Poisson Lie groups and their relations to quantum groups

Abstract

The notion of Poisson Lie group (sometimes called Poisson Drinfel’d group) was first introduced by Drinfel’d [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic facts about Poisson Lie groups together with some relations to the recent work on quantum groups. 1. Poisson structures on manifolds. The basic mathematical structures of classical mechanics are the algebra C∞(N) of all smooth functions on the phase space N under ordinary multiplication and the Lie structure on C∞(N) induced by the Poisson bracket {·, ·} defined by the symplectic form on N . The Poisson bracket is a biderivation of the associative algebra, i.e. (1.1) {f, gh} = {f, g}h+ g{f, h} and in so called canonical coordinates (q,p) = (q1, ..., qn, p1, ..., pn) has the form