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 $\mon{L_K(E,C)}$ 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 $E^0$ and a subset of $C$. Necessary conditions for $\mon{L_K(E,C)}$ to be a refinement monoid are developed, together with a construction that embeds $(E,C)$ in a separated graph $(E_+,C^+)$ such that $\mon{L_K(E_+,C^+)}$ has refinement.

💡 Research Summary

The paper introduces a substantial generalization of Leavitt path algebras (LPAs) by incorporating a family of partitions of the edge sets emanating from each vertex of a directed graph. Given a directed graph (E=(E^{0},E^{1},s,r)) and, for every vertex (v\in E^{0}), a partition (C_{v}) of the set (s^{-1}(v)) of edges leaving (v), the pair ((E,C)) with (C=\bigcup_{v}C_{v}) is called a separated graph. The associated algebra (L_{K}(E,C)) over a field (K) is defined by generators ({v\mid v\in E^{0}}\cup{e,e^{*}\mid e\in E^{1}}) and the usual relations (V), (E1), (E2) together with a refined Cuntz–Krieger family of relations for each partition block (X\in C_{v}): \