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.

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 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 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.

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 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.

On graded primitive leavitt path algebras

2020 ◽  
pp. 2150173
Author(s):  
Kulumani M. Rangaswamy
Keyword(s):  
Directed Graphs ◽  
Sufficient Conditions ◽  
Path Algebra ◽  
Prime Rings ◽  
Leavitt Path Algebra ◽  
Von Neumann ◽  
Path Algebras ◽  
Von Neumann Regular ◽  
Leavitt Path Algebras ◽  
Necessary And Sufficient

Graded primitive Leavitt path algebras of arbitrary directed graphs over a field [Formula: see text] are completely characterized by means of graphical conditions. Necessary and sufficient conditions are given under which a graded prime Leavitt path algebra becomes graded primitive and this leads to answering the graded version of a question of Kaplansky on von Neumann regular prime rings in the context of Leavitt path algebras.

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