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 $\ast$-algebra of $\mathfrak{u}$ for some $0<q$ distinct from $1$. Let ${\mathcal{O}}_q(U)$ be the coquasitriangular quantized function Hopf $\ast$-algebra of $U$, whose Drinfeld double ${\mathcal{O}}_q(G_{\mathbb{R}})$ we view as the quantized function $\ast$-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 $\epsilon :I\to \mathbb{R}$ can be used to deform ${\mathcal{O}}_q(G_{\mathbb{R}})$ into a $\ast$-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 $\ast$-subalgebra ${\mathcal{O}}_q(G_{\nu }\backslash \backslash G_{\mathbb{R}})$ of ${\mathcal{O}}_q^{\nu ,\mathrm{id}}(G_{\mathbb{R}})$ admits $\ast$-homomorphisms into both $U_q(\mathfrak{u})$ and ${\mathcal{O}}_q(U)$, the images being coideal $\ast$-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 Balagović and Kolb.

arXiv comment66 pages; some typos were corrected and more references were added. An extra appendix (Appendix A) was added to explain in more detail connections to the work of Semenov-Tian-Shansky

References in corpus (22)

• Quantum symmetric Kac-Moody pairs
• Braided Momentum Structure of the q-Poincare Group
• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• Coideal Subalgebras and Quantum Symmetric Pairs
• Quantum symmetric spaces and related q-orthogonal polynomials
• On idempotent states on quantum groups
• The bar involution for quantum symmetric pairs
• Quantized algebras of functions on homogeneous spaces with Poisson stabilizers
• q-difference intertwining operators for $U_q(sl(n))$: general setting and the case $n=3$
• Quantized flag manifolds and irreducible *-representations
• Embeddable Quantum Homogeneous Spaces
• Method of quantum characters in equivariant quantization
• Compact and discrete subgroups of algebraic quantum groups I
• K-homology class of the Dirac operator on a compact quantum group
• Braided module categories via quantum symmetric pairs
• Non-Levi closed conjugacy classes of SP_q(2n)
• Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
• Ribbon braided module categories, quantum symmetric pairs and Knizhnik-Zamolodchikov equations
• Quantization of borderline Levi conjugacy classes of orthogonal groups
• On representations of quantum conjugacy classes of GL(n)
• Quantum sphere S^4 as a non-Levi conjugacy class
• Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras

Cited by in corpus (4)

• Universal K-matrices for quantum Kac-Moody algebras
• Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals
• Quantum $SL(2,\mathbb{R})$ and its irreducible representations
• Invariant integrals on coideals and their Drinfeld doubles