Using Steinberg algebras to study decomposability of Leavitt path algebras

TL;DR

This paper explores the connection between graph properties, groupoid structures, and the decomposability of Leavitt path algebras, providing new characterizations and conditions for algebraic decomposability.

Contribution

It establishes a lattice isomorphism linking graph-theoretic data to groupoid invariants and characterizes when Leavitt path algebras are decomposable.

Findings

Lattice isomorphism between hereditary sets and open invariant subsets

Graph conditions for closedness of invariant subsets

Criteria for decomposability of Leavitt path algebras

Abstract

Given an arbitrary graph $E$ we investigate the relationship between $E$ and the groupoid $G_E$. We show that there is a lattice isomorphism between the lattice of pairs $(H, S)$, where $H$ is a hereditary and saturated set of vertices and $S$ is a set of breaking vertices {associated to $H $}, onto the lattice of open invariant subsets of $G_E^{(0)}$. We use this lattice isomorphism to characterize the decomposability of the Leavitt path algebra $L_K(E)$, where $K$ is a field. First we find a graph condition to characterise when an open invariant subset of $G_E^{(0)}$ is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to $L_K(E)$ being decomposable {in the sense that it can be written as a direct sum of two nonzero ideals}.