Flow invariants in the classification of Leavitt path algebras

Abrams, Gene; Louly, A.; Pardo, Enrique; Smith, Christopher

doi:10.1016/j.jalgebra.2011.01.022

Cited by 58 publications

(109 citation statements)

References 20 publications

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“…Let E = (E 0 , E 1 , r E , s E ) be an arbitrary directed graph. 1 We denote the infinite-path space by E ∞ and the finite-path space by E * . We use the convention that a path x is a sequence of edges x i in which each s E (x i ) = r E (x i+1 ) and we write |x| for the length of x.…”

Section: Preliminariesmentioning

confidence: 99%

Equivalent groupoids have Morita equivalent Steinberg algebras

Clark

Sims

2015

Journal of Pure and Applied Algebra

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Abstract. Let G and H be Hausdorff ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly-RenaultWilliams, then the associated Steinberg algebras of locally constant R-valued functions with compact support are Morita equivalent. We deduce that collapsing a "collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path R-algebra, and therefore a number of graphical constructions which yield Morita equivalent C

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Section: Preliminariesmentioning

confidence: 99%

Equivalent groupoids have Morita equivalent Steinberg algebras

Clark

Sims

2015

Journal of Pure and Applied Algebra

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“…The classification problem of Leavitt path algebras (up to isomorphisms) has been present in the literature since the pioneering works [1] and [2]. The study of the classification of Leavitt path algebras associated to small graphs was started in [6], where the authors considered graphs with at most 3 vertices satisfying Condition (Sing), i.e, there is at most one edge between two vertices.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Classification of Leavitt Path Algebras with Two Vertices

Kanuni̇¹,

Barquero²,

González³

et al. 2019

MMJ

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We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the K 0 group, detpN 1 E q (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated by vertices in extreme cycles. The starting point is a simple linear algebraic result that determines when a Leavitt path algebra is IBN.An interesting result that we have found is that the ideal generated by extreme cycles is invariant under any isomorphism (for Leavitt path algebras whose associated graph is finite).We also give a more specific proof of the fact that the shift move produces an isomorphism when applied to any row-finite graph, independently of the field we are considering.

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“…This question has been investigated in numerous articles, see e.g. [6] and [32]. In large part this research effort has been motivated by the goal of resolving the "Morita equivalence conjecture": if the Leavitt path algebras of the graphs E and F are Morita equivalent, must the corresponding graph C * -algebras for E and F also be (strongly) Morita equivalent?…”

Section: Introductionmentioning

confidence: 99%