Vol. 12, No. 1, 2018

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Graded Steinberg algebras and their representations

Pere Ara, Roozbeh Hazrat, Huanhuan Li and Aidan Sims

Vol. 12 (2018), No. 1, 131–172
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 K0-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.

Keywords
Steinberg algebra, Leavitt path algebra, skew-product, smash product, graded irreducible representation, annihilator ideal, effective groupoid
Mathematical Subject Classification 2010
Primary: 22A22
Secondary: 16G30, 18B40
Milestones
Received: 16 April 2017
Revised: 6 November 2017
Accepted: 8 November 2017
Published: 13 March 2018
Authors
Pere Ara
Department of Mathematics
University Autònoma de Barcelona
BGSMath (at CRM)
Campus de Bellaterra
Edifici C 08193 Bellaterra (Barcelona) Spain
Roozbeh Hazrat
Centre for Research in Mathematics
Western Sydney University
South Parramatta Campus, Western Sydney University
Sydney NSW 2150
Australia
Huanhuan Li
Centre for Research in Mathematics
Western Sydney University
South Parramatta Campus, Western Sydney University
Sydney NSW 2150
Australia
Aidan Sims
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong NSW 2522
Australia