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}}.