Algebraic bivariant $K$-theory and Leavitt path algebras

TL;DR

This paper explores how algebraic bivariant $K$-theory can classify Leavitt path algebras of graphs, revealing that certain homology invariants depend only on specific matrix cokernels, thus limiting their distinguishing power.

Contribution

It establishes a structure theorem for unital Leavitt path algebras in bivariant $K$-theory and shows these invariants depend solely on cokernel classes of the incidence matrix.

Findings

$kk$-theory depends only on cokernels of $I-A_E$ and its transpose.

Homology theories with certain properties cannot distinguish some Leavitt path algebras.

$kk$-theory satisfies analogues of UCT and K"unneth theorems for Leavitt path algebras.

Abstract

This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic $K$-theory $kk$ is the universal homology theory with the properties above; we prove a structure theorem for unital Leavitt path algebras in $kk$. We show that under very mild assumptions on $\ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ depends only on the isomorphism classes of the cokernels of the matrix $I-A_E$ and of its transpose, which are respectively the $kk$ groups $KH^1(L(E))=kk_{-1}(L(E),\ell)$ and $KH_0(L(E))=kk_0(\ell,L(E))$. Hence if $L(E)$ and $L(F)$ are unital Leavitt path algebras such that $KH_0(L(E))\cong KH_0(L(F))$ and $KH^1(L(E))\cong KH^1(L(F))$ then no homology theory with the above properties can distinguish them. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^*$-graph algebras, including analogues of the Universal coefficient and K\"unneth theorems of Rosenberg and Schochet.