A characterization of coboundary Poisson Lie groups and Hopf algebras

Abstract

We show that a Poisson Lie group (G, π) is coboundary if and only if the natural action of G×G on M = G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known π+). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the π+ structure on SU(N) is described in terms of generators and relations as an example. 1. Preliminaries. For the theory of Poisson Lie groups we refer to [1, 2, 3, 4, 5]. We follow the notation used in our previous papers [6, 7]. A Poisson Lie group is a Lie group G equipped with a Poisson structure π such that the multiplication map is Poisson. The latter property is equivalent to the following property (called multiplicativity of π): (1) π(gh) = π(g)h+ gπ(h) for g, h ∈ G. Here π(g)h denotes the right translation of π(g) by h etc. This notation will be used throughout the paper. A Poisson Lie group is said to be coboundary if (2) π(g) = rg − gr for a certain element r ∈ g∧g. Here g denotes the Lie algebra of G. Any bivector field of the form (2) is multiplicative. It is Poisson if and only if [r, r] ∈ (g ∧ g ∧ g)inv (the Schouten bracket [r, r] is g-invariant). In this case the element r is said to be a classical r-matrix (on g). 1991 Mathematics Subject Classification: Primary 17B37, 16W30; Secondary 81S05. Research supported by KBN grant 2 P301 020 07. The paper is in final form and no version of it will be published elsewhere. [273]