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Using Steinberg algebras to study decomposability of Leavitt path algebras. (English) Zbl 1401.16036

If \(E\) is an oriented graph, the groupoid \(G_E\) is appropriately defined by certain paths of \(E\). If \(K\) is a field, the Leavitt path algebra \(L_K(E)\) of \(E\) is isomorphic to the Steinberg algebra \(A_K(G_E)\) of \(G_E\). In the present paper, the authors continue to study relations between \(E\) and \(G_E\) and show that the lattice of pairs \((H, S)\), where \(H\) is a hereditary and saturated set of vertices and \(S\) is a set of breaking vertices of \(H,\) is isomorphic to the lattice of open invariant subsets of \(G_E\). As a corollary, these lattices are isomorphic to the lattice of graded ideals of \(A_K(G_E)\) (as well as to the lattice of graded ideals of \(L_K(E)\)).
Using these results, the authors present both graph and groupoid conditions characterizing the decomposability of \(L_K(E)\). In particular, the decomposability of \(L_K(E)\) is equivalent to the topological decomposability of \(G_E\) as well as to the following condition on \(E:\) there exists a nonempty, proper, hereditary and saturated set \(H\) such that conditions (1) and (2), described as follows, hold. (1) Every infinite path whose vertices are outside of \(H\) eventually does not connect to \(H\). (2) Every infinite emitter having an infinite number of edges connecting to \(H\) must be either in \(H\) or a breaking vertex of \(H\).

MSC:

16S99 Associative rings and algebras arising under various constructions
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
06B10 Lattice ideals, congruence relations
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References:

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