The Dixmier-Moeglin equivalence for Leavitt path algebras.
Given an algebra \(A\) and \(P\in\text{Spec}(A)\) a prime ideal, it is said to be locally closed when the intersection of all prime ideals in \(A\) which properly contain \(P\), strictly contains \(P\). An ideal \(P\in\text{Spec}(A)\) is rational when the extended centroid of \(A/P\) is an algebraic extension of the ground field of \(A\). Let \(K\) be a field and \(A\) a \(K\)-algebra, we say that \(A\) satisfies the Dixmier-Moeglin equivalence on prime ideals if for \(P\in\text{Spec}(A)\) the following conditions are equivalent: \(P\) is primitive. \(P\) is rational. \(P\) is locally closed. In the work under review it is proved that for a finite directed graph \(E\) the Leavitt path algebra \(L_K(E)\) satisfies the Dixmier-Moeglin equivalence on prime ideals. It is also proved that the spectrum \(\text{Spec}(L_K(E))\) decomposes into a finite disjoint union of subsets each of which is isomorphic to \(\text{Spec}(K)\) or to \(\text{Spec}(K[x,x^{-1}])\). Furthermore when \(K\) is infinite it is proved that \(L_K(E)\) has a rational \(K^\times\)-action and that the decomposition on the spectrum is induced by this action.
scientific article; zbMATH DE number 6038211
scientific article