# 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…

## 46 Citations

### Sectional algebras of semigroupoid bundles

- MathematicsInt. J. Algebra Comput.
- 2020

This article generalizes the definition of smash products to groupoid graded algebras and proves that whenever $\theta$ is a $\land$-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid $\mathcal{G}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinburg algebra of $\math Calculus$ by $S.

### On induced graded simple modules over graded Steinberg algebras with applications to Leavitt path algebras

- Mathematics
- 2020

For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$, Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra…

### Graded K-theory and Leavitt path algebras

- MathematicsJournal of Algebraic Combinatorics
- 2022

Let G be a group and l a commutative unital ∗-ring with an element λ ∈ l such that λ + λ = 1. We introduce variants of hermitian bivariant K-theory for ∗-algebras equipped with a G-action or a…

### Graded Cancellation Properties of Graded Rings and Graded Unit-regular Leavitt Path Algebras

- MathematicsAlgebras and Representation Theory
- 2020

We raise the following general question regarding a ring graded by a group: “If P is a ring-theoretic property, how does one define the graded version P gr of the property P in a meaningful way?”.…

### Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

- Mathematics
- 2022

. We show that any pointed, peordered module map BF gr ( E ) → BF gr ( F ) between Bowen-Franks modules of ﬁnite graphs can be lifted to a unital, graded, diagonal preserving ∗ -homomorphism L ℓ ( E…

### Graded Cancellation Properties of Graded Rings and Graded Unit-regular Leavitt Path Algebras

- MathematicsAlgebras and Representation Theory
- 2020

We raise the following general question regarding a ring graded by a group: “If P is a ring-theoretic property, how does one define the graded version Pgr of the property P in a meaningful way?”.…

### Graded semigroups

- MathematicsIsrael Journal of Mathematics
- 2022

We systematically develop a theory of graded semigroups, that is semigroups S partitioned by groups G, in a manner compatible with the multiplication on S. We define a smash product S#G, and show…

### TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

- MathematicsJournal of the Australian Mathematical Society
- 2021

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that…

## References

SHOWING 1-10 OF 54 REFERENCES

### Graphs with relations, coverings and group-graded algebras

- Mathematics
- 1983

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…

### Group Graded Rings, Smash Products and Additive Categories

- Mathematics
- 1988

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

- MathematicsTransactions of the American Mathematical Society
- 2018

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…

### Nonstable K-theory for Graph Algebras

- Mathematics
- 2004

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

- Mathematics
- 2016

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

- Mathematics
- 2010

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

- Mathematics
- 1983

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…