Finitely presented simple modules over Leavitt path algebras.
Recently, X. W. Chen has provided a method of constructing simple modules \(V_{[p]}\) over a Leavitt path algebra \(L_K(E)\) of an arbitrary graph \(E\) by using the equivalence class \([p]\) of infinite paths tail-equivalent to a fixed infinite path \(p\) in \(E\) (definitions included in the paper under review). He also constructed simple modules corresponding to various sinks \(w\) in \(E\).NEWLINENEWLINE In this paper, the authors introduce additional classes of simple modules using vertices which are infinite emitters and also exclusive cycles, and call all these simple modules over \(L_K(E)\) Chen modules. They give a description of the annihilators of the various Chen modules and, as a consequence, they show that every primitive ideal of \(L_K(E)\) can be realized as the annihilator of a Chen module. They also prove, that the Chen modules are pairwise non-isomorphic.NEWLINENEWLINE For an algebra \(A\), denote by \(\widetilde A\) the set of isoclasses of simple left \(A\)-modules, and by \(\text{Prim}(A)\) the set of primitive ideals of \(A\). There is a canonical map \(\widetilde A\to\text{Prim}(A)\) mapping \([N]\) to \(\text{Ann}_A(N)\), the annihilator of \(N\).NEWLINENEWLINE Then it is proved that: If \(E\) is finite graph, \(K\) an arbitrary field and \(L:=L_K(E)\), then the following conditions are equivalent:NEWLINENEWLINE (1) Every simple left \(L\)-module is finitely presented.NEWLINENEWLINE (2) Every simple Chen module is finitely presented.NEWLINENEWLINE (3) Every vertex \(v\) in \(E\) is the base of at most one cycle.NEWLINENEWLINE (4) The map \(\widetilde L\to\text{Prim}(L)\) is a bijection.NEWLINENEWLINE (5) All simple left \(L\)-modules are Chen modules.NEWLINENEWLINE (6) The Gelfand-Kirillov dimension of \(L_K(E)\) is finite.
scientific article; zbMATH DE number 6339760
scientific article