Weighted Leavitt Path Algebras that are Isomorphic to Unweighted Leavitt Path Algebras

Preusser, Raimund

doi:10.1007/s10468-020-09953-1

Cited by 3 publications

(5 citation statements)

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“…They are a model for the algebras L K (n, m) which could not be obtained via the classical theory of Leavitt path algebras. The structure of weighted Leavitt path algebras was further investigated in [18,27,26,29]. Further generalisations of these algebras were also carried out in [28,24].…”

Section: Weighted Leavitt Path Algebras and Their Representationsmentioning

confidence: 99%

Irreducible representations of Leavitt algebras

Hazrat¹,

Preusser²,

Щеголев³

2021

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For a weighted graph E, we construct representation graphs F , and consequently, L K (E)modules V F , where L K (E) is the Leavitt path algebra associated to E, with coefficients in a field K. We characterise representation graphs F such that V F are simple L K (E)-modules. We show that the category of representation graphs of E, RG(E), is a disjoint union of subcategories, each of which contains a unique universal object T which gives an indecomposable L K (E)-module V T and a unique irreducible representation graph S, which gives a simple L K (E)-module V S .Specialising to graphs with one vertex and m loops of weight n, we construct irreducible representations for the celebrated Leavitt algebras L K (n, m). On the other hand, specialising to graphs with weight one, we recover the simple modules of Leavitt path algebras constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable modules. 2020 Mathematics Subject Classification. 16S88.

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Section: Weighted Leavitt Path Algebras and Their Representationsmentioning

confidence: 99%

Irreducible representations of Leavitt algebras

Hazrat¹,

Preusser²,

Щеголев³

2021

Preprint

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“…If all the weights are equal to 1, then the weighted Leavitt path algebras reduce to the usual Leavitt path algebras. While their unweighted cousins only include the Leavitt algebras L(1, n) (n > 1) as special cases, the weighted Leavitt path algebras embrace all Leavitt algebras (see [29]).…”

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confidence: 99%

“…In [29] the weighted graphs (E, w) having the property that the weighted Leavitt path algebra L(E, w) is isomorphic to an unweighted Leavitt path algebra were characterised. Moreover, it was shown that if a weighted Leavitt path algebra is Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then it is isomorphic to an unweighted Leavitt path algebra.…”

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confidence: 99%

“…In Section 4 we present the Gröbner basis result from [15] and the computation of the Gelfand-Kirillov dimension from [26]. Section 5 contains the characterisation of weighted Leavitt path algebras that are isomorphic to unweighted Leavitt path algebras from [29]. In Section 6 we give graph-theoretical criteria for finite-dimensionality, Noetherianess and von Neumann regularity of weighted Leavitt path algebras.…”

mentioning

confidence: 99%

“…Noetherian in this case. If one of the sets Γ and Λ i (i ∈ Γ) is infinite, then one can use[29, Lemma 20] to obtain a contradiction.6. Finite-dimensionality, Noetherianess and von Neumann regularityThroughout this section (E, w) denotes a weighted graph.…”

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Weighted Leavitt path algebras -- an overview

Preusser

2021

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Weighted Leavitt path algebras were introduced in 2013 by Roozbeh Hazrat. These algebras generalise simultaneously the usual Leavitt path algebras and William Leavitt's algebras L(m, n). In this paper we try to give an overview of what is known about the weighted Leavitt path algebras. We also prove some new results (in particular on the graded K-theory of weighted Leavitt path algebras) and mention open problems. Contents 1. Introduction 1 2. Preliminaries 3 3. Unweighted and weighted Leavitt path algebras 4 4. Linear bases and the Gelfand-Kirillov dimension 6 5. Weighted Leavitt path algebras that are isomorphic to unweighted Leavitt path algebras 11 6. Finite-dimensionality, Noetherianess and von Neumann regularity 15 7. Realisation as generalised corner skew Laurent polynomial rings 21 8. Local valuations 26 9. The V-monoid and K 0 28 10. The graded V-monoid and K gr 0 33 11. Representations 40 12. Open problems 50 References 51 1. Introduction Leavitt path algebras are algebras associated to directed graphs. They were introduced by G. Abrams and G. Aranda Pino in 2005 [1] and independently by P. Ara, M. Moreno and E. Pardo in 2007 [7]. The Leavitt path algebras turned out to be a very rich and interesting class of algebras, whose studies so far have comprised over 150 research papers and counting. A comprehensive treatment of the subject can be found in the book [2]. The definition of the Leavitt path algebras was inspired by the algebras L(m, n) studied by W. Leavitt in the 1950's and 60 's [19, 20, 21, 22]. Recall that for positive integers m < n, the Leavitt algebra L(m, n) is universal with the property that L(m, n) m ∼ = L(m, n) n as left L(m, n)-modules.Weighted Leavitt path algebras are algebras associated to weighted graphs, i.e. directed graphs with a weight map associating to each edge a positive integer. These algebras were introduced by R. Hazrat in 2013 [13]. If all the weights are equal to 1, then the weighted Leavitt path algebras reduce to the usual Leavitt path algebras. While their unweighted cousins only include the Leavitt algebras L(1, n) (n > 1) as special cases, the weighted Leavitt path algebras embrace all Leavitt algebras (see [29]).

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