Homotopy classification of Leavitt path algebras

TL;DR

This paper demonstrates that the pair (K_0, [1]) serves as a complete invariant for classifying purely infinite simple Leavitt path algebras of finite graphs up to polynomial homotopy equivalence, advancing the understanding of their algebraic structure.

Contribution

The paper introduces the bivariant algebraic K-theory for Leavitt path algebras and proves its effectiveness in classifying these algebras up to polynomial homotopy.

Findings

K_0 and [1] form a complete invariant for classification

Development of bivariant algebraic K-theory for Leavitt path algebras

Results applicable to purely infinite simple Leavitt path algebras

Abstract

In this paper we address the classification problem for purely infinite simple Leavitt path algebras of finite graphs over a field $\ell$. Each graph $E$ has associated a Leavitt path $\ell$-algebra $L(E)$. There is an open question which asks whether the pair $(K_0(L(E)), [1_{L(E)}])$, consisting of the Grothendieck group together with the class $[1_{L(E)}]$ of the identity, is a complete invariant for the classification, up to algebra isomorphism, of those Leavitt path algebras of finite graphs which are purely infinite simple. We show that $(K_0(L(E)), [1_{L(E)}])$ is a complete invariant for the classification of such algebras up to polynomial homotopy equivalence. To prove this we develop the bivariant algebraic $K$-theory of Leavitt path algebras and obtain several results of independent interest.