Homotopy classification of 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 ℓ. Each graph E has associated a Leavitt path ℓ-algebra L(E). There is an open question which asks whether the pair (K0(L(E)),[1L(E)]), consisting of the Grothendieck group together with the class [1L(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 (K0(L(E)),[1L(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.