The Dixmier-Moeglin equivalence for Leavitt path algebras

Abstract

Let $K$ be a field, let $E$ be a finite directed graph, and let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$. We show that for a prime ideal $P$ in $L_K(E)$, the following are equivalent: \begin{enumerate} \item $P$ is primitive; \item $P$ is rational; \item $P$ is locally closed in $\mathrm{Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum $\mathrm{Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to $\mathrm{Spec}(K)$ or to $\mathrm{Spec}(K[x,x^{-1}])$. In the case that $K$ is infinite, we show that $L_K(E)$ has a rational $K^{\times }$-action, and that the indicated decomposition of $\mathrm{Spec}(L_K(E))$ is induced by this action.

17 pages. Updated version, in which an erroneous proof (of "not primitive implies not rational" in the main theorem) has been replaced by a correct one. All original results from first version remain valid