Reflection Equation

Reflection equation algebras and related Uq(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 Bf of Uq(g) for each character f of a covariantized algebra. The locally finite part Fl(Uq(g)) of Uq(g) with respect to the left adjoint action is a special example of a covariantized algebra. We show that for each character f of Fl(Uq(g)) the centre Z(Bf ) canonically contains the representation ring Rep(g) of the semisimple Lie algebra g. We show moreover that for g = sln(C) such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(sln(C)) inside Uq(sln(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.