Leavitt path algebras of separated graphs (Q2911000)

The present paper initiates the study of the Leavitt path algebras of separated graphs \(L_K(E,C)\) based on the concept of a separated graph \((E,C)\), namely a directed graph \(E\) together with a family \(C\) that gives partitions of the set of edges departing from each vertex of \(E\). This class of algebras generalizes the Leavitt path algebras and the type \((m,n)\) Leavitt algebras, and also includes free products of type \((1,n)\) Leavitt algebras.NEWLINENEWLINEAdditionally, a class of Cohn path algebras is studied, and these algebras are called Cohn-Leavitt algebras \(CL_K(E, C, S)\) associated to \((E, C, S)\), where \(S\) is a family consisting of some of the finite sets in \(C\). The Cohn-Leavitt algebras not only provide an interesting larger class of algebras that can be analysed by the same techniques, but they allow us to write all Leavitt path algebras (in fact, all Cohn-Leavitt algebras) as direct limits of algebras based on finite graphs. The algebras \(L_K(E, C)\) and \(CL_K(E, C, S)\) are homologically well behaved: all are hereditary.NEWLINENEWLINEThe main motivation for the introduction of this class of algebras is \(K\)-theoretical. Much effort is devoted to compute \(V(CL_K(E, C, S))\) (using Bergman's machinery), that is, the monoid of finitely generated projective modules of \(CL_K(E, C, S)\). These monoids are entirely determined by graph-theoretical data. The authors prove that, for every conical monoid \(M\), there exists \((E, C)\) such that \(M\) is isomorphic to \(V(L_K(E, C))\). They also find conditions for the monoid \(V(L_K(E, C))\) to have refinement, and develop a construction by which a separated graph \((E, C)\) can be embedded in a separated graph \((E_+, C^+)\) such that \(V(L_K(E_+, C^+))\) has refinement and preserves key properties of \(V(L_K(E, C))\). Further, if \(M (E, C)\) is simple, they can arrange the construction so that it embeds in the monoid of order-units of \(V(L_K(E_+, C^+))\) together with zero and so that the latter is a simple, divisible, refinement monoid. With these techniques, the authors attack the question of whether an exchange ring can have both finite and properly infinite full idempotents, and that enables them to construct Leavitt path algebras \(L_K(E_+, C^+)\) of separated graphs \((E_+, C^+)\) having finite and properly infinite full idempotents, and such that the monoids \(V(L_K(E_+, C^+))\) satisfy the refinement property.NEWLINENEWLINEFor ordinary Leavitt path algebras \(L_K(E)\), the lattice of graded ideals (and even the lattice of all ideals, if condition (K) holds) is isomorphic to a lattice formed from graph-theoretic data. Such a result does not hold for separated graphs, since the algebras \(L_K(E, C)\) and \(CL_K(E, C, S)\) typically have far more complicated ideal structure than \(L_K(E)\). Nonetheless, the authors are able to capture the lattice of trace ideals of \(CL_K(E, C, S)\) (these coincide with the idempotent generated ideals). This lattice is isomorphic to the lattice of order-ideals of \(V(CL_K(E, C, S))\) and to a certain lattice of pairs \((H, G)\), where \(H\) is a hereditary subset of \(E^0\) and \(G\) a subset of \(C\). Consequently, they derive necessary and sufficient conditions for \(V(CL_K(E, C, S))\) to be simple, equivalently, for \(CL_K(E, C, S)\) to be trace-simple.