The V-monoid of a weighted Leavitt path algebra

Preusser, Raimund

doi:10.1007/s11856-019-1915-1

Cited by 10 publications

(9 citation statements)

References 7 publications

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“…Although the computation of the V-monoid can be carried out for the Leavitt path algebra of arbitrary (finite) weighted graphs (see e.g. [20]), for the current work we need only describe the V-monoid of weighted Leavitt path algebras associated to vertex weighted graphs, which we now do. Definition 5.5.…”

Section: Weighted Leavitt Path Algebrasmentioning

confidence: 99%

“…(cf. [20], [21]) Let (E, w) be a row-finite vertex weighted graph. Then there is an isomorphism of monoids…”

Section: Weighted Leavitt Path Algebrasmentioning

confidence: 99%

“…The weighted Leavitt path algebras L k (E, w) provide a natural context in which all of Leavitt's algebras (corresponding to any pair n, k ∈ N) can be realised as a specific example. As well, the monoid V(L k (E, w)) and the corresponding Grothendieck group K 0 (L k (E, w)) have been completely described in the two works [15] and [20].…”

Section: Introductionmentioning

confidence: 99%

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Connections between Abelian sandpile models and the $K$-theory of weighted Leavitt path algebras

Abrams¹,

Hazrat²

2021

Preprint

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In our main result, we establish that any conical sandpile monoid M = SP(G) of a directed sandpile graph G can be realised as the V-monoid of a weighted Leavitt path algebra L k (E, w), and consequently, the sandpile group as the Grothendieck group K 0 (L k (E, w)). We show how to explicitly construct (E, w) from G. Additionally, we describe the conical sandpile monoids which arise as the V-monoid of a standard (i.e., unweighted) Leavitt path algebra.

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Section: Weighted Leavitt Path Algebrasmentioning

confidence: 99%

“…(cf. [20], [21]) Let (E, w) be a row-finite vertex weighted graph. Then there is an isomorphism of monoids…”

Section: Weighted Leavitt Path Algebrasmentioning

confidence: 99%

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Connections between Abelian sandpile models and the $K$-theory of weighted Leavitt path algebras

Abrams¹,

Hazrat²

2021

Preprint

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“…For a ring R with local units the Grothendieck group K 0 (R) is the group completion of V(R). In [25] the V-monoid and the Grothendieck group of a weighted Leavitt path algebra were computed.…”

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confidence: 99%

“…In Section 8 we describe the local valuations for weighted Leavitt path algebras found in [15]. Section 9 contains the computation of the V -monoid and the Grothendieck group of a weighted Leavitt path algebra from [25]. In Section 10, we compute the graded V -monoid and the graded Grothendieck group of a weighted Leavitt path algebra using the Leavitt path algebras of bi-separated graphs, which were recently introduced by R. Mohan and B. Suhas [23].…”

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Weighted Leavitt path algebras -- an overview

Preusser

2021

Preprint

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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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Leavitt Path Algebras of Hypergraphs

Preusser

2019

Bull Braz Math Soc, New Series

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The V-monoid of a weighted Leavitt path algebra

Cited by 10 publications

References 7 publications

Connections between Abelian sandpile models and the $K$-theory of weighted Leavitt path algebras

Connections between Abelian sandpile models and the $K$-theory of weighted Leavitt path algebras

Weighted Leavitt path algebras -- an overview

Leavitt Path Algebras of Hypergraphs

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