The Dixmier-Moeglin equivalence for Leavitt path algebras.
type: Item
instance of: scholarly article
title: The Dixmier-Moeglin equivalence for Leavitt path algebras.
zbMATH Open document ID: 1250.16012
author: Jason P. Bell
author: Kulumani M. Rangaswamy
author: Gene Abrams
published in: Algebras and Representation Theory
publication date: 2012-05-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.4321
review text: 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.
Mathematics Subject Classification ID: 16S88
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 6038211
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: primitive ideals
zbMATH Keywords: rational ideals
zbMATH Keywords: locally closed ideals
zbMATH Keywords: Dixmier-Moeglin equivalence
zbMATH Keywords: prime ideals
reviewed by: Candido Martín González
MaRDI profile type: Publication
OpenAlex ID: W2031658508
arXiv ID: 1005.4321
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DOI: 10.1007/S10468-010-9245-3
description: scientific article; zbMATH DE number 6038211