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

Lauren Grimley, Naomi Krawzik, Colin M. Lawson, Christine Uhl

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.

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

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.

Paper Structure

This paper contains 8 sections, 14 theorems, 86 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Resolutions and chain maps for the group
  4. Twisted product resolutions and chain maps
  5. Deformation cohomology for cyclic transvection groups
  6. Further restricting candidates for parameter functions
  7. Relating system of equations to an algebra homomorphism
  8. Classifying the Drinfeld orbifold algebras

Key Result

Theorem 2.2

Let $G$ be a finite group acting linearly on $V$, a finite dimensional $\mathbb F$-vector space. Then $\mathcal{H}_{\lambda, \kappa}$ is a PBW deformation of $S(V)\rtimes G$ if and only if in $S(V)\rtimes G$, for all $g,h \in G$ and all $u,v,w,v_1,v_2,v_3 \in V$.

Theorems & Definitions (28)

  • Theorem 2.2: SW18 Theorem 6.1
  • Remark 2.3
  • Theorem 2.5: LS24 Theorem 8.1
  • Remark 3.6
  • Lemma 4.3
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.5
  • proof
  • ...and 18 more