BALOG ET AL.

to know the complete set of transformations generated by the iST-algebra in this case, from which we can of course again (as for A2) read off the iST-relations them­ selves. By applying the algorithm presented above one finds after lengthy but straightforward calculations that

{U2(x), U2(y)}* = \ Z VF2i+l(x) + F2i+i(yn x5(2i+i\xx-yl)-KsP5(1\xl-yl)

(3.67)

on Mp, where &l = Q\(U2)" + Q[UlU2 + Q\(W)m + Q\U\U1)", + Ql((uiy)2 + Ql6(ul)\ F3 = QlU2 + Ql(Ul)" + Ql(U1)2, P = p2 + {p-q)2,

and

P5 = Q5Ul.

(3.68)

q = K-p.

Here Q5, Q{ are polynomials of the parameter p, given explicitly as Q\=-2K2P, 2

Q2 = SP2-\6KP

Q\ = 2K (3p-q),

4

Q\=-2K [2P

+ 4K2,

Q\ = 2K\P+2pq-]

+ 3pql,

Ql6 = 2(q + K) pq2

(3.69a)

Ql=-K(q + K)2P-2K2(2q + K)pq Q5 = 2K3t(q + K)P+Kpq] Q\ = 2tc2(q + K)P + 2K2(q + 2K) pq

(3.69b)

Q\ = K\_3q2 + Anq + 2K 2 ]P + 2>c2{q + 2K) pq. Observe that unlike for the A2-mode\, there is now also a cubic term, (t/ 1 ) 3 in J^1. The coefficient Q\ of this single cubic term vanishes in the special cases when p = 0, K or p = 2K. In other words, the B2 T^-algebra is given by quadratic relations in that li^-bases which are associated to the particular DS gauges of parameter p = 0, K or p = 2K. These "quadratic gauges" could be useful in the quantization of the liT-algebra, since normal ordering is more complicated when the order of the poly­ nomials involved gets larger. In contrast, the conformal properties are hidden in these gauges and are not as transparent as in the highest weight gauge (which belongs to 5/» = 2K in the B2 example). III.4. Other Convenient Gauges In the previous subsections we have discussed the DS type gauges and have shown that choosing a DS gauge naturally leads to a corresponding choice of basis for the niT-algebra, by relating the iT-generators to the non-vanishing current com­ ponents in that gauge. The highest weight gauge plays a particular role because the

142

TODA THEORY AND 1^-ALGEBRA

107

corresponding "^-generators are conformal primary fields (with the exception of the conformal generator W'). In the examples of A2 and B2 (C2) we have shown that it is possible to choose such DS gauges in which the generating relations of the ^-algebra are quadratic. These gauges are also important because the quadratic closure of the algebra simplifies the quantization. In this section we show that such gauges exist for the algebras A,, B,, and C,. We will see in the next section that they are not available for the rest of the Lie algebras. We start by considering A,, i.e., sl(l+ 1, R) and will use the defining representa­ tion. Here (and also for Bh C,, and D, later) we shall use the conventions [20] in which the positive and negative step-operators are upper and lower triangular matrices, respectively, and the elements of the Cartan subalgebra are diagonal matrices. For simplicity, now we choose all T, in (3.2) to be equal to 1, and then the matrix /_ reads /0 0 0 ••• 0 0\ ' 1 0 0 ••• 0 0 /_ =

0 1 0 ■•• 0 0

(3.70)

i 0 0 0 ••• 0 0 , \ 0 0 0 ••• 1 0 / The elements of 1fh are matrices with non-zero entries only in the slanted row h steps above the diagonal. The image / _ ( ^ + 1 ) (for A>0) consists of those matrices in % for which the sum of the matrix elements is zero. Fixing a DS gauge means choosing a single matrix in % for which the sum of the matrix elements is different from zero. The simplest choice yields the " Wronskian" gauge defined by fo u1 ■■■ u'\

vo

o ... o .

This gauge is a special example of the more general block gauges for which

/=/_+, = / _ + Q

(3.72)

and U is a pxq block (p + q = l+l) containing the / DS currents. The "Wronskian" gauge is the special case when p = 1 and q = /. In general these "block" gauges are not unique: we are still free to distribute the DS currents in a number of different ways along the intersections of the slanted rows with the block. Now we are going to show that the TfT-algebra closes quadratically in any of these "block" gauges. According to the results developed in the previous subsection,

143 108

BALOGETAL.

we can derive the #"-relations by determining thefielddependent KM transforma­ tion R(a, J) in (3.45) which implements the induced ^-transformations on Mv. To do this we first rewrite the defining Eq. (3.39) of R(a, J) in the form [/?,/_ ]

(3.73)

+ KR' = SJ+U,RI

Now, since we know that the unique solution of (3.73) for R = R(a,J) and SJ is linear in the infinitesimal parameters of the transformation, i.e., in the functions a, introduced in (3.45), and polynomial in the given gaugefixedcurrent /_ +j, we can expand both R and 8J in powers of the DS currents {j), R = R0 + Rl + R2+ ■■■ <5/=(<5/)o +

(3.74)

(<5./)1 + (<5./)2+ •••

and solve (3.73) perturbatively, lRm,I^

+ KR'm = (SJ)m +[y,/J m _ 1 ],

m = 0,l,2,....

