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.

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 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.

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 Irreducible representations of Leavitt path algebras

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Xiao-Wu Chen
Keyword(s):  
Path Algebra ◽  
Irreducible Representations ◽  
Leavitt Path Algebra ◽  
Reducible Representation ◽  
Path Algebras ◽  
Completely Reducible ◽  
Leavitt Path Algebras

AbstractWe construct some irreducible representations of the Leavitt path algebra of an arbitrary quiver. The constructed representations are associated to certain algebraic branching systems. For a row-finite quiver, we classify algebraic branching systems, to which irreducible representations of the Leavitt path algebra are associated. For a certain quiver, we obtain a faithful completely reducible representation of the Leavitt path algebra. The twisted representations of the constructed ones under the scaling action are studied.

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.

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.

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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