scholarly journals Morita Equivalence and Morita Invariant Properties: Applications in the Context of Leavitt Path Algebras

2016 ◽  
pp. 83-103 ◽  
Author(s):  
Mercedes Siles Molina ◽  
José F. Solanilla Hernández
Keyword(s):  
Morita Equivalence ◽  
Path Algebras ◽  
Morita Invariant ◽  
Leavitt Path Algebras ◽  
Invariant Properties

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 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 BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

scholarly journals On the representations of Leavitt path algebras

2011 ◽  
Vol 333 (1) ◽  
pp. 258-272 ◽  
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals FP-injective semirings, semigroup rings and Leavitt path algebras

2016 ◽  
Vol 45 (5) ◽  
pp. 1893-1906 ◽  
Author(s):  
Marianne Johnson ◽  
Tran Giang Nam
Keyword(s):  
Semigroup Rings ◽  
Path Algebras ◽  
Leavitt Path 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 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.

Sign in / Sign up

Export Citation Format

Share Document