Exchange Leavitt path algebras and stable rank

By G. Pino, Enrique Pardo, Mercedes Siles Molina. 2005

90 citations recorded by the source corpus.

Abstract We characterize those Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 or ∞. Concrete criteria in terms of properties of the underlying graph are given for each case.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e054bc84408a167ff8a25e096be5e9757703e98b

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The socle of a Leavitt path algebra (2007, background): “Concretely, necessary and sufficient conditions on a graph E have been given so that the corresponding Leavitt path algebra LK(E) is simple [1], purely infinite simple [2], exchange [7], finite dimensional [3], and locally finite (equivalently noetherian) [4].”
Socle theory for Leavitt path algebras of arbitrary graphs (2008, background): “s were determined; in the algebraic one, the graded ideals were described. In addition, several other ring properties were studied in the case of row-ﬁnite Leavitt path algebras, such as beingexchange[10], ﬁnite dimensional[5], noetherian[6] , semisimple [4] or having stable rank [10, 8]. It has bee”
Chain conditions for Leavitt path algebras (2009, background): “For example, conditions on the graph allow us to decide when the corresponding Leavitt path algebra is simple [1], purely infinite simple [2], exchange [9], finite-dimensional [4].”
Leavitt path algebras of separated graphs (2010, background): “Leavitt path algebras LK(E), natural algebraic versions of graph C -algebras, were introduced and studied firstly in [1] and [9] for row-finite graphs, and in [2] and [39] for general graphs; see also [4], [5], [10], [11], [26].”
Uniqueness theorems and ideal structure for Leavitt path algebras (2006, background): “ven only in the case that Eis a row-ﬁnite graph. Abrams and Aranda-Pino established basic properties of these Leavitt path algebras, and soon after there was a ﬂurry of activity as many other authors [4, 6, 2, 7, 8, 3] investigated their structure and found applications to various topics in algebra.”
Leavitt path algebras and direct limits (2007, background): “In particular, some natural candidates can be found in [2, 3, 10, 11, 13].”