scholarly journals Using the Steinberg algebra model to determine the center of any Leavitt path algebra

2019 ◽  
Vol 230 (1) ◽  
pp. 23-44 ◽  
Author(s):  
Lisa Orloff Clark ◽  
Dolores Martín Barquero ◽  
Cándido Martín González ◽  
Mercedes Siles Molina
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Algebra Model

scholarly journals The socle series of a Leavitt path algebra

2011 ◽  
Vol 184 (1) ◽  
pp. 413-435 ◽  
Author(s):  
Gene Abrams ◽  
Kulumani M. Rangaswamy ◽  
Mercedes Siles Molina
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra

An AF Algebra Associated with the Farey Tessellation

2008 ◽  
Vol 60 (5) ◽  
pp. 975-1000 ◽  
Author(s):  
Florin P. Boca
Keyword(s):  
Half Plane ◽  
Path Algebra ◽  
Cutting Sequences ◽  
Upper Half Plane ◽  
Af Algebras ◽  
Algebra Model

AbstractWe associate with the Farey tessellation of the upper half-plane an AF algebra encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen AF algebras arise as quotients of . Using the path algebra model for AF algebras we construct, for each τ ∈ ( 0, ¼], projections (En) in such that EnEn±1En ≤ τ En.

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.

The decomposition of cyclic modules in weighted Leavitt path algebra of reducible graph

2021 ◽  
Vol 10 (5) ◽  
pp. 10-14
Author(s):  
Phuc Ngo Tan ◽  
Thanh Tran Ngoc ◽  
Trung Tang Vo Nhat
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Reducible Graph

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 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 ALGEBRAIC CUNTZ–KRIEGER ALGEBRAS

2019 ◽  
Vol 109 (1) ◽  
pp. 93-111
Author(s):  
ALIREZA NASR-ISFAHANI
Keyword(s):  
Directed Graph ◽  
Finite Graph ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Group Of Units ◽  
Unital Ring ◽  
Morita Equivalent

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

scholarly journals The center of a Leavitt path algebra

2011 ◽  
pp. 621-644 ◽  
Author(s):  
Gonzalo Aranda Pino ◽  
Kathi Crow
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra
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