Homotopy classification of Leavitt path algebras

DIEGO MONTERO; GUILLERMO CORTIÑAS

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2020 vol. 362

0001-8708

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 further develop the study of bivariant algebraic $K$-theory of Leavitt path algebras started in a previous paper and obtain several other results of independent interest.