Homotopy classification of Leavitt path algebras (Q2287949)

The paper under review is using algebraic homotopy and the techniques that the same authors develop in the paper [``Algebraic bivariant \(K\)-theory and Leavitt path algebras'', Preprint, \url{arXiv:1806.09204}] on bivariant algebraic \(K\)-theory to answer an open and well-known classification problem of purely infinite simple Leavitt path algebras on finite graphs and over a field. The main result is Theorem 6.1 of the paper and shows that \(K_0\) group and the isomorphism class of the identity element of the Leavitt path algebra is a complete invariant for the classification (of the Leavitt path algebra defined above) up to polynomial homotopy equivalence. The paper is organized in 8 sections which starts with preliminaries in the introduction. Section 2 is titled ``Idempotents, units and the groups \(K_0\) and \(K_1\) in the purely infinite simple unital case''. Section 3 and 4 studies lifting of \(K\)-theory maps to algebra maps, concentrating on \(K_0\) and \(K_1\) groups and Section 5 is devoted to lifting of \(kk\)-maps to algebra maps which provides the necessary tools. Corollary 3.2 is worthwhile to mention here: When the Leavitt path algebra \(L(E)\) is simple, the corollary gives the necessary condition on the graph \(E\) so that \(L(E)\) is embedded in a unital purely infinite algebra or the infinite dimensional matrix algebra over a unital purely infinite algebra. The main result Theorem 6.1 is stated in Section 6 Homotopy classification theorem and Section 7 and 8 studies some further results derived as applications. Section 7 is devoted to algebra extensions and Section 8 is concerned with maps to a Leavitt path algebra over a graph with one vertex and two loops and to a tensor product of this Leavitt path algebra in Theorem 8.3.