Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related U_q(mathfrak g){U{_q}(mathfrak 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 (mathfrak g)){F_l(U{_q} (mathfrak g))} of U_q(mathfrak g){U{_q}(mathfrak 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(mathfrak g){U{_q}(mathfrak g)} for each character f of a covariantized algebra. We show that for any character f of F_l(U_q(mathfrak g)){F_l(U{_q}(mathfrak g))} the centre Z(B _f ) canonically contains the representation ring Rep(mathfrak g){{rm Rep}(mathfrak g)} of the semisimple Lie algebra mathfrak g{mathfrak g} . We show moreover that for mathfrak g = mathfrak sl_n(mathbb C){mathfrak g = {mathfrak sl}_n(mathbb C)} such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(mathfrak sl_n(mathbb C)){{rm Rep}({mathfrak sl}_n(mathbb C))} inside U_q(mathfrak sl_n(mathbb C)){U_q({mathfrak sl}_n(mathbb 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 mathbb C^2m{{mathbb C}^{2m}}.