The dynamics of Leavitt path algebras

TL;DR

This paper explores the deep connections between Leavitt path algebras and symbolic dynamics, showing how conjugacy relates to graded Morita theory and proving classification results using graded Grothendieck groups.

Contribution

It establishes a link between conjugacy in symbolic dynamics and graded Morita equivalence in Leavitt path algebras, leading to a classification result for purely infinite simple unital Leavitt path algebras.

Findings

Graded Grothendieck group coincides with Krieger dimension group.

Conjugacy corresponds to graded Morita equivalence.

Isomorphic graded dimension groups imply algebra isomorphism.

Abstract

Recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the Morita theory and the Grothendieck group in the theory of Leavitt path algebras \cite{flowa}. In this paper we show that the notion of conjugacy of shifts of finite type is closely related to the {\it graded} Morita theory and consequently the {\it graded} Grothendieck group. This fits into the general framework we have in these two theories: Conjugacy yields the flow equivalence and the graded Morita equivalence can be lifted to the Morita equivalence. Starting from a finite directed graph, the observation that the graded Grothendieck group of the Leavitt path algebra associated to $E$ coincides with the Krieger dimension group of the shift of finite type associated to $E$ provides a link between the theory of Leavitt path algebras and symbolic dynamics. It has been conjectured that the ordered graded Grothendieck group as $\mathbb Z[x,x^{-1}]$-module (we call this the graded dimension group) classifies the Leavitt path algebras completely \cite{hazann}. Via the above correspondence, utilising the results from symbolic dynamics, we prove that for two purely infinite simple unital Leavitt path algebras, if their graded dimension groups are isomorphic, then the algebras are isomorphic.