Equivariant Quantizations of Symmetric Algebras

TL;DR

This paper investigates equivariant deformations of symmetric algebras of modules over Lie bialgebras, classifies certain quantizations for simple complex Lie algebras, and introduces co-Poisson module algebras to unify known examples.

Contribution

It classifies equivariant quantizations for simple complex Lie algebras and introduces co-Poisson module algebras to unify various quantized symmetric algebra examples.

Findings

Classification of equivariant quantizations for simple complex Lie algebras.

Introduction of co-Poisson module algebras as a unifying framework.

Many known quantized symmetric algebras fit into the co-Poisson algebra framework.

Abstract

Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra of g. In particular we investigate such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V. We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce more a general notion, co-Poisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language.