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arXiv · 1603.01033

Using Steinberg algebras to study decomposability of Leavitt path algebras

Lisa Orloff Clark·Dolores Martin Barquero·Candido Martin Gonzalez·Mercedes Siles Molina
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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}.

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Lisa Orloff Clark, Dolores Martin Barquero, Candido Martin Gonzalez, Mercedes Siles Molina. 2016-03-03. Using Steinberg algebras to study decomposability of Leavitt path algebras. https://arxiv.org/abs/1603.01033

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