The Gelfand–Kirillov dimension of a weighted Leavitt path algebra

Preusser, Raimund

doi:10.1142/s0219498820500590

Cited by 10 publications

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“…Moreover, there is no selfconnected quasicycle and the maximal length of a chain of quasicycles is 2, see [26,Example 31]. Thus GKdim L(E, w) = 2 by Theorem 4.2.3.…”

Section: 2mentioning

confidence: 95%

“…. x n is a quasicycle, then x i = x j for all i = j by [26,Remark 16(a)]. It follows that there is only a finite number of quasicycles if (E, w) is finite.…”

Section: 2mentioning

confidence: 98%

“…We denote by P ′ the set of all nod-paths which are composed from elements of E ′ . If (E, w) is finite, then |P ′ | < ∞ by [26,Lemma 21]. Lemma 4.2.2 ([26, Lemma 22]).…”

Section: 2mentioning

confidence: 99%

“…In [26] the Gelfand-Kirillov dimension of a weighted Leavitt path algebra was determined. Moreover, it was shown that a finite-dimensional weighted Leavitt path algebra is isomorphic to an unweighted Leavitt path algebra.…”

mentioning

confidence: 99%

“…In Section 3 we recall the definitions of the unweighted and weighted Leavitt path algebras. 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].…”

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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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“…Moreover, there is no selfconnected quasicycle and the maximal length of a chain of quasicycles is 2, see [26,Example 31]. Thus GKdim L(E, w) = 2 by Theorem 4.2.3.…”

Section: 2mentioning

confidence: 95%

“…. x n is a quasicycle, then x i = x j for all i = j by [26,Remark 16(a)]. It follows that there is only a finite number of quasicycles if (E, w) is finite.…”

Section: 2mentioning

confidence: 98%

“…We denote by P ′ the set of all nod-paths which are composed from elements of E ′ . If (E, w) is finite, then |P ′ | < ∞ by [26,Lemma 21]. Lemma 4.2.2 ([26, Lemma 22]).…”

Section: 2mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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