Algebras of functions on compact quantum groups, Schubert cells and quantum tori

Abstract

The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R,\(u \in \Lambda ^2 \mathfrak{h}_R\), and\(\mathfrak{h}_R\) is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a,u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universalR-matrix, are given.