Leavitt path algebras of weighted Cayley graphs $$\varvec{C_n(S,w)}$$

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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Group gradations on Leavitt path algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501650 ◽

2019 ◽

Vol 19 (09) ◽

pp. 2050165 ◽

Cited By ~ 1

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.

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Representations of Leavitt path algebras

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2019.07.018 ◽

2020 ◽

Vol 224 (3) ◽

pp. 1297-1319

Author(s):

Ayten Koç ◽

Murad Özaydın

Keyword(s):

Path Algebras ◽

Leavitt Path Algebras

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