Graded Steinberg algebras and their representations

@article{Ara2017GradedSA,
  title={Graded Steinberg algebras and their representations},
  author={Pere Ara and Roozbeh Hazrat and Huanhuan Li and Aidan Sims},
  journal={Algebra \& Number Theory},
  year={2017},
  volume={12},
  pages={131-172}
}
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… 

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References

SHOWING 1-10 OF 54 REFERENCES

Graphs with relations, coverings and group-graded algebras

The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic

On graded irreducible representations of Leavitt path algebras

Group Graded Rings, Smash Products and Additive Categories

For rings graded by a finite group the smash products of the rings with the grading groups play an important part in the duality theory that allows to relate properties of graded nature to ungraded

Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed

A GENERALIZATION OF THE SMASH PRODUCT OF A GRADED RING

A Groupoid Approach to Discrete Inverse Semigroup Algebras

Nonstable K-theory for Graph Algebras

We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and

A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras

Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra and

Leavitt path algebras of separated graphs

Abstract The construction of the Leavitt path algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices

Group graded rings

Let R be a group graded ring . The map ( , ): R × R →1 defined by: (x,y) = (xy)1 , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a
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