Leavitt path algebras over arbitrary unital rings and algebras

2019 ◽  
Vol 19 (06) ◽  
pp. 2050107
Author(s):  
Hans Nordstrom ◽  
Jennifer A. Firkins Nordstrom
Keyword(s):  
Commutative Ring ◽  
General Setting ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Unital Rings ◽  
Rings And Algebras

We expand the work of Tomforde by further extending the construction of Leavitt path algebras (LPAs) over arbitrary associative, unital rings. We show that many of the results over a commutative ring hold in the more general setting, provide some useful generalizations of prior results, and give a definition for an iterated Leavitt path extension in our context.

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.

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 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.

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