(3.75)

Since in the "block" gauge both J and SJ are upper triangular in the block sense,

If we write out (3.73) in "block" components it is not difficult to see that the first order solution must be of the form

•-G3

(3.77)

where the p x p block A and the q x q block C are further restricted by Api = 0

for i^p-l

and

C,,=0

for / > 2

(3.78) i

and that the second order solution is of the form

-Q

with Z>„ = £>.. = 0,i=1.2,..., p; j = 1, 2,.... q.

(3.79)

For the "block" gauges the expansion stops here and, by the results of Subsec­ tion III.3, this implies that the algebra of the ^-generators corresponding to any DS gauge from the family of block gauges closes quadratically indeed. Note that the "Wronskian" gauge is special since D = 0 in this case and thus the KM transformation R = R(a, j) is only linear in the DS currents. The algebra is still quadratic, since (SJ)2 = [ / ? „ y ] .

(3.80)

144 TODA THEORY AND li^-ALGEBRA

109

For the other matrix algebras, B, and C,, one can define analogous "block" gauges by embedding them into appropriate ^4-type algebras. For Ct~sp(2l,R) we can use the 2/-dimensional defining representation. We write the C, matrices in terms of four Ixl square blocks. In this notation the symplectic metric is given by (3.81) - ( - . : > where the only nonvanishing entries of e are in the second diagonal (the diagonal from bottom-left to top-right), and these entries are all 1. The elements of the Lie algebra are represented by matrices of the form K =

\r

21'

where

B = B>€ =

C

(3.82)

and ~ means reflection with respect to the second diagonal. Positive (negative) step-operators are again upper (lower) triangular matrices and elements of the Cartan subalgebra are diagonal. By a convenient choice of the (irrelevant) parameters T,, /_ is now given by the 2/x2/ matrix: 0 00 1 00 0 10

(3.83)

0 0 0 0 0 0

It has / 1 entries and (/— 1) ( — l)'s. The "block" gauges, in which the algebra closes quadratically are characterized by 7=/

- + (oo)

(184)

where U= U and it has non-vanishing components along every second slanted row, corresponding to the exponents of this algebra. Finally, for Bt~so(l + 1, /) we take the (2/+ 1 )-dimensional vector representa­ tion. In a 3 x 3 block matrix notation corresponding to the partition / + 1 + / the Lorentzian metric is (3.85)

145 110

BALOG ET AL.

and the elements of the Lie-algebra are of the form K=\ Y'

0

-X'\,

where S=-B,

C=-C.

(3.86)

The matrix /_ is again similar to (3.83) but it is now a (2/+ 1) x (2/+ 1) matrix and has / upper entries 1 and / lower entries (-1). The "block" gauges for this algebra are defined by (3.87)

where 5= —b and the DS currents are again distributed along every second slanted row. An other convenient gauge is what we will call the diagonal gauge. It is defined by J(x) = F_+t

6>,(x)//,,

(3.88)

i = i

(Here we choose the {//,} to form an orthonormal basis for the Cartan subalgebra.) Note that this is a new type of gauge fixing, not a member of the family of the DS gauges, but it will turn out to be very useful in applications and it is most useful in the quantum theory. Before we start discussing the gauge choice (3.88) in detail, we mention two difficulties connected with it. We will illustrate these difficulties on the simplest example, 5/(2, R). In this case the gauged fixed current in the diagonal gauge is parametrized by a single real field 8(x), 6 0

•/<.,* = ( . i

J -e

(3.89)

and it is easy to see that the transformation from the "Wronskian" gauge

to the diagonal gauge amounts to solving the Riccati equation 92-K0'

= U.

(3.91)

Now, if we require all fields to be periodic and integrate the Riccati equation over

146 TODA THEORY AND TiT-ALGEBRA

111

the period, the derivative term drops out and we see that (3.91) has no solution unless f " U(x)dxx>0.

(3.92)

In other words, the diagonal gauge can only be reached from that part of the phase space where (3.92) is satisfied. A related difficulty is that when the Riccati equation can be solved, its solution is not unique, it in fact has two independent solutions. (For an arbitrary Lie algebra, the number of independent solutions of the analogous equations is equal to the order of the Weyl-group.) However, note that when available the diagonal gauge is locally well defined (the ambiguities mentioned above correspond to finite gauge transformations) and therefore the corresponding Dirac brackets are also well defined. Since the i^-algebra is determined by polynomial relations, its structure can be analysed by restricting the considerations to that part of the phase space where the diagonal gauge is available and we will see that this is often convenient. Expanding the general KM current, /, in the Cartan-Weyl basis as J= £

r * £ - , + I 0tH,+

£

C'£„

(3-93)

the set of constraints defining the diagonal gauge can naturally be divided into two parts, * = ([)•

(3.94)

The diagonal gauge is defined by constraining the £~9 by imposing the original constraints (2.8) and, in addition, setting the f to 0. Since on the corresponding constraint surface K,?}=0

and

{C,C}=0,

(3.95)

the C operator, whose inverse enters the formula for the Dirac bracket can schematically be written as C = { X , * } * ( _ ° B o),

(3-96)

where B= {
(3.97)

147

112

BALOG ET AL.

The important property of the diagonal fields O^x) that makes the diagonal gauge extremely simple is that they (weakly) commute with the additional constraints (*": {0,(x),C}*O.

