Cycles in Leavitt path algebras by means of idempotents.
This paper studies two graph properties known as Condition (NE) and Condition (L). The first of them asks for all the cycles of the graph to have no exits, while the second one demands that every cycle has an exit. The authors present ring theoretic characterizations of both conditions for arbitrary graphs, in terms of idempotents. Concretely, they establish that \(E\) satisfies Condition (NE) if and only if \(L_K(E)\) does not present infinite idempotents (and identify the algebraic structure of \(L_K(E)\)). Then, they prove that \(E\) satisfies Condition (L) if and only if \(L_K(E)\) has no non-minimal primitive idempotents. A refining of the so-called Reduction Theorem is achieved and is used to prove that \(I(P_c(E))\), the ideal of the vertices which are base of cycles without exits of the graph \(E\), is a ring isomorphism invariant for arbitrary Leavitt path algebras. The authors also determine its structure in any case. To achieve the main results, two different tools are developed: the study of primitive idempotents and the notion of dual graph. For the result related to \(I(P_c(E))\), a combination of both tools is required.
scientific article; zbMATH DE number 6387824