# Graded Steinberg algebras and their representations

## TL;DR

This paper explores graded Steinberg algebras and their modules, establishing isomorphisms with skew-product groupoid algebras, analyzing minimal representations, and connecting algebraic structures with groupoid effectiveness and $K_0$-group ideals.

## Contribution

It introduces a graded isomorphism between Steinberg algebras of skew-product groupoids and smash products, and links minimal representations to groupoid effectiveness, with applications to Leavitt path and Kumjian--Pask algebras.

## Key findings

- Isomorphism between graded modules and skew-product groupoid modules
- Connection between minimal representations and groupoid effectiveness
- Structural results on graded von Neumann regularity of Kumjian--Pask algebras

## Abstract

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the $K_0$-group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian--Pask algebras of row-finite $k$-graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.