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 $\mathfrak{u}$ and $\mathfrak{g}$ be the associated Lie algebras. Let $\Gamma$ be the Dynkin diagram of $\mathfrak{g}$ with underlying set $I$, and let $U_q(\mathfrak{u})$ be the associated quantized universal enveloping $*$-algebra of $\mathfrak{u}$ for some $0<q$ distinct from $1$. Let $\mathcal{O}_q(U)$ be the coquasitriangular quantized function Hopf $*$-algebra of $U$, whose Drinfeld double $\mathcal{O}_q(G_{\mathbb{R}})$ we view as the quantized function $*$-algebra of $G$ considered as a real algebraic group. We show how the datum $\nu = (\tau,\epsilon)$ of an involution $\tau$ of $\Gamma$ and a $\tau$-invariant function $\epsilon: I \rightarrow \mathbb{R}$ can be used to deform $\mathcal{O}_q(G_{\mathbb{R}})$ into a $*$-algebra $\mathcal{O}_q^{\nu,\mathrm{id}}(G_{\mathbb{R}})$ by a modification of the Drinfeld double construction. We then show how, by a generalized theory of universal $K$-matrices, a specific $*$-subalgebra $\mathcal{O}_q(G_{\nu}\backslash \backslash G_{\mathbb{R}})$ of $\mathcal{O}_q^{\nu,\mathrm{id}}(G_{\mathbb{R}})$ admits $*$-homomorphisms into both $U_q(\mathfrak{u})$ and $\mathcal{O}_q(U)$, the images being coideal $*$-subalgebras of respectively $U_q(\mathfrak{u})$ and $\mathcal{O}_q(U)$. 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.