Strongly graded groupoids and strongly graded Steinberg algebras

TL;DR

This paper investigates strongly graded topological groupoids and their Steinberg algebras, establishing conditions for strong grading and providing a graphical characterization of related Leavitt path and Kumjian-Pask algebras.

Contribution

It introduces the concept of strongly graded groupoids, proves an analogue of Dade's Theorem for these structures, and characterizes strongly graded Steinberg algebras and related graph algebras.

Findings

Strongly graded groupoids are characterized by their associated sheaves.

Steinberg algebra is strongly graded if and only if the groupoid is.

Graphical criteria for strong grading in Leavitt path and Kumjian-Pask algebras.

Abstract

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta) = \kappa^{-1}(\gamma \delta)$, for all $\gamma, \delta \in \Gamma$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced by a $\mathcal G_{\epsilon}$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.