The Dixmier-Moeglin equivalence for Leavitt path algebras.

DOI10.1007/S10468-010-9245-3zbMATH Open1250.16012arXiv1005.4321OpenAlexW2031658508MaRDI QIDQ421482

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Publication date: 24 May 2012

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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: �egin{enumerate} item $P$ is primitive; item $P$ is rational; item $P$ is locally closed in $mSpec(L_K(E))$. end{enumerate} We show that the prime spectrum $mSpec(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to $mSpec(K)$ or to $mSpec(K[x,x^{-1}])$. In the case that $K$ is infinite, we show that $L_K(E)$ has a rational $K^{imes}$-action, and that the indicated decomposition of $mSpec(L_K(E))$ is induced by this action.

Full work available at URL: https://arxiv.org/abs/1005.4321

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