Projective modules and Gr\"obner bases for skew PBW extensions

TL;DR

This paper explores the structure of finitely generated projective modules and develops Gr"obner basis theory for skew PBW extensions, which are important in non-commutative algebra and quantum mechanics applications.

Contribution

It introduces a matrix-constructive approach to projective modules and constructs Gr"obner bases for left ideals and modules in skew PBW extensions, advancing computational methods in non-commutative algebra.

Findings

Developed a matrix-constructive method for projective modules.

Established Gr"obner basis theory for skew PBW extensions.

Potential applications in non-commutative geometry and quantum systems.

Abstract

Many rings and algebras arising in quantum mechanics, algebraic analysis, and non-commutative algebraic geometry can be interpreted as skew PBW (Poincar\'e-Birkhoff-Witt) extensions. In the present paper we study two aspects of these non-commutative rings: its finitely generated projective modules from a matrix-constructive approach, and the construction of the Gr\"obner theory for its left ideals and modules. These two topics could be interesting in future eventual applications of skew $PBW$ extensions in functional linear systems and in non-commutative algebraic geometry.