Extensions and duality

TL;DR

This paper explores the relationship between strongly graded algebras and gerbes on groupoids, extending the concept to Hopf algebras and revealing dualities especially in the case of abelian groups.

Contribution

It introduces a new notion of extensions for Hopf algebras and clarifies the duality between extensions with a Hopf algebra and its dual, generalizing previous work.

Findings

Established a correspondence between strongly graded algebras and gerbes on groupoids.

Extended the concept of extensions to Hopf algebras, including the case of group algebras.

Identified a symmetric duality for abelian group cases.

Abstract

For a fixed finite group $Q$ and semi-simple finite dimensional algebra $S$, we examine an equivalence between strongly $Q$-graded algebras (extensions) with identity component $S$ and $S^1$-gerbes on action groupoids of $Q$ on the set of isomorphism classes of simple objects of the category of $S$-modules. This clarifies the nature of the map considered in arXiv:1312.7316. Motivated by this and arXiv:0909.3140(2) we suggest and study a notion of extensions suitable to the case when $S$ is replaced by a Hopf algebra, in the sense that there is a bijection between extensions with "fiber" $H$ and $H^*$. In particular we focus on the case of $H$ equal to the group algebra of a finite group. When $K$ is abelian, the answer is particularly symmetric as duality of Hopf algebras does not take us outside of the category of groups.