Leavitt path algebras of separated graphs

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, L K{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(L K(E, C)) of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of L K(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(L K(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(L K(E +, C +)) has refinement. © Walter de Gruyter 2012.