Reflection equation algebras, coideal subalgebras, and their centres

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. 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. Generalising 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. We show that for any 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.