More on quantum groups from the the quantization point of view

TL;DR

This paper explores the quantization of classical structures like Lie groups and algebras into quantum groups using star products, providing new insights into their properties and representations within the deformation framework.

Contribution

It introduces new star product constructions for classical doubles and symplectic groupoids, and discusses the quantization of Poisson structures leading to quantum groups, including an analogue of Kirillov's formula.

Findings

Star products on classical doubles yield quantum doubles.

Quantization of Poisson structures produces the quantum enveloping algebra.

Classical properties of compact groups are derived as simple consequences.

Abstract

Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex" quantum groups and bicovariant quantum Lie algebras are discused from this point of view. Further we discuss the quantization of the Poisson structure on symmetric algebra $S(g)$ leading to the quantized enveloping algebra $U_{h}(g)$ as an example of biquantization in the sense of Turaev. Description of $U_{h}(g)$ in terms of the generators of the bicovariant differential calculus on $F(G_q)$ is very convenient for this purpose. Finally we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducible representation in the compact case.