(3.98)

Because of (3.98), the Dirac bracket of two diagonal currents is the same as their original KM Poisson bracket: {0M.6j(y)}**{0i(x),ejiy)} = K5v8'(x1-yl).

(3.99)

In other words, the diagonal components of the current are a set of free fields. Therefore in the diagonal gauge the 1^-generators are given as differential poly­ nomials in free fields and these differential polynomials are simply obtained by restricting the full (gauge-invariant) differential polynomials to the "diagonal currents" of the form (3.88). This free-field representation of the iF-generators is called the Miura-transformation and has been used to quantize the theory [5].

IV. CASIMIR ALGEBRA

IV. 1. Leading Terms and Casimir Algebra We have already seen that any DS gauge defines a basis of the li^-algebra, and that there is a one-to-one correspondence between the conformal weights of the ^-generators associated to the highest weight gauge (or the scale dimensions of the "^-generators associated to any DS gauge) and the orders of the independent Casimirs of the underlying simple Lie algebra. In this section we shall elaborate on this connection further, by showing that the leading terms of the ^-generators (i.e., terms without any derivatives) are always Casimirs (restricted to P). Then we demonstrate that the Casimirs themselves form a polynomial algebra under the Poisson bracket, which is a truncated version of the full 1^-algebra. This Casimir algebra, in its quantum version, has been studied in [15]. We shall denote the leading terms of the "^-generators, W>, by W{. Since these leading terms contain no derivatives, they are invariant under rigid gauge transfor­ mations, that is Wi(JA) = W]0(J)

for AeN, where JA = AJA ~'

for any constrained current (JeP). On the other hand, an arbitrary Casimir C' is a group-invariant polynomial, that is, for any KM current J and an arbitrary Be G one has Cj(JB) = CJ\J),

where

JB =

BJBl.

148

TODA THEORY AND 1^-ALGEBRA

113

First we want to show that the leading terms of the "^"-generators are restricted Casimirs, or in other words that Wi(J) = CJ(J),

JeP

(4.1)

for some CJ. To do this we shall use the theorem of Chevalley from the theory of invariant polynomials [18], which we now recall. This theorem states that there is a one-toone correspondence between the Casimirs and the Weyl-invariant polynomials on the Cartan subalgebra, and that the correspondence is simply given by restriction. That is, first, if CJ(J) is an arbitrary group invariant polynomial (Casimir) on #, then its restriction to the Cartan subalgebra, CJ\H), is a Weyl-invariant polyno­ mial. (We shall denote the Cartan subalgebra by Jf and the restriction of any func­ tion to 3HC by an overbar.) Conversely, from any given Weyl-invariant polynomial on Jf, a corresponding full group invariant can be reconstructed in a unique way. For later use we also recall that the uniqueness of the reconstruction is proven by "diagonalization." First note that for any Lie algebra element J in the compact form of # there exists a group element g e G that "diagonalizes" /: J* = gJg-l = H(J)eX. (The use of the compact form is justified here since the problem is purely algebraic.) Using the group invariance of the Casimir CJ we see that C\J) = CJ(JS) = CJ(H(J)) = Cj{H(J)) so CJ determines the full Casimir CJ uniquely indeed. By using Chevalley's theorem, (4.1) will follow if we can prove that the restriction of Wl(J) to currents / in the diagonal gauge (cf. (3.88)) is a Weyl-invariant polyno­ mial of the Cartan components of J. To do this we only have to show that for any "diagonal" constrained current J it its possible to find such rigid gauge transforma­ tions AeN, whose action on the Cartan components 0, of J coincide with the action of the Weyl-group on the 0,. To show this, let us choose a simple root
with a = wE:,k

(4.2a)

on a constrained current JeP, J^J^

= eaJe-a = J+\_a,J'\ + \\_a, [ a , 7 ] ] + •••,

(4.2b)

where a> is an arbitrary real parameter. Parametrizing the constrained current JeP

J = I + I 0,H« + I £,£„,+£ tvEv, f'= 1

1 = 1

(p

149

114

BALOGETAL.

where

a* l j

t'

where the precise form of the coefficients V w and
which implies that the transformation (4.2) applied to the "diagonal" current / i= I

takes it into another current which is also in the diagonal gauge. Moreover, with this choice of co the action of the gauge transformation A = e" (4.2a) on the the Cartan components 0, of this particular diagonal current is

I «* I

j

which is precisely the same as the effect of the Weyl-reflection corresponding to the simple root a* on the Cartan components 0,. This implies that every Weyl-transformation of the Cartan components of the diagonal currents can indeed be implemen­ ted by rigid gauge transformations. (Since the Weyl-group is not a subgroup of N, the particular rigid gauge transformation A which "implements" a given Weyltransformation on the components 0, of a "diagonal" current ydjag must depend on the particular current on which it acts, and is really field-dependent according to the above construction.) Since the leading term W0 is invariant under rigid gauge transformations, it follows that its restriction WJ0 to the diagonal gauge is a Weylinvariant polynomial of the current components 0,. Chevalley's theorem then tells us that W'0 is the restriction of a uniquely determined Casimir CJ to the diagonal currents (note that /_ has no contribution in CJ(JdMg) because of the neutrality of

