The Classical Dynamical Yang-Baxter Equation; A geometric interpretation
The Classical Yang Baxter Equation (CYBE) is an algebraic equation for a meromorphic function with values in the tensor square of a Lie algebra. The equation is central in the theory of integrable systems. Drinfeld, following work of Sklyanin, discovered the geometric meaning of the CYBE: It corresponds to a Poisson structure on the loop group. Quantization of the CYBE led to the theory of Quantum groups. Felder (1994) discovered a differential equation (CDYBE), which is a generalization of the CYBE arising naturally in Conformal Field Theory. We will review work of Etingof and Varchenko on the geometric interpretation of the CDYBE in terms of Poisson groupoids. We will then provide a simple explanation of the geometry in terms of moduli spaces of principal G-bundles on an elliptic curve, for any reductive group G (Joint work with J. Hurtubise).