Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices

Abstract

Let U be a connected, simply connected compact Lie group with complexification G. Let u and g be the associated Lie algebras. Let Γ be the Dynkin diagram of g with underlying set I, and let Uq(u) be the associated quantized universal enveloping ∗-algebra of u for some 0 < q distinct from 1. Let Oq(U) be the coquasitriangular quantized function Hopf ∗-algebra of U, whose Drinfeld double Oq(GR) we view as the quantized function ∗-algebra of G considered as a real algebraic group. We show how the datum ν = (τ,ε) of an involution τ of Γ and a τ-invariant function ε : I → R can be used to deform Oq(GR) into a ∗-algebra Oν,id(G ) by a modification of the Drinfeld double construction. We then show how, by qR a generalized theory of universal K-matrices, a specific ∗-subalgebra O (G \\G ) of Oν,id(G ) admits qνRqR ∗-homomorphisms into both Uq(u) and Oq(U), the images being coideal ∗-subalgebras of respectively Uq(u) and Oq(U). We illustrate the theory by showing that two main classes of examples arise by such coideals, namely quantum flag manifolds and quantum symmetric spaces (except possibly for certain exceptional cases). In the former case this connects to work of the first author and Neshveyev, while for the latter case we heavily rely on recent results of Balagovi ́c and Kolb.