150 TODA THEORY AND ^-ALGEBRA

115

the group invariant CJ). To finish the proof of (4.1) one has to show that the leading term WJ0 itself is the restriction of the same Casimir CJ to P. This last step follows from the fact that WJ0 and the restriction of C' to P are the same (namely W{) when restricted to "diagonal" currents, since an TV-invariant on P can uniquely be reconstructed from its Weyl-invariant restriction to the space of diagonal currents. (The uniqueness of this reconstruction can be shown by an argument similar to the one that was used in the case of the Chevalley theorem.) It is not hard to see that the Casimirs {CJ} corresponding to the leading terms of a i^-basis {WJ} form a basis in the ring of group-invariant polynomials. (It is enough to prove this for a i^-basis constructed by means of some DS gauge, but in this case these / Casimirs are independent even if restricted to the gauge section Mv, where they simply coincide with the / DS currents {Uj}.) So we can associate a Casimir basis to any i+1,

y= 1,2,...,/.

(4.3)

Then we can define ^-generators corresponding to these Casimirs by the formula 7+1 where J=AJA1

+

KA'A-1

is the representative of the gauge orbit of the constrained current JeP in some particular DS gauge. (Remember that both JeMy and the gauge transformation A are uniquely determined by JeP. ) It follows that we have WJ

X

--TT JJ+X =^—Tt{J+KA-lA')J+i ^ - T r / ' + 1 + •••, (4.4) J+l J+l 7+1 J that is the leading terms of the W are indeed the Casimirs CJ. It is also easy to see that the {W'\ associated by this method to a set of independent Casimirs form a basis of "^-algebra. (The ^-generators associated to a given Casimir by means of different DS gauges differ in their derivative, non-leading terms.) In the SL(3, R) example, choosing the "Wronskian" gauge / 0 Wl W1 /= 1 0 0 \0 1 0

151 116

BALOGETAL.

we have Wl = \lxJ2

W2 = \TxP.

and

(4.5)

By using the results of Subsection III.3 on the A2 example we can derive the relations {W'(*), W\y)} = K( W1)' (X) b + 2KW\X)

b' - 2K3 b'"

{Wl(x), W2(y)} = 2K(W2)' (X) b + 3KW2{X) 6' -K2[W1{X)5]" 2

2

+ K*5"" 1

1

K(W2)"-IK2(W1)'"](X)S

{W (x), W (y)} = K\_1(W )' W + + Kll(Wi)2 + -2K\W1)'

(X)

2K(W2y-2K2(W1)"l(x)6' 5"-^W^x)

S'" + %KS 5'""

where <5 = d(xl — yl) and x° = y°. On the other hand, it is not difficult to verify that the corresponding Casimirs Cl = {lxJ2

and

C2 = i T r / 3

(4.7)

satisfy the following algebra under Poisson bracket, {Cl(x),Ci(y)}

= K(Cl)'(x)d + 2KCl(x)5'

{C'ix), C2(y)} = 2K{C2)' (X) b + 3KC2(X) b' 2

2

(4.8) 2

{C (x), C (y)} = K [ § ( C ' ) ' C'](x) S + K [ | ( C ' ) ] ( X ) S'y which is nothing but the leading term (in K) of the full liT-algebra for 51.(3, R). In fact we will show that in general, if the TT-generators W' and WJ satisfy {Wi{x),Wi(y)}=Y,fA{W)(x)b«\xx-yi\ A

A

where the "structure functions" f (W) are differential polynomials in {Wj}, then the corresponding Casimirs C and C satisfy the simplified (truncated) algebra {C'(x), CJ(y)}=rt(C)(x)b,+f°l(C)(x)b,

(4.9a)

where fl0 and f\ are the leading terms of / ' and f° in the number of derivatives, which are 0 and 1, respectively. To show this, let us first note that from the form of the KM Poisson brackets and group-invariance of the Casimirs it already follows that the commutator (4.9a) must be of the form {C'W, 0{y)} = gi(J)(x) b' + *?(/, J')(x) b

(4.9b)

152 TODA THEORY AND HiT-ALGEBRA

117

with some group-invariant functions gl and g°. (g° is polynomial in J, but is linear in J'.) Now we have to demonstrate that go(J) = fo(C(J))

and

g?(J, •/')=/?(C(/)).

(4.10)

We will make use of the diagonal gauge and the Chevalley theorem once more. In the diagonal gauge the leading terms of W' and C' coincide and therefore we have gtiH) = fl0(€(H))

and

fi(H,H')=f°(C(H)).

(4.11)

Now applying the Chevalley theorem to g^, the first equation in (4.10) follows from the first one in (4.11). Before one is able to apply the theorem also to g°, one first has to generalize it for the case of operators containing one derivative. This is possible and the proof is basically the same as for operators without any derivatives. Let us define the group invariant <5°(J, / ' ) by g°l(J,J')=f°i(C(J)) + S0l(J,J'). From the second equation in (4.11) we see that d°x(H, H') vanishes, but then the full (5°(7, J) must vanish too since «?(/, J') = <5?(//(/), (J')') = *?(#(/), (//')*) = <5°(//(/), (/T)«) = 0, where the second step follows from the neutrality of the group-invariant 5°. This way we have shown that the set of Casimirs closes to form a polynomial algebra under the Poisson bracket and that this algebra is a truncated version of the corresponding "^-algebra. Since the completely local Casimirs {C} are more elementary objects than the {W) which contain derivatives as well, one can ask wether the closure of the Casimir algebra can be shown without any reference to the more complicated ^-algebra. In other words, one has to show that (4.9a) holds with some functions fl0 and f°. It is trivial that gl in (4.9b) depends on J only through the Casimirs, since this merely expresses the fact that the {C} form a basis for the completely local group-invariants. To show that g° is also a function of the Casimirs we go to the diagonal gauge again. In this gauge the restriction of g°t must be of the form * ? ( # , # ' ) = I Afi\,

