International Congress of Mathematicians

On the Dynamical Yang-Baxter Equation 561Remark 2 As in the quantum case, Example 3 can be degenerated intoits trigonometric and rational versions where 9(x) is replaced by sin(x) and x,respectively, and Examples 1 and 2 can be obtained from Example 3 by a limit.Remark 3 The classical limit of the basic rational and trigonometric solutionsof QDYBE (modified by A —¥ X/K) is the basic rational, respectively trigoinometric,solutions of CDYBE for g = si n (in the trigonometric case we should set q = e^hl 2 ).The same is true for the basic elliptic solution (with 7 = h).Remark 4 These examples make sense for any reductive Lie algebra g.2.8. Gauge transformations and classification of solutions forCDYBEIt is clear from the above that it is interesting to classify solutions of CDYBE.As in the quantum case, it should be done up to gauge transformations. Thesetransformations are classical analogs of the gauge transformations in the quantumcase, and look as follows.1. r —¥ r + oj, where OJ = VA • CìJ(X)XìAXJ is a meromorphic closed differential2-form on if*.2. r(u, X) —¥ ar(aX — v); Weyl group action.In the case of spectral parameter, there are additional transformations:3. u —¥ bu.4. Yet r = £. . Syar, ® Xj + £ Q (p a e a ® e_ Q . The transformation is S'y —¥Sij + u () X .Q X . , a e u9aV , where tp is a function on h* with meromorphic dtp.Theorem 2.3 [EVI] (i) Any classical dynamical r-matrix with zero coupling constantis a gauge transformation of the basic rational solution for a reductive subalgebraof g containing if, or its limiting case.(ii) Any classical dynamical r-matrix with nonzero coupling constant is a gaugetransformation of the basic trigonometric solution for g, or its limiting case.(iii) Any classical dynamical r-matrix with spectral parameter and nonzerocoupling constant is a gauge transformation of the basic elliptic solution for g, orits limiting case.Remark One may also classify dynamical r-matrices with nonzero couplingconstant (without spectral parameter) defined on I* for a Lie subalgebra I C h, onwhich the inner product is nondegenerate ([Sch]). Up to gauge transformations theyare classified by generalized Belavin-Drinfeld triples, i.e. triples (Fi,F 2 ,T), whereFj are subdiagrams of the Dynkin diagram F of g, and T : Y\ —t F 2 is a bijectionperserving the inner product of simple roots (so this classification is a dynamicalanalog of the Belavin-Drinfeld classification of r-matrices on simple Lie algebras,and the classification of [EVI] is the special case Y\ =Y 2 = Y,T = 1). The formulafor a classical dynamical r-matrix corresponding to such a triple given in [Sch] worksfor any Kac-Moody algebra, and in the case of affine Lie algebras yields classicaldynamical r-matrices with spectral parameter (however, the classification is thiscase has not been worked out). Explicit quantization of the dynamical r-matricesfrom [Sch] (for any Kac-Moody algebra) is given in [ESS].