Quantization of pencils with a gl-type Poisson center and braided geometry

TL;DR

This paper explores the quantization of Poisson pencils with a GL-type center within braided geometry, focusing on their structures, super-analogs, and quantum versions, with detailed examples provided.

Contribution

It introduces a method for quantizing Poisson pencils with a GL-type center using braided affine geometry, including super-analogs and explicit examples.

Findings

Poisson brackets can be restricted to GL(m)-orbits

Quantization frameworks extend to super-analogs

Explicit examples illustrate the quantization process

Abstract

In the algebra Sym(gl(m)) we consider Poisson pencils generated by the linear Poisson-Lie bracket {,}_{gl(m)} and that corresponding to the so-called Reflection Equation Algebra. Each bracket of such a pencil has the Poisson center coinciding with that of the bracket {,}_{gl(m)}. Consequently, any bracket from this pencil can be restricted to a generic GL(m)-orbit O in gl(m)*. Quantization of such a restricted bracket can be done in the frameworks of braided affine geometry. In the paper we consider these Poisson structures, their super-analogs as well as their quantum (braided) counterparts. Also, we exhibit some detailed examples.