Quantization of Poisson CGL extensions

TL;DR

This paper develops a method to quantize Poisson CGL extensions, connecting classical Poisson structures with their quantum counterparts, and proves the uniqueness of this quantization process.

Contribution

It introduces quantum-CGL extensions as $L$-forms of CGL extensions, providing an explicit construction and establishing the uniqueness of their quantization from Poisson-CGL extensions.

Findings

Constructed symmetric quantum-CGL extensions from Poisson-CGL extensions.

Established the uniqueness of the quantization process.

Applied results to coordinate rings of matrix affine Poisson spaces.

Abstract

CGL extensions, named after G. Cauchon, K. Goodearl, and E. Letzter, are a special class of noncommutative algebras that are iterated Ore extensions of associative algebras with compatible torus actions. Examples of CGL extensions include quantum Schubert cells and quantized coordinate rings of double Bruhat cells. CGL extensions have been studied extensively in connection with quantum groups and quantum cluster algebras. For a field $\mathbf{k}$ of characteristic $0$, let $L=\mathbf{k}[q^{\pm 1}]$ be the $\mathbf{k}$-algebra of Laurent polynomials in the single variable $q$ and let $\mathbb{K}=\mathbf{k}(q)$ be the fraction field of $L$. We introduce quantum-CGL extensions as certain $L$-forms of CGL extensions over $\mathbb{K}$, which have Poisson-CGL extensions as their semiclassical limits. Poisson-CGL extensions, recently introduced and systematically studied by K. Goodearl and M. Yakimov, are certain Poisson polynomial algebras which admit presentations as iterated Poisson-Ore extensions with compatible torus actions. Examples of Poisson-CGL extensions include the coordinate rings of matrix affine Poisson spaces and more generally those of Schubert cells. We describe an explicit procedure for constructing a symmetric quantum-CGL extension from a symmetric integral Poisson-CGL extension and establish the uniqueness of such a quantization in a proper sense.