scholarly journals Purely infinite simple Kumjian–Pask algebras

2018 ◽  
Vol 30 (1) ◽  
pp. 253-268 ◽  
Author(s):  
Hossein Larki
Keyword(s):  
Commutative Ring ◽  
Path Algebras ◽  
Higher Rank ◽  
Leavitt Path Algebras ◽  
Unital Simple

Abstract Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian–Pask algebra {\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if {\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then {\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and {\mathrm{KP}_{R}(\Lambda)} is simple, then {\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.

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.

A note on the centralizer of a subalgebra of the Steinberg algebra

2021 ◽  
Vol 33 (1) ◽  
pp. 179-184
Author(s):  
R. Hazrat ◽  
Huanhuan Li
Keyword(s):  
Commutative Ring ◽  
Invariant Subset ◽  
Content Type ◽  
Path Algebras ◽  
Unit Space ◽  
Leavitt Path Algebras

For an ample Hausdorff groupoid G \mathcal {G} , and the Steinberg algebra A R ( G ) A_R(\mathcal {G}) with coefficients in the commutative ring R R with unit, the centralizer is described for the subalgebra A R ( U ) A_R(U) with U U an open closed invariant subset of the unit space of  G \mathcal {G} . In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.

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 CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS

2018 ◽  
Vol 104 (3) ◽  
pp. 403-411 ◽  
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebras ◽  
Chain Conditions ◽  
Path Algebras ◽  
Finite Dimensional ◽  
Unit Space ◽  
Leavitt Path Algebras ◽  
Étale Groupoid ◽  
Totally Disconnected

The author has previously associated to each commutative ring with unit$R$and étale groupoid$\mathscr{G}$with locally compact, Hausdorff and totally disconnected unit space an$R$-algebra$R\,\mathscr{G}$. In this paper we characterize when$R\,\mathscr{G}$is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

scholarly journals Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 203-245 ◽  
Author(s):  
P. Ara ◽  
E. Pardo
Keyword(s):  
Weak Version ◽  
Graded Algebras ◽  
Path Algebras ◽  
Theoretic Characterization ◽  
Leavitt Path Algebras

AbstractIn Hazrat gave a K-theoretic invariant for Leavitt path algebras as graded algebras. Hazrat conjectured that this invariant classifies Leavitt path algebras up to graded isomorphism, and proved the conjecture in some cases. In this paper, we prove that a weak version of the conjecture holds for all finite essential graphs.

scholarly journals GCR and CCR Steinberg Algebras

2019 ◽  
Vol 72 (6) ◽  
pp. 1581-1606 ◽  
Author(s):  
Lisa O. Clark ◽  
Benjamin Steinberg ◽  
Daniel W. van Wyk
Keyword(s):  
Representation Theory ◽  
Locally Compact ◽  
Path Algebras ◽  
Leavitt Path Algebras

AbstractKaplansky introduced the notions of CCR and GCR $C^{\ast }$-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.

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.

Sign in / Sign up

Export Citation Format

Share Document