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 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.

scholarly journals Full idempotents in Leavitt path algebras

2019 ◽  
Vol 18 (04) ◽  
pp. 1950062
Author(s):  
Ekrem Emre
Keyword(s):  
Directed Graph ◽  
Sufficient Conditions ◽  
Path Algebra ◽  
Necessary And Sufficient Conditions ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Necessary And Sufficient

We give necessary and sufficient conditions on a directed graph [Formula: see text] for which the associated Leavit path algebra [Formula: see text] has at least one full idempotent. Also, we define [Formula: see text] sub-graphs of [Formula: see text] and show that [Formula: see text] has at least one full idempotent if and only if there is a sub-graph [Formula: see text] such that the associated Leavitt path algebra [Formula: see text] has at least one full idempotent.

TWO-SIDED CHAIN CONDITIONS IN LEAVITT PATH ALGEBRAS OVER ARBITRARY GRAPHS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250044 ◽  
Author(s):  
GENE ABRAMS ◽  
JASON P. BELL ◽  
PINAR COLAK ◽  
KULUMANI M. RANGASWAMY
Keyword(s):  
Directed Graph ◽  
Path Algebra ◽  
Explicit Description ◽  
Leavitt Path Algebra ◽  
Chain Conditions ◽  
Finite Case ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Arbitrary Graphs

Let E be any directed graph, and K be any field. For any ideal I of the Leavitt path algebra LK(E) we provide an explicit description of a set of generators for I. This description allows us to classify the two-sided noetherian Leavitt path algebras over arbitrary graphs. This extends similar results previously known only in the row-finite case. We provide a number of additional consequences of this description, including an identification of those Leavitt path algebras for which all two-sided ideals are graded. Finally, we classify the two-sided artinian Leavitt path algebras over arbitrary graphs.

scholarly journals SUBSETS OF VERTICES GIVE MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS

2017 ◽  
Vol 96 (2) ◽  
pp. 212-222
Author(s):  
LISA ORLOFF CLARK ◽  
ASTRID AN HUEF ◽  
PAREORANGA LUITEN-APIRANA
Keyword(s):  
Directed Graph ◽  
Morita Equivalence ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Algebraic Version ◽  
Morita Equivalent ◽  
Path Algebras ◽  
Leavitt Path Algebras

We show that every subset of vertices of a directed graph$E$gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of$E$can be contracted to a new graph$G$such that the Leavitt path algebras of$E$and$G$are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.

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 STABLE RANK OF LEAVITT PATH ALGEBRAS OF ARBITRARY GRAPHS

2012 ◽  
Vol 88 (2) ◽  
pp. 206-217 ◽  
Author(s):  
HOSSEIN LARKI ◽  
ABDOLHAMID RIAZI
Keyword(s):  
Directed Graph ◽  
Path Algebra ◽  
Stable Rank ◽  
Leavitt Path Algebra ◽  
Finite Graphs ◽  
Finite Case ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Arbitrary Graphs ◽  
Finite Setting

AbstractThe stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.

scholarly journals Irreducible and permutative representations of ultragraph Leavitt path algebras

2020 ◽  
Vol 32 (2) ◽  
pp. 417-431
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Path Algebra ◽  
Irreducible Representations ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Permutative Representations ◽  
Branching System ◽  
Leavitt Path Algebras ◽  
The Way

AbstractWe completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this, we extend to ultragraph Leavitt path algebras Chen’s construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching system and characterize irreducible representations associated to perfect branching systems. Along the way, we improve the characterization of faithfulness of Chen’s irreducible representations.

Leavitt path algebras for power graphs of finite groups

2021 ◽  
pp. 2250209
Author(s):  
Sumanta Das ◽  
M. K. Sen ◽  
S. K. Maity
Keyword(s):  
Finite Group ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Cyclic Groups ◽  
Power Graph ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
A Finite Group ◽  
Generalized Quaternion

The aim of this paper is the characterization of algebraic properties of Leavitt path algebra of the directed power graph [Formula: see text] and also of the directed punctured power graph [Formula: see text] of a finite group [Formula: see text]. We show that Leavitt path algebra of the power graph [Formula: see text] of finite group [Formula: see text] over a field [Formula: see text] is simple if and only if [Formula: see text] is a direct sum of finitely many cyclic groups of order 2. Finally, we prove that the Leavitt path algebra [Formula: see text] is a prime ring if and only if [Formula: see text] is either cyclic [Formula: see text]-group or generalized quaternion [Formula: see text]-group.

scholarly journals A class of corners of a Leavitt path algebra

2019 ◽  
Vol 2 (4) ◽  
pp. 75-81
Author(s):  
Deo Thanh Trinh
Keyword(s):  
Directed Graph ◽  
Finite Subset ◽  
Path Algebra ◽  
Graph Algebras ◽  
Leavitt Path Algebra ◽  
C Algebras ◽  
Path Algebras ◽  
Leavitt Path Algebras

Let E be a directed graph, K a field and LK(E) the Leavitt path algebra of E over K. The goal of this paper is to describe the structure of a class of corners of Leavitt path algebras LK(E). The motivation of this work comes from the paper “Corners of Graph Algebras” of Tyrone Crisp in which such corners of graph C*-algebras were investigated completely. Using the same ideas of Tyrone Crisp, we will show that for any finite subset X of vertices in a directed graph E such that the hereditary subset HE(X) generated by X is finite, the corner ( ) ( )( )     K v X v X v L E v is isomorphic to the Leavitt path algebra LK(EX) of some graph EX. We also provide a way how to construct this graph EX.

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.

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