Convolution algebras for Relational Groupoids and Reduction

TL;DR

This paper introduces relational groupoids and convolution algebras, providing examples, conditions for Haar systems, and a reduction theorem connecting to Lie groupoid convolution.

Contribution

It defines relational groupoids and convolution algebras, offering new frameworks and results that generalize classical groupoid convolution theory.

Findings

Examples from group algebras and normal subgroups

Conditions for Haar systems on relational groupoids

A reduction theorem linking to Lie groupoid convolution

Abstract

We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a Haar system of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.