Projective modules and Gr\"obner bases for skew PBW extensions
Oswaldo Lezama, Claudia Gallego

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 extensions in functional linear systems and in non-commutative algebraic geometry.
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