(4.12)

;= 1

where the {0,} are coordinates with respect to some basis in the Cartan subalgebra and the coefficients {^4,} can be considered as an /-component vector in the Cartan subalgebra and can be expanded as

153 118

BALOG ET AL.

simply because the / vectors {dCJ/dd,} are linearly independent. (This is the analytic expression of the fact that the / invariants {C7} are functionally independent.) Substituting (4.13) into (4.12) we find

and we see that the coefficients Bj must be Weyl-invariants: g°(ff, H') = I j - 1 —

Bt(C{H))[C\H)-y.

Now using the generalized Chevalley theorem for g° again we have

*?< y ' J>) =I irrr Bj(c(j))ia(j)J. After this digression we return to the question of the quadratic closure of the T^-algebra. We have shown in the previous section that the nT-algebras for A,, Bh and C, are quadratic in a suitably chosen basis. As an application of the relation between the ^-algebras and the algebras of the corresponding Casimirs we now prove that no such basis exists for £>, and the exceptional algebras. In fact we show this for the Casimir algebras, from which the analogous result for the li^-algebras immediately follows. Let CH be the highest order Casimir, of order H (see Appendix A), and let us consider the Poisson bracket of CH with itself, {C„{x), CH{y)} = r2H_2(C)(x)3' + kr2H_2(C)(x)3.

(4.14)

(Here the two structure functions are not independent of each other due to the antisymmetry of the Poisson bracket.) The structure function r2H_2(C) is a Casimir of order (2H — 2) and by inspecting the list of group-invariants for the case of the exceptional groups it is easy to see that it can never be expressed as a quadratic function of the basic Casimirs {CJ} for these groups. The situation for D, is more complicated. Here we can show that the set of Casimirs {C\ C2,..., C'} defined by

d e t ( l - ^ 7 ) = l - t Hncn>

( 415 )

n= 1

where the determinant is taken in the 2/-dimensional vector representation of Dh forms a quadratic algebra under Poisson bracket. (This is actually the same algebra as is formed by the corresponding Casimirs of the B, and C, groups.) However, as

154

TODA THEORY AND iS^-ALGEBRA

119

is well known, (4.15) is not a correct choice of basic Casimirs for D,, the latter is given by the set {C\ C 2 ,..., C ' " 1 ; Ca = sfc'}. By introducing the "spinorial" invariant C„, we destroy the quadratic property of the algebra. We find that r4/_6, the structure function in the commutator of two highest Casimirs C'~l is given by (see Appendix C) r4/_6=-12K(CJ2C'-3-4icC/-1C/-2

(4.16)

which is indeed cubic for / > 3. IV.2. Explicit Casimir Algebras In Subsection IV. 1 we have shown that the Casimir operators, C , form a closed, polynomial algebra under Poisson bracket, which is a truncated version of the ^-algebra. These Casimir algebras are interesting in their own right and they are also useful for studying the related "^-algebras. In this subsection we exhibit their structure in some detail. First, it is obvious that the Casimirs are conformal primary fields with respect to the Sugawara energy-momentum tensor. Next we want to determine the non-trivial Poisson bracket relations describing this algebra. What we are actually going to calculate is the Poisson bracket of the generating polynomials / £ fin+iC"(x)

A(n,x) = dct(l-iiJ(x))=l-

(4.17a)

«= i

and B(n, x) = det(l - J~n Ax)) = 1 - 1

fi"C(x)

(4.17b)

n- 1

for the / independent Casimirs C 1 ,..., C of the A, and B, (C,) algebras, respectively. One first observes that the overcomplete set of group-invariant polynomials « = 2,3,...

Q"(X) = -TTJ"(X),

(4.18)

n which are related to the / independent Casimirs C" via Id"

/

°°

for

A,,

»-0■ o

c

Id"

"=-^

exp

f

2

°°

e2

\

(- ,?/ v

for p-0

B„ C„

155

120

BALOG ET AL.

satisfy the Poisson brackets (see Appendix C)

{Qn(x),Q'"(y)}=KUp-2)Q''-2-^Q"-,Qm-l](x)S' + KUm-l)(Q"-2y~^Q"-l(Q"'-ly^{x)d,

(4.19)

where S stands for <5(JC' - yl) as before, p = n + m, q = (n - 1 )(m - 1), and N is the dimension of the defining representation. Note in particular that for the B and C algebras both n and m must be even integers (since for odd n the Q" vanish identi­ cally) and, as a consequence, the quadratic terms on the right hand side of (4.19) are automatically absent. However, formula (4.19) is only the first step in finding the explicit Casimir algebras. For example, in the case of A2 one obtains

{Q\x), Q\y)} = K ( V - \ Q2Q2) &' + 1 ( 4 ( 2 4 - \ Q2Q2)' 8 and only after expressing Q* in terms of the independent Casimirs Q2 and (?3 via 2Q* = (Q2)2 does one find the result (4.8) (note that Q2 = C1 and Q3 = C2 there). More generally, if one computes the Poisson brackets of the highest Casimirs for an algebra of rank /, one has to use the characteristic polynomial 0(1/2) times to express the right hand side of (4.19) in terms of the independent Casimirs. Clearly this method becomes soon cumbersome and another algorithm is needed. As a first step to calculate the Poisson bracket of the generating polynomial (with itself) we expand its logarithm, oo

log det( 1 - nJ(x)) = - X /i"C"(x)

(4.20)

n= 2

and use (4.19) to calculate the Poisson bracket of logdet(l — nJ). This then allows for the computation of the Poisson brackets of the determinant. After some algebra one finds that this Poisson bracket can be reexpressed in terms of the determinant and its derivatives. For the details of the derivation we refer the reader to Appendix C. The final results are A, algebras: {A(fi, x), A(v, y)} = K / iV (^~-v (dv - ^ ) - j-J-j- d„ 8^ A(ii, x) A(v, x) S' + K/iV ((0, - d„) - ~ - - ^ d„ 3 y ) A(n, x) BxA(v, x) 8 + - ^ - L2 j (A(n, x) dxA{v, x) - A(v, x) dxA(n, x)) S, (H-v)

(4.21)

156

TOD A THEORY AND "W- ALGEBRA

121

Bh C, algebras: {B{\i, x), B(v, y)}=4Knv

( v 5 v - ^ ) B(n, x) B(v, x)8' fi — V

+

4K//V(V5V

- iidj

(B(/i, x) dxB(v, x)) 3 fl—V

+ 2K/iv /

+ V

(fi-v)2

.2 (B(n, x) dxB(v, x)- B(v, x) dxB(ti, x)) d. (4.22)

The algebra of the Casimirs can now be computed by inserting the expansions (4.17) into both sides of (4.21), resp. (4.22), and comparing coefficients in the resulting polynomials in n and v. One sees, in particular, that with respect to the Casimirs defined by the determinant the algebras close always quadratically. For example, for the highest Casimirs of A,, 1^2 one obtains {C'(jc),C'O0} = Ka / (x)*' + ! f l ; ( x ) $ (4.23) 2

1 2

a/ = - 2 C ' C ' - 0 ( / - 3 ) + - — r ( C ' - ) and for the highest Casimirs of B, and Ct with / > 2 one finds {C\x\

C'(y)} =

-4KC'C'-

'

8'-2K(C'C'-1)'

S.

(4.24)

The corresponding results for the lower Casimirs are presented in Appendix C.

V. DIFFERENTIAL AND PSEUDO-DIFFERENTIAL OPERATORS AND TODA FIELDS

The aim of this last section is to demonstrate that the differential and pseudo-dif­ ferential operators studied in [10], and taken as a starting point for the quantiza­ tion of ^-algebra in [ 5 ] , arise naturally in our framework. These operators appear in the differential equations satisfied by the gauge-invariant components of the WZNW field. Let us recall that the solution of the field equations for the group-valued field g is g(x\x-)

= gL(x

+

)gR(x-)

(5.1)

with gLgL1=J

and

gilg'*

=Z

(5.2)

157 122

BALOGETAL.

where the currents J and J are subject to the constraints (2.8). (In this section prime means 2K d/dxl.) We will consider the simplest case, SL(n, R) first, and concentrate on the left-moving part of the theory. (We omit the subscript L.) In order to reconstruct the group-valued field g(x+) from the current J(x+) (satisfying the constraints (2.8)), one has to solve the set of linear differential equations g' = Jg-

(5.3)

Obviously this is a separate set of equations for each column-vector of the matrix g, which are of the form

=( / - + » . •

(5.4)

Solving (5.4) is the simplest in the "Wronskian" gauge, (3.71). In this gauge one can easily express all components of g in terms of the bottom components, gnj, denoted by i^,, gin-1)1

=

^

(5.5)

leading to a single nth-order differential equation satisfied by tpn

,/,(«>= £ uty—J

".

(5.6)

J-I

The group-valued field g can now be built from the n independent solutions of (5.6),

^1

"h

•"

4>n

where the set of solutions {i/^,} must satisfy the Wronskian constraint (hence the name of the gauge): detg=l in order that the matrix g be an element of the group SL(n, R).

(5.8)

158 TODA THEORY AND iF-ALGEBRA

123

We note that if the DS currents {UJ} are regular functions then so are the solu­ tions {\pt} of the generalized Schrodinger equation (5.6). By combining gL given by (5.7) with the similarly constructed right-moving solution g R , the resulting WZNW field g(x+,x~) is also regular, as are the globally defined Toda fields, being subdeterminants of the latter (according to (2.23)). Furthermore, if the niT-generators corresponding to a given (say, the "Wronskian") DS gauge are given by regular functions for a Toda solution, then by this procedure one can always construct a regular WZNW representative of that Toda solution, whether or not the solution appears to be regular in terms of the traditional local Toda variables, ". We also remark that once the solutions of the "right handed" analogue of (5.6), {Xi}, are known, then as a consequence of (2.24) and (5.1), the Toda fields can be expressed in terms of the {^}'s and the {x}'s as e-'"-
\^-x

vxJ

= W-X')WX)-W-X){>I>X')

(5-9)

(rr r-x' r-x\ e-*-«2=®„_2=deti r-xK r-x' r-x\ =(r-x")(r-x'M-x)+--where *-X«5>rZ„

r-X = Y.r-Xi,

(5-10)

and so on. In fact Eq. (5.9) was the starting point of the analysis of Toda theory in [8]. Without going into details we note that the above results can easily be generalized for the B, and C, series. So far we have studied (5.4) in a definite DS gauge. Let us now try to solve it for g without gauge-fixing the current J. It is easy to see that starting from the bottom row, it is always possible to eliminate all higher components of g successively, even without any gauge fixing. This elimination leads to a differential equation of the form n— 1

^ V , . = 5 V , - Z Wi\_J(x + )-\d"-i-^i

= Q.

(5.11)

y-i

Here d = K d/dx + and the coefficient functions { W'} are automatically obtained as some differential polynomials in the current components. Moreover, they are gauge invariant, since the original equation (5.3) was gauge-covariant and the bottom components gni — ^t are gauge-invariant (with respect to left-moving upper

159 124

BALOGETAL.

triangular gauge transformations). This implies that the WJ'& in (5.11) are nothing but the ^-generators associated to the "Wronskian" gauge, since they reduce to the DS currents {UJ} in this gauge. To summarize, if the 1^-densities associated to a DS gauge (here the "Wronskian" gauge) are known, then one can reconstruct the corresponding WZNW solution by solving (5.2) for g = gLgR in that DS gauge. In the reconstruction procedure one obtains a higher order differential equation (here (5.6)) satisfied by the gauge-invariant (bottom row) components of gL (and an analogous equation for the last column of gK). The same equation can also be derived from (5.4) by elimination without any gauge fixing. Since the resulting dif­ ferential equation is gauge-invariant, one can read off the explicit formula of the ^"-generators corresponding to the given DS gauge by comparing the coefficients in the differential equations obtained with and without gauge-fixing. By a similar argument, one can also establish the transformation rules relating the iC-bases corresponding to different DS gauges. The elimination is also simple in the diagonal gauge. In this gauge

(

0, 0 ... 0 (5.12)

0 0 2 ••• 0 and the differential operator takes the form 0 0 .. d„ and the differential operator takes the form By rearranging this product as a sum corresponding to (5.11), one can read off the ^)=(d-e (5.13) i)(d-e2)---(d-dn). By rearranging this product as a sum corresponding to (5.11), one can read off the expression of the iF-generators in this gauge. Note that the diagonal fields (5.12) are not independent, because 0, + 02 + ■■■ + 0„ = 0. This is the original form of the Miura-transformation [21] and the operator (5.13) is the starting point for the Lukyanov-Fateev free-field construction of quantized T^-algebra [ 5 ] . The derivation of the gauge-invariant higher order differential operators and the reconstruction of the matrix-valued field g from the constrained currents (or from iT-generators) proceeds analogously for the Lie algebras B„ and C„. The resulting gauge-invariant differential operators are of order (2« + 1) and (2/i), respectively, according to the dimensions of the defining representations. Due to the restrictions (3.86) and (3.82), the differential operators 3i(nB) and 2>l,C) are (formally) antiselfadjoint and selfadjoint, respectively. Without going into detail, we give these operators in the factorized form corresponding to the diagonal gauge: ®inB) = (d-0l)(d-02)...(d-e„)d(d + d„)--(d + 82)(d + dl) (5.14a)

&nC) = (d-el)(d-e2)...(d-9„\d

Here the 0/s are independent free fields.

+ en).-(d + 92)(d + ei).

(5.i4b)

160

TODA THEORY AND 1^-ALGEBRA

125

The case of the algebras D„ is more complicated. As an example, let us consider D3 first. We use the six dimensional vector representation and go to the diagonal gauge, where (with a convenient choice of the T,)

J=I_+J=\

0, i 0 n 0 0 0

0 0 0 0 0 e2 0 0 0 0 0 0 0 1 03 \ t n n 0 -03 0 n0 1 0 0 - 1 - 1 -02 0 - 1 -0. 0 0

•

(5.15)

If we write out (5.4) in components we have (suppressing the index i) g5=-(d

+ 9l)g6

(5.16a)

£3 + £4 =~(d + 92)g5

(5.16b)

g2 = (d + 03)g4

(5.16c)

g2 = (d-d3)g,

(5.16d)

gi = (8-e2)g2

(5.16c)

O = (3-0,)g,.

(5.16f)

From (5.16) we see that the elimination is blocked here after the second step, since the combination g3 — g4 never occurs on the left hand side. On the other hand, its derivative can be expressed, combining (5.16c) and (5.16d), as d(g3-g*)

= e3(g3 + g4).

(5.17)

One can go on with the elimination by integrating (5.17) (formally) using the "antiderivation" symbol d~l: (gi-g4)

= d-%(g3

+ g4).

(5.18)

One then finds &3D) = (d-dl)(d-82)(d-03d-lO3)(d+

6^(8 + 6^

= (d- 0,)(3 - 62)(d - 03) d~\d + 93)(d + e2)(d + 0,).

(5.19)

Similarly, by performing the elimination in the (2«)-dimensional vector representa­ tion, one can associate a pseudo-differential operator to any £>„ algebra:

®inD) = (d-ei)(d-e2)---(d-en)d-\d

+ en)..-(d + e2)(d + el).

(5.20)

This not only shows that it is impossible to obtain a differential operator for D„ in the vector representation, but from the example of D3 ~ A3 we also see that the type

161 126

BALOG ET AL.

of pseudo-differential operator depends on the representation in which (5.4) is taken. (For Z)3 there is an ordinary differential operator corresponding to the four dimensional representation, but this is the spinor of Z)3.) For the case of A„, B„, and C„ what makes the elimination simple is that the matrix /_ (see (3.70) and (3.83)) has non-zero entries immediately below the diagonal and only there. Since /_ is the negative step-operator of the special 5/(2, R) subalgebra Sf introduced in Subsection III.l, this fact means that the defining representations of these algebras are still irreducible with respect to this 5/(2, R) subalgebra. For D„, the vector representation is reducible with respect to £f with branching 2n = (2/i- 1)+ 1. This is why one has a pseudo-differential, rather than a differen­ tial, operator after eliminating the higher components from the system of first order differential equations (5.4). (The spinor representations of D„ are even worse from this point of view, except for n = 3.) Turning to the exceptional algebras, we find that the seven dimensional represen­ tation of G2 is irreducible with respect to if and therefore the elimination for G2 will result in a 7th order differential operator (see Appendix B). The corresponding branching rule for FA is [22] 26 = 17 + 9, so in this case we have a pseudo-differen­ tial operator, containing one integration. Finally, for E6, E7, and Es the branching rules are [22] E6: 27 = 17 + 9 + 1 £ 7 : 56 = 28 + 18+10

(5.21)

£ 8 : 248 = 59 + 47 + 39 + 35 + 27 + 23 + 15 + 3 and therefore in these cases the elimination leads to pseudo-differential operators, containing 2, 2, and 7 integrations, respectively. VI. CONCLUSIONS

In this paper we have shown that extended conformal algebras, "^"-algebras arise naturally in the constrained WZNW formulation of Toda field theories. Our main results are the following: We have given an ambidextrous generalization of the usual gauged WZNW models to derive Toda theories. Using the embedding WZNW phase space, we have shown how to implement the action of the ^-algebra generators as certain field dependent Kac-Moody transformations. This led us to a powerful algorithm to calculate the iT-algebra relations. Using this algorithm we calculated the so far unknown Poisson bracket algebra of ifr(G2) explicitly. We exhibited a particular basis where all the ^-generators are conformal primary fields. We have also shown that for the A, B, C series there is always a basis in which the iS^-algebra closes quadratically, and that is not true for the rest

162

TODA THEORY AND "W-ALGEBRA

127

of the simple Lie algebras. Finally we have proved that the leading terms of the ^-generators (i.e., terms without derivatives) are restricted Casimir operators. We exhibited the Casimir algebra relations in detail for the A, B, and C series and have given a general proof of closure of their Poisson bracket algebra for any simple Lie algebra. As found in [15] the quantum version of the Casimir algebra does not close in general (it has been shown to close for SU{3) only when the level is equal to one) hence it is natural to ask whether an extension of the nT-generators to the full KacMoody phase space (in the sense discussed in the introduction of Section III) exists, with the full, unrestricted Casimirs as leading terms. If one could find such an extension it would make it possible to associate a representation of the T^-algebra in terms of unrestricted Kac-Moody generators to any Kac-Moody algebra. As such an extension would also be a deformation of the Casimir algebra, it could possibly survive quantization. This problem is certainly very interesting as it would also clarify the origin of li^-algebras, without making reference to any particular model. A detailed investigation of such deformations of the Casimir algebras is out­ side the scope of this paper. After some preliminary investigation of the problem we found that at least for A2 one cannot extend the ^-algebra to the whole phase space of the (chiral) Kac-Moody currents, at least with the assumption of the W"s being differential polynomials with unrestricted Casimirs as leading terms. However, by giving up the polynomial nature of the W"s we have found such an extension of the generators of the 1^-algebra for A2. In fact this result can be generalized for an arbitrary A„ algebra. This problem is under investigation.

APPENDIX A: CONVENTIONS

Here we give our conventions and present some formulae which are used in the paper. Space-Time and Poisson Brackets. »foo= —1n = l.

x±={(x°±xl),

d±=80±dl.

(A.1)

We use equal time Poisson brackets and spatial ^-distributions. At fixed x° all quantities are supposed to be periodic with period 2n. Prime means "twice spatial derivative" everywhere, even on Dirac <5's. Note that this is equal to d+ on quantities depending on x through x+ only. Conformal Primary Fields. The left-moving conformal transformations are generated by the conserved moments Qa=f

dxia(x)L(x)

(A.2)

of the Virasoro density L(x) = & + +(x), for any periodic test function a(x) for

163

128

BALOGETAL.

which d_a(x) = 0. A conformal primary field W of left conformal weight A trans­ forms as (SL
+ Jny)d

+ a(y).

(A.3)

If d _ V = 0 then this is equivalent to {L(x),ny)}ixo.yo

= A.nx)S'(xl-yl)

+ (A-l).(d+nx))d(xl-yl).

(A.4)

Lie Algebras. Let #c be a complex simple Lie algebra, <& the set of roots with respect to some Cartan subalgebra, and A a set of simple roots. There is a Cartan element Hv associated to every