Decomposable Leavitt path algebras for arbitrary graphs.
Given any field \(K\) and a completely arbitrary graph \(E\), the paper studies those Leavitt path algebras \(L_K(E)\) that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. One of the notions involved is that of direct connectedness of a graph. This, takes into account the direction of the edges (that is, for every \(v,w\in E_0\) there exists a path \(\mu\) with \(s(\mu)=v\) and \(r(\mu)=w\)). However the concept of connectedness of a graph, does not (i.e., for every \(v,w\in E_0\) there exists a walk from \(v\) to \(w\)). It is proved that if \(E\) is a directly connected graph, then \(L_K(E)\) is a connected algebra, and that if \(L_K(E)\) is connected, then \(E\) is connected. Furthermore the implications can not be reversed. The paper uses a notion of \(X\)-compatible path, for a hereditary saturated set \(X\). By using this, a Leavitt path algebra \(L_K(E)\) is decomposable if and only if there exist disjoint hereditary and saturated subsets \(X\) and \(Y\) such that there are a finite and nonzero number of paths that are either \(X\)-compatible paths or \(Y\)-compatible paths. The work also studies the indecomposable ``atoms'' and proves that when the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists.
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