## Algebra & Number Theory

### Graded Steinberg algebras and their representations

#### 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 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.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 1 (2018), 131-172.

Dates
Revised: 6 November 2017
Accepted: 8 November 2017
First available in Project Euclid: 4 April 2018

https://projecteuclid.org/euclid.ant/1522807233

Digital Object Identifier
doi:10.2140/ant.2018.12.131

Mathematical Reviews number (MathSciNet)
MR3781435

Zentralblatt MATH identifier
06861738

#### Citation

Ara, Pere; Hazrat, Roozbeh; Li, Huanhuan; Sims, Aidan. Graded Steinberg algebras and their representations. Algebra Number Theory 12 (2018), no. 1, 131--172. doi:10.2140/ant.2018.12.131. https://projecteuclid.org/euclid.ant/1522807233

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