Reflection equation algebras, coideal subalgebras, and their centres

Abstract

Reflection equation algebras and related $${U{_q}(\mathfrak g)}$$ -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 $${F_l(U{_q} (\mathfrak g))}$$ of $${U{_q}(\mathfrak g)}$$ 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 $${U{_q}(\mathfrak g)}$$ for each character f of a covariantized algebra. We show that for any character f of $${F_l(U{_q}(\mathfrak g))}$$ the centre Z(B f ) canonically contains the representation ring $${{\rm Rep}(\mathfrak g)}$$ of the semisimple Lie algebra $${\mathfrak g}$$ . We show moreover that for $${\mathfrak g = {\mathfrak sl}_n(\mathbb C)}$$ such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of $${{\rm Rep}({\mathfrak sl}_n(\mathbb C))}$$ inside $${U_q({\mathfrak sl}_n(\mathbb C))}$$ . As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in $${{\mathbb C}^{2m}}$$ .