Non-stable K-theory for Leavitt path algebras

Hay, Damon M.; Loving, Marissa; Montgomery, M.G.; Ruiz, Efren; Todd, Katherine

doi:10.48550/arxiv.1211.1102

Cited by 1 publication

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“…The computation above then follows from the row-finite case and [13, Lemma 2.3]. Moreover, the claim about the positive cone of K alg 0 is established in [17,Theorem 4.9]. Also, since K is a field, we have (K alg 1 (K), +) ∼ = (K × , •).…”

Section: Computing the Algebraic K-theory Of Leavitt Path Algebrasmentioning

confidence: 88%

Classification of unital simple Leavitt path algebras of infinite graphs

Ruiz

Tomforde

2013

Journal of Algebra

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We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if LK (E) and LK (F ) are simple Leavitt path algebras, then LK (E) is Morita equivalent to LK (F ) if and only if K alg 0 (LK (E)) ∼ = K alg 0 (LK (F )) and the graphs E and F have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K alg 1 (LK(E)) ∼ = K alg 1 (LK (F )), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras -the only missing part is determining whether the "sign of the determinant condition" is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.

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Section: Computing the Algebraic K-theory Of Leavitt Path Algebrasmentioning

confidence: 88%