The Dixmier-Moeglin equivalence for Leavitt path algebras

TL;DR

This paper establishes the equivalence of primitivity, rationality, and local closedness for prime ideals in Leavitt path algebras over finite graphs, and describes the structure of their prime spectrum.

Contribution

It proves the Dixmier-Moeglin equivalence for Leavitt path algebras and characterizes the prime spectrum's decomposition related to the base field.

Findings

Prime ideals are equivalent in being primitive, rational, and locally closed.

The prime spectrum decomposes into parts homeomorphic to spectra of K or K[x,x^{-1}].

For infinite fields, a rational action induces this spectrum decomposition.

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 ${\rm Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum ${\rm Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to ${\rm Spec}(K)$ or to ${\rm Spec}(K[x,x^{-1}])$. 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 ${\rm Spec}(L_K(E))$ is induced by this action.