Universal property of skew $PBW$ extensions

TL;DR

This paper establishes the universal property of skew PBW extensions, generalizing skew polynomial rings, and provides a new proof of the Poincaré-Birkhoff-Witt theorem for Lie algebra enveloping algebras.

Contribution

It introduces the universal property for skew PBW extensions, broadening the understanding of their algebraic structure and applications.

Findings

Proves the universal property of skew PBW extensions.

Includes various examples like Weyl algebras and quantum matrices.

Provides a new proof of the PBW theorem.

Abstract

In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examples 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. As a corollary we will give a new short proof of the Poincar\'e-Birkhoff-Witt theorem about the bases of enveloping algebras of finite-dimensional Lie algebras.