Graded deformations of skew group algebras for cyclic transvection groups acting on polynomial rings in positive characteristic

TL;DR

This paper characterizes and constructs deformations of skew group algebras for cyclic groups acting on polynomial rings in positive characteristic, revealing new algebraic structures and conditions.

Contribution

It provides a complete characterization and explicit formulas for PBW deformations of skew group algebras in positive characteristic, including a practical method to generate examples.

Findings

Full classification of PBW deformations in this setting

Explicit formulas for the deformations

A combinatorial analysis of the algebra classes

Abstract

We investigate deformations of skew group algebras that arise from a finite cyclic group acting on a polynomial ring in positive characteristic, where characteristic divides the order of the group. We allow deformations which deform both the group action and the vector space multiplication. We fully characterize the Poincare-Birkhoff-Witt deformations which arise in this setting from multiple perspectives: a necessary and sufficient condition list, a practical road map from which one can generate examples corresponding to any choice of group algebra element, an explicit formula, and a combinatorial analysis of the class of algebras.