Distinguishing Leavitt algebras among Leavitt path algebras of finