Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. (English) Zbl 0915.17018

Etingof, Pavel; Varchenko, Alexander

Commun. Math. Phys. 192, No. 1, 77-120 (1998).

Author’s abstract: The classical Yang-Baxter equation (CYBE) is an algebraic equation central in the theory of integrable systems. Its nondegenerate solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CYBE was given by Drinfeld and gave rise to the theory of Poisson-Lie groups.
The classical dynamical Yang-Baxter equation (CDYBE) is an important differential equation analogous to CYBE and introduced by Felder as the consistency condition for the differential Knizhnik-Zamolodchikov-Bernard equations for correlation functions in conformal field theory on tori. Quantization of CDYBE allowed Felder to introduce an interesting elliptic analog of quantum groups. It becomes clear that numerous important notions and results connected with CYBE have dynamical analogs.
In this paper we classify solutions to CDYBE and give geometric interpretation to CDYBE. The classification and interpretation are remarkably analogous to the Belavin-Drinfeld picture.

Reviewer: A.N.Pressley (London)

MSC:

17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Keywords:

dynamical Poisson groupoid; classical dynamical Yang-Baxter equation