Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 093, 26 pages      arXiv:2312.00341      https://doi.org/10.3842/SIGMA.2024.093

Convolution Algebras of Double Groupoids and Strict 2-Groups

Angel Román ^a and Joel Villatoro ^b
a) Washington University in St. Louis, USA
^b) Indiana University Bloomington, USA

Received February 12, 2024, in final form October 09, 2024; Published online October 19, 2024

Abstract
Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally, strict 2-groups. Groupoid structures give rise to convolution operations on the space of arrows. Therefore, a double groupoid comes equipped with two product operations on the space of functions. In this article we investigate in what sense these two convolution operations are compatible. We use the representation theory of compact Lie groups to get insight into a certain class of 2-groups.

Key words: Lie groupoids; convolution; double groupoids; 2-groups; Haar systems.

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