Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related -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 of 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 for each character f of a covariantized algebra. We show that for any character f of the centre Z ( B f ) canonically contains the representation ring of the semisimple Lie algebra . We show moreover that for such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of inside . As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr ( m ,2 m ) of m -dimensional subspaces in .