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Licensed Unlicensed Requires Authentication Published by De Gruyter September 27, 2011

Leavitt path algebras of separated graphs

Pere Ara and Kenneth R. Goodearl

Abstract

The construction of the Leavitt path algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. The new algebras, LK(E, C), are analyzed in terms of their homology, ideal theory, and K-theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid 𝒱(LK(E, C)) of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of LK(E, C) is determined by graph-theoretic data, namely as a lattice of certain pairs consisting of a subset of E0 and a subset of C. Necessary conditions for 𝒱(LK(E, C)) to be a refinement monoid are developed, together with a construction that embeds (E, C) in a separated graph (E+, C+) such that 𝒱(LK(E+, C+)) has refinement.

Received: 2010-04-27
Revised: 2011-03-28
Published Online: 2011-09-27
Published in Print: 2012-08

©[2012] by Walter de Gruyter Berlin Boston