$\Uh$ invariant Quantization of Coadjoint Orbits and Vector Bundles over them

Abstract

Let M be a coadjoint semisimple orbit of a simple Lie group G. Let $U_h(\g)$ be a quantum group corresponding to G. We construct a universal family of $U_h(\g)$ invariant quantizations of the sheaf of functions on M and describe all such quantizations. We also describe all two parameter $U_h(\g)$ invariant quantizations on M, which can be considered as $U_h(\g)$ invariant quantizations of the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M. We also consider how those quantizations relate to the natural polarizations of M with respect to the KKS bracket. Using polarizations, we quantize the sheaves of sections of vector bundles on M as one- and two-sided $U_h(\g)$ invariant modules over a quantized function sheaf.

Latex2e, 26 pp

References in corpus (4)

• Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation
• Quantization of Lie bialgebras, V
• Double quantization on the coadjoint representation of sl(n)
• Double quantization on coadjoint representations of simple Lie groups and its orbits

Cited by in corpus (6)

• Explicit equivariant quantization on (co)adjoint orbits of $GL(n,\C)$
• $\U_q(sl(n))$-invariant quantization of symmetric coadjoint orbits via reflection equation algebra
• Equivariant quantization on quotients of simple Lie groups by reductive subgroups of maximal rank
• Dynamical Yang-Baxter equation and quantum vector bundles
• Quantum G-manifolds
• Quantum coadjoint orbits of GL(n) and generalized Verma modules