Reflection equation algebras, coideal subalgebras, and their centres

Abstract

Reflection equation algebras and related U_q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called `covariantized' algebras, in particular concerning their centres, invariants, and characters. Generalising M. Noumi's construction of quantum symmetric pairs we define a coideal subalgebra B_f of U_q(g) for each character f of a covariantized algebra. The locally finite part F_l(U_q(g)) of U_q(g) with respect to the left adjoint action is a special example of a covariantized algebra. We show that for each character f of F_l(U_q(g)) the centre Z(B_f) canonically contains the representation ring Rep(g) of the semisimple Lie algebra g. We show moreover that for g=sl_n(C) such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(sl_n(C)) inside U_q(sl_n(C)). As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in C^{2m}.