A note on the Diximier-Mouglin Equivalence in Leavitt path algebras of arbitrary graphs over a field

TL;DR

This paper investigates the Dixmier-Moeglin equivalence in Leavitt path algebras of arbitrary graphs, extending previous results from finite graphs and exploring properties of prime ideals.

Contribution

It demonstrates that locally closed prime ideals in Leavitt path algebras of arbitrary graphs satisfy the DM-equivalence and are strongly primitive and completely irreducible.

Findings

Locally closed prime ideals are strongly primitive.

Prime ideals satisfy the DM-equivalence in arbitrary graphs.

Examples illustrate the constraints and properties.

Abstract

The Dixmier-Moeglin Equivalence (for short, the DM-equivalence) is the equivalence of three distinguishing properties of prime ideals in a non-commutative algebra A. These properties are of (i) being primitive, (ii) being rational, and (iii) being locally closed in the Zariski topology of Spec(A). The DM-equivalence holds in many interesting algebras over a field. Recently, it was shown that the prime ideals of a Leavitt path algebra of a finite graph satisfy the DM-equivalence, In this note, we investigate the occurrence of the DM-equivalence in a Leavitt path algebra L of an arbitrary directed graph E. Our analysis shows that locally closed prime ideals of L satisfy interesting equivalent properties such as being strongly primitive and completely irreducible. Examples illustrate the results and the constraints.