A Categorical Approach to Groupoid Frobenius Algebras

TL;DR

This paper establishes a correspondence between $ ext{G}$-Frobenius algebras for finite groupoids and a special class of Frobenius objects in the representation category of the Drinfeld double of the quantum groupoid, enriching the algebraic understanding of groupoid-based structures.

Contribution

It introduces a categorical framework linking $ ext{G}$-Frobenius algebras to Frobenius objects in the Drinfeld double's representation category, providing new insights into quantum groupoid algebra.

Findings

$ ext{G}$-Frobenius algebras correspond to Frobenius objects in the Drinfeld double category

The paper characterizes the structure of these Frobenius objects

Provides a categorical equivalence relating algebraic and quantum groupoid structures

Abstract

In this paper, we show that $\C{G}$-Frobenius algebras (for $\C{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\C{G}])$, where $D(k[\C{G}])$ is the Drinfeld double of the quantum groupoid $k[\C{G}]$.