Leavitt Path Algebras of Hypergraphs

Preusser, Raimund

doi:10.1007/s00574-019-00150-3

Cited by 9 publications

(6 citation statements)

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“…We also obtain several other results about (hermitian) graded K-theory of ( * -) algebras in general and Leavitt path algebras in particular which we think are of independent interest. Building upon work of Ara, Hazrat, Li and Sims in [2] and Preusser in [15], we show in Theorems 3.1.8 and 3.2.6 that if R is a ( * -) ring, graded over a group G, and having (self-adjoint) graded local units, then the (hermitian) graded K-theory of R is the (hermitian) K-theory of the crossed product (1.5)…”

Section: Introductionmentioning

confidence: 82%

“…The rest of this article is organized as follows. In Section 2 we recall basic definitions, notations and properties for algebras equipped with an action of, or a grading over a group G. Section 3 contains the proof of the identities (1.5) in Theorems 3.1.8 and 3.2.6; the basic idea is to use the category isomorphism between graded R-modules and G ⋉ R-modules due to Ara, Hazrat, Li and Sims [2] and the fact that the latter preserves finite generation, proved in Preusser's article [15], and to check that the category isomorphism intertwines the relevant duality functors. As an application we also establish Theorem 4.3, which is a hermitian variant of Dade's theorem.…”

Section: Introductionmentioning

confidence: 99%

“…There is an isomorphism of G- * -algebras L E∼ −→ G ⋉ L ω (E).Proof. By[15, Example 7], this is a particular case of[15, Proposition 74].3. Graded K-theory and crossed products 3.1.…”

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confidence: 99%

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Graded $K$-theory and Leavitt path algebras

Arnone¹,

Cortiñas²

2022

Preprint

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Let G be a group and ℓ a commutative unital * -ring with an element λ ∈ ℓ such that λ + λ * = 1. We introduce variants of hermitian bivariant K-theory for * -algebras equipped with a G-action or a G-grading. For any graph E with finitely many vertices and any weight function ω :Ggr is obtained, describing L(E) as a cone of a matrix with coefficients in Z[G] associated to the incidence matrix of E and the weight ω. In the particular case of the standard Z-grading, and under mild assumptions on ℓ, we show that the isomorphism class of L(E) in kk hZgr is determined by the graded Bowen-Franks module of E. We also obtain results for the graded and hermitian graded K-theory of * -algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence relating graded and ungraded K-theory.Again the same formulas hold for homotopy algebraic and homotopy hermitian K-theory without any unitality assumptions on R. We also consider gradings on L(E) with values on a group G for an arbitrary graph E.

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

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Graded $K$-theory and Leavitt path algebras

Arnone¹,

Cortiñas²

2022

Preprint

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“…In [28] Leavitt path algebras of hypergraphs were introduced, which include as special cases the vertexweighted Leavitt path algebras, i.e. weighted Leavitt path algebras of weighted graphs having the property that any two edges emitted by the same vertex have the same weight.…”

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confidence: 99%

“…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]. By doing so, we generalise the graded V-monoid result for vertex-weighted Leavitt path algebras obtained in [28] to arbitrary weighted Leavitt path algebras. In Section 11 we present the modules for weighted Leavitt path algebras found in [16].…”

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confidence: 99%

Weighted Leavitt path algebras -- an overview

Preusser

2021

Preprint

Self Cite

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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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Modules for Leavitt Path Algebras via Extended Algebraic Branching Systems

Preusser

2023

Bull Braz Math Soc, New Series

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Leavitt Path Algebras of Hypergraphs

Cited by 9 publications

References 20 publications

Graded $K$-theory and Leavitt path algebras

Graded $K$-theory and Leavitt path algebras

Weighted Leavitt path algebras -- an overview

Modules for Leavitt Path Algebras via Extended Algebraic Branching Systems

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