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Graded relations on crossed products

Ofir Schnabel

Abstract

We classify crossed product gradings for arbitrary groups and fields up to several equivalence relations in terms of group actions and their orbits.

Graded relations on crossed products

Abstract

We classify crossed product gradings for arbitrary groups and fields up to several equivalence relations in terms of group actions and their orbits.
Paper Structure (11 sections, 8 theorems, 77 equations)

This paper contains 11 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Relations on graded algebras
  3. The group $K$ and its action on $\Gamma$
  4. $K$ is a group which acts on $\Gamma$
  5. Proofs of the results from the previous section
  6. Proof of Lemma \ref{['lemma:Kisagroup']}
  7. Proof of Corollary \ref{['cor:concreteactionofK']}
  8. Proof of Theorem \ref{['th:GammaisKinvariant']}
  9. Proof of Theorem \ref{['th:KisactingasagrouponGamma']}
  10. Proof of Theorem A
  11. Examples

Key Result

Corollary 2.6

Two crossed products $R^{\alpha _1}_{\eta _1}G$ and $R^{\alpha _2}_{\eta _2}G$ are equivalent as Clifford systems if and only there exist a graded isomorphism $\psi :R^{\alpha _1}_{\eta _1}G\rightarrow R^{\alpha _2}_{\eta _2}G$ such that the restriction of $\psi$ to $R$ is trivial, that is $\psi (r)

Theorems & Definitions (22)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Corollary 2.6
  • Definition 2.7
  • Lemma 3.1
  • Corollary 3.2
  • ...and 12 more