Ore and Goldie theorems for skew PBW extensions

TL;DR

This paper extends classical Ore and Goldie theorems to skew PBW extensions, a broad class of non-commutative rings relevant in quantum mechanics, and proves a quantum Gelfand-Kirillov conjecture.

Contribution

It generalizes fundamental theorems to skew PBW extensions and applies these results to quantum polynomial algebras.

Findings

Ore and Goldie theorems are valid for skew PBW extensions.

Proves the quantum Gelfand-Kirillov conjecture for skew quantum polynomials.

Establishes a framework connecting skew PBW extensions with quantum algebra structures.

Abstract

Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincar\'e-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand-Kirillov conjecture for the skew quantum polynomials.