Yoshida algebra for groupoids

TL;DR

This paper generalizes the Yoshida algebra concept from finite groups to finite groupoids, exploring its properties and connections to the groupoid algebra and crossed Burnside ring.

Contribution

It introduces the Yoshida algebra for finite groupoids and establishes key isomorphisms and homomorphisms involving its center and related algebraic structures.

Findings

Center of Yoshida algebra is isomorphic to the center of the groupoid algebra.

Existence of a surjective ring homomorphism from the crossed Burnside ring to the Yoshida algebra's center.

Abstract

In this paper, we extend the notion of the Yoshida algebra of a finite group introduced in \cite{Yos83} to finite groupoids and investigate its fundamental properties. Our main results show that the center of the Yoshida algebra of a finite groupoid is isomorphic to the center of the corresponding groupoid algebra, and that there exists a surjective ring homomorphism from the crossed Burnside ring of a finite groupoid, introduced in \cite{Shi26+}, onto the center of the Yoshida algebra of a finite groupoid.