Partial Skew Group Rings Research Articles

Algebras of one-sided subshifts over arbitrary alphabets

We introduce two algebras associated with a subshift over an arbitrary alphabet. One is unital, and the other not necessarily. We focus on the unital case and describe a conjugacy between Ott–Tomforde–Willis subshifts in terms of a homeomorphism between the Stone duals of suitable Boolean algebras, and in terms of a diagonal-preserving isomorphism of the associated unital algebras. For this, we realise the unital algebra associated with a subshift as a groupoid algebra and a partial skew group ring.

Partial Group Representations on Semialgebras

Let A bean additively cancellative semialgebra over an additively cancellative semifield K as defined in [9]. For a given partial action α of a group G on an algebra, the associativity of partial skew group ring together with the existence and uniqueness of enveloping (global) action were studied by M. Dokuchaev and R. Exel [2] which were extended for semialgebras with some restriction by Sharma et. al. using the ring of differences. In a similar way, we extend the results of [2,3] for semialgebras regarding partial representations.

Leavitt path algebras of labelled graphs

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.

Ultragraph algebras via labelled graph groupoids, with applications to generalized uniqueness theorems

Ultragraphs give rise to labelled graphs. We realize algebras associated to such labelled graphs as groupoid algebras, generalizing a known groupoid algebra realization of ultragraph C*-algebras to any ultragraph. Then, we characterize the shift space of an ultragraph as the tight spectrum of the inverse semigroup associated with an ultragraph via its labelled graph. In the purely algebraic setting, we show that the algebraic partial action used to describe an ultragraph Leavitt path algebra as a partial skew group ring is equivalent to a dual of a topological partial action, and we use this to describe ultragraph Leavitt path algebras as Steinberg algebras. Finally, we prove generalized uniqueness theorems for both ultragraph C*-algebras and ultragraph Leavitt path algebras and characterize their abelian core subalgebras.

On partial skew groupoid rings

Given a partial action [Formula: see text] of a connected groupoid [Formula: see text] on an associative ring [Formula: see text] we investigate under what conditions the partial skew groupoid ring [Formula: see text] can be realized as a partial skew group ring. In such a case, applications concerning to the separability, semisimplicity and Frobenius property of the ring extension [Formula: see text] as well as to the artinianity of [Formula: see text] are given.

Representations of relative Cohn path algebras

We study relative Cohn path algebras, also known as Leavitt-Cohn path algebras, and we realize them as partial skew group rings. To do this we prove uniqueness theorems for relative Cohn path algebras. Furthermore, given any graph E we define E-relative branching systems and prove how they induce representations of the associated relative Cohn path algebra. We give necessary and sufficient conditions for faithfulness of the representations associated to E-relative branching systems. This improves previous results known to Leavitt path algebras of row-finite graphs with no sinks. To prove this last result we show first a version, for relative Cohn-path algebras, of the reduction theorem for Leavitt path algebras.

Induced quotient group gradings of epsilon-strongly graded rings

Let [Formula: see text] be a group and let [Formula: see text] be a [Formula: see text]-graded ring. Given a normal subgroup [Formula: see text] of [Formula: see text], there is a naturally induced [Formula: see text]-grading of [Formula: see text]. It is well known that if [Formula: see text] is strongly [Formula: see text]-graded, then the induced [Formula: see text]-grading is strong for any [Formula: see text]. The class of epsilon-strongly graded rings was recently introduced by Nystedt et al. as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and sufficient conditions for the induced [Formula: see text]-grading of an epsilon-strongly [Formula: see text]-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units ([Formula: see text]-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings.

SIMPLICITY AND CHAIN CONDITIONS FOR ULTRAGRAPH LEAVITT PATH ALGEBRAS VIA PARTIAL SKEW GROUP RING THEORY

AbstractWe realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial C⁎-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.

Artinian and noetherian partial skew groupoid rings

Let α={αg:Rg−1→Rg}g∈mor(G) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S=R⋆αG be the associated partial skew groupoid ring. We show that if α is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and Rg={0}, for all but finitely many g∈mor(G). We use this result to prove that if α is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and Rg={0}, for all but finitely many g∈mor(G). This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R. We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings.

The interplay between Steinberg algebras and skew rings

We study the interplay between Steinberg algebras and skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorff ample groupoid, can be seen as a partial skew inverse semigroup ring.

Ring theoretic properties of partial crossed products and related themes

In this paper we work with partial actions and unital twisted partial actions of groups. We investigate ring theoretic properties of partial crossed products and partial skew group rings as artinianity, noetherianity, perfect property, semilocal property, semiprimary property and Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products. Finally we study when the partial crossed products are Frobenius algebras.

Partial crossed products as equivalence relation algebras

For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial skew group ring. This generalizes, to the purely algebraic setting, the known characterization of partial $C^*$-crossed products as groupoid $C^*$-algebras. For completeness we include a new proof of the $C^*$ result for free partial actions.

Outer Partial Actions and Partial Skew Group Rings

AbstractWe extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Simple semigroup graded rings

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and Re has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring f ReGe f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Gonçalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics

Let R0 be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of R0, all of which have local units. We show that R0 is maximal commutative in the partial skew group ring R0⋊αG if and only if R0 has the ideal intersection property in R0⋊αG. From this we derive a criterion for simplicity of R0⋊αG in terms of maximal commutativity and G-simplicity of R0. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.

Simplicity of Partial Skew Group Rings of Abelian Groups

AbstractLet A be a ring with local units, E a set of local units for A, G an abelian group, and α a partial action of G by ideals of A that contain local units. We show that A*αG is simple if and only if A is G-simple and the center of the corner eδ0(A*αGe)eδ0 is a field for all e ∊ E. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.

Partial Skew Group Rings Over Polycyclic by Finite Groups

In this paper we consider a partial action α of a polycyclic by finite group G on a ring R. We prove that if R is right noetherian, then the partial skew group ring R ⋆ αG is also right noetherian. Extending the methods of Passman in Passman (Trans Am Math Soc 301:737–759, 1987), we obtain a description of the prime spectrum of R ⋆ αG. The results obtained are applied to get bounds for the Krull dimension and the classical Krull dimension of R ⋆ αG.

Partial actions of groups on algebras, a survey

In this survey paper we define partial actions of groups on algebras and give several related results. For any partial action a partial skew group ring is defined. This partial skew group ring is not always associative. Conditions under which associativity holds are studied. Several other questions are considered like, for example, enveloping actions and properties of partial skew group rings.