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Graded Steinberg algebras and their representations

Journal Article


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.

UOW Authors


  •   Ara, Pere (external author)
  •   Hazrat, Roozbeh (external author)
  •   Li, Huanhuan (external author)
  •   Sims, Aidan

Publication Date


  • 2018

Citation


  • Ara, P., Hazrat, R., Li, H. & Sims, A. (2018). Graded Steinberg algebras and their representations. Algebra & Number Theory, 12 (1), 131-172.

Scopus Eid


  • 2-s2.0-85046767621

Number Of Pages


  • 41

Start Page


  • 131

End Page


  • 172

Volume


  • 12

Issue


  • 1

Place Of Publication


  • United States

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.

UOW Authors


  •   Ara, Pere (external author)
  •   Hazrat, Roozbeh (external author)
  •   Li, Huanhuan (external author)
  •   Sims, Aidan

Publication Date


  • 2018

Citation


  • Ara, P., Hazrat, R., Li, H. & Sims, A. (2018). Graded Steinberg algebras and their representations. Algebra & Number Theory, 12 (1), 131-172.

Scopus Eid


  • 2-s2.0-85046767621

Number Of Pages


  • 41

Start Page


  • 131

End Page


  • 172

Volume


  • 12

Issue


  • 1

Place Of Publication


  • United States