scholarly journals Crossed product Leavitt path algebras

2021 ◽  
pp. 1-21
Author(s):  
Roozbeh Hazrat ◽  
Lia Vaš
Keyword(s):  
Group Ring ◽  
Matrix Ring ◽  
Finite Graph ◽  
Path Algebra ◽  
Crossed Product ◽  
Ordered Group ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Skew Group Ring

If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text]

scholarly journals Centers of Leavitt path algebras and their completions

2017 ◽  
Vol 16 (05) ◽  
pp. 1750090 ◽  
Author(s):  
Adel Alahmadi ◽  
Hamed Alsulami
Keyword(s):  
Boolean Algebra ◽  
Finite Graph ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Underlying Graph ◽  
Path Algebras ◽  
Leavitt Path Algebras

In [9,10] Corrales Garcia, Barquero, Martin Gonzalez, Siles Molina, Solanilla Hernandez described the center of a Leavitt path algebra and characterized it in terms of the underlying graph. We offer a different characterization of the center. In particular, we prove that the Boolean algebra of central idempotents of a Leavitt path algebra of a finite graph is isomorphic to the Boolean algebra of finitary annihilator hereditary subsets of the graph.

scholarly journals Cohn–Leavitt path algebras and the invariant basis number property

2019 ◽  
Vol 18 (05) ◽  
pp. 1950086 ◽  
Author(s):  
Müge Kanuni̇ ◽  
Murad Özaydin
Keyword(s):  
Finite Graph ◽  
Path Algebra ◽  
Necessary Condition ◽  
Leavitt Path Algebra ◽  
Morita Equivalent ◽  
Path Algebras ◽  
Invariant Basis ◽  
K Theory ◽  
Leavitt Path Algebras ◽  
Invariant Basis Number

We give the necessary and sufficient condition for a separated Cohn–Leavitt path algebra of a finite digraph to have Invariant Basis Number (IBN). As a consequence, separated Cohn path algebras have IBN. We determine the non-stable K-theory of a corner ring in terms of the non-stable K-theory of the ambient ring. We give a necessary condition for a corner algebra of a separated Cohn–Leavitt path algebra of a finite graph to have IBN. We provide Morita equivalent rings which are non-IBN, but are of different types.

scholarly journals Prüfer modules over Leavitt path algebras

2019 ◽  
Vol 18 (08) ◽  
pp. 1950154
Author(s):  
Gene Abrams ◽  
Francesca Mantese ◽  
Alberto Tonolo
Keyword(s):  
Simple Module ◽  
Finite Graph ◽  
Path Algebra ◽  
Injective Hull ◽  
Closed Path ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Simple Factor ◽  
Leavitt Path Algebras

Let [Formula: see text] denote the Leavitt path algebra associated to the finite graph [Formula: see text] and field [Formula: see text]. For any closed path [Formula: see text] in [Formula: see text], we define and investigate the uniserial, artinian, non-Noetherian left [Formula: see text]-module [Formula: see text]. The unique simple factor of each proper submodule of [Formula: see text] is isomorphic to the Chen simple module [Formula: see text]. In our main result, we classify those closed paths [Formula: see text] for which [Formula: see text] is injective. In this situation, [Formula: see text] is the injective hull of [Formula: see text].

scholarly journals A note on the regular ideals of Leavitt path algebras

2021 ◽  
pp. 2250225
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Finite Graph ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
C Algebras ◽  
Path Algebras ◽  
Regular Ideal ◽  
Vertex Set ◽  
Leavitt Path Algebras

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition (L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.

scholarly journals TWO-SIDED IDEALS IN LEAVITT PATH ALGEBRAS

2011 ◽  
Vol 10 (05) ◽  
pp. 801-809 ◽  
Author(s):  
PINAR COLAK
Keyword(s):  
Finite Graph ◽  
Chain Condition ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Ascending Chain Condition

We explicitly describe two-sided ideals in Leavitt path algebras associated with a row-finite graph. Our main result is that any two-sided ideal I of a Leavitt path algebra associated with a row-finite graph is generated by elements of the form [Formula: see text], where g is a cycle based at vertex v. We use this result to show that a Leavitt path algebra is two-sided Noetherian if and only if the ascending chain condition holds for hereditary and saturated closures of the subsets of the vertices of the row-finite graph E.

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Grzegorz Bajor ◽  
Leon van Wyk ◽  
Michał Ziembowski
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Unital Ring ◽  
Commutative Subalgebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Commutative Subalgebras

Abstract Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

scholarly journals Group gradations on Leavitt path algebras

2019 ◽  
Vol 19 (09) ◽  
pp. 2050165 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Directed Graph ◽  
Path Algebra ◽  
Graded Rings ◽  
Leavitt Path Algebra ◽  
Arbitrary Group ◽  
Graded Ring ◽  
Identity Component ◽  
Path Algebras ◽  
Leavitt Path Algebras

Given a directed graph [Formula: see text] and an associative unital ring [Formula: see text] one may define the Leavitt path algebra with coefficients in [Formula: see text], denoted by [Formula: see text]. For an arbitrary group [Formula: see text], [Formula: see text] can be viewed as a [Formula: see text]-graded ring. In this paper, we show that [Formula: see text] is always nearly epsilon-strongly [Formula: see text]-graded. We also show that if [Formula: see text] is finite, then [Formula: see text] is epsilon-strongly [Formula: see text]-graded. We present a new proof of Hazrat’s characterization of strongly [Formula: see text]-graded Leavitt path algebras, when [Formula: see text] is finite. Moreover, if [Formula: see text] is row-finite and has no source, then we show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] has no sink. We also use a result concerning Frobenius epsilon-strongly [Formula: see text]-graded rings, where [Formula: see text] is finite, to obtain criteria which ensure that [Formula: see text] is Frobenius over its identity component.

scholarly journals Leavitt path algebras satisfying a polynomial identity

2016 ◽  
Vol 15 (05) ◽  
pp. 1650084 ◽  
Author(s):  
Jason P. Bell ◽  
T. H. Lenagan ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Polynomial Identity ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Graph Theoretic ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Dimension One ◽  
Kirillov Dimension

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

scholarly journals Finite-dimensional representations of Leavitt path algebras

2018 ◽  
Vol 30 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Ayten Koç ◽  
Murad Özaydın
Keyword(s):  
Full Subcategory ◽  
Morita Equivalence ◽  
Path Algebra ◽  
Reduction Algorithm ◽  
Quiver Representations ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Finite Dimensional ◽  
Leavitt Path Algebras ◽  
Finite Digraph

Abstract When Γ is a row-finite digraph, we classify all finite-dimensional modules of the Leavitt path algebra {L(\Gamma)} via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph Γ. The category of (unital) {L(\Gamma)} -modules is equivalent to a full subcategory of quiver representations of Γ. However, the category of finite-dimensional representations of {L(\Gamma)} is tame in contrast to the finite-dimensional quiver representations of Γ, which are almost always wild.

scholarly journals Strongly graded Leavitt path algebras

2021 ◽  
pp. 2250141
Author(s):  
Patrik Lundström ◽  
Johan Öinert
Keyword(s):  
Recent Result ◽  
Directed Graph ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Unital Ring ◽  
Path Algebras ◽  
Leavitt Path Algebras

Let [Formula: see text] be a unital ring, let [Formula: see text] be a directed graph and recall that the Leavitt path algebra [Formula: see text] carries a natural [Formula: see text]-gradation. We show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.

Sign in / Sign up

Export Citation Format

Share Document