A construction of an iterated Ore extension

TL;DR

This paper constructs an iterated skew polynomial algebra whose semiclassical limit recovers a specific Poisson algebra with a structured Poisson bracket, providing explicit examples and methods for such constructions.

Contribution

It introduces a method to build iterated skew polynomial algebras corresponding to a class of Poisson algebras with structured brackets, expanding the tools for quantization.

Findings

Explicit construction of iterated skew polynomial algebra

Semiclassical limit matches the given Poisson algebra

Illustrative examples demonstrating the construction

Abstract

Let $B$ be a Poisson algebra $\Bbb C[x_1,\ldots, x_k]$ with Poisson bracket such that $$\{x_j,x_i\}=c_{ji}x_ix_j+p_{ji}$$ for all $j>i$, where $c_{ji}\in\Bbb C$ and $p_{ji}\in\Bbb C[x_1,\ldots,x_i]$. Here we obtain an iterated skew polynomial algebra such that its semiclassical limit is equal to $B$ and the results are illustrated by examples.