Volume 243, number 1,2 PHYSICS LETTERS B 21 June 1990 R-matrices and symmetric spaces ~" J. Avan, J.M. MaiUard and M. Talon Laboratoire de Physique Th~orique et des Hautes Energies 1, Universit~ Paris 6, Tour 16, let ~tage, 4 Place Jussieu, F- 75252 Paris Cedex 05, France Received 1 March 1990 Non-antisymmetric R-matrices which play a central role in the description of the Poisson structure of integrable models, are investigated. A wide class of such solutions is described. They are associated with the symmetric Lie algebras constructed from simple Lie algebras. This extends the classification of skew-symmetric matrices achieved by Belavin and Drinferd. I. Introduction Recent studies have enabled us to construct a num- ber of examples of non-skew-symmetric solutions of the classical Yang-Baxter equation, describing the Poisson bracket structure of the Lax operator for an integrable classical model [ 1-3 ]. We recall that such models are described by a so-called Lax equation of motion [ 5 ] dL dt -[L'M]' (1) L, M belonging to a Lie algebra ~, for example sl (n, C) represented by n × n matrices. The Poisson alge- bra of L is described by an operator R living in ~q® and acting on a tensor product of vector spaces of representation 8® ¢ such that {L~'~@,L~2)}= [R,L<')®I]- [R ~, I®L ~2)] , (2) where {L~l)@,L~Z)}k!={L<~)i k, L<2)/} and H is the permutation operator on f~® f¢: H(x®y) =y®x and RII-HRH. As a matter of fact, eq. (2) is equivalent to the integrability of eq. ( 1 ), see ref. [ 6 ]. When L lives in a loop algebra if®C[2, 2-1 ], 2 being the so- called spectral parameter of the Lax pair, R depends on the two spectral parameters 2, # ofL ~ ~ ) and L ~2), and of course, Rr/(2,/t) =HR(It, 2)/7. Work supported by CNRS. i U.R.A. 280. The Jacobi identity on the Poisson brackets in eq. (2) induces the classical Yang-Baxter equation on R as [RI2 , RI3 ] d- [RI2 , R23 ] + [R32 , RI3 ] + (terms) =0. (3) The (terms) in eq. (3) only appear when R depends on the dynamical variables of the problem, as can be seen in refs. [6,7]. We shall not consider this case here, restricting ourselves to constant R-matrices. Of course, Yang-Baxter equations also appear with per- muted indexations of { 1, 2, 3}, but they are equiva- lent to eq. (3) under conjugation by the idempotent permutation operators /7o. As an example, taking H12 eq. (3) H~2, noting that (/~12)2=1 can be in- serted in the commutators, and /-IiERl3Hl2 •R23 , etc., eq. (3) is obtained with 1 ~ 2. When in addi- tion R is supposed to be antisymmetric: R~V= -R, the Yang-Baxter equation (3) can be rewritten as [RI2,RI3]+[RI2, R23]+[R13,R23]=O. (4) It has been thoroughly studied in a series of papers by Belavin and Drinfel'd [8-10 ]. They showed that such an R could only depend on the difference (2-#) [ 8], and that the classification of rational solutions to eq. (4) was essentially based on the classification of simple Lie algebras. The trigonometric and elliptic R-matrices could be obtained by suitably defined for- mal series of rational solutions, given a so-called Coxeter automorphism of the associated simple Lie algebra [ 9,10 ]. 1 16 0370-2693/90/$ 03.50 © 1990 - ElsevierSciencePublishers B.V. ( North-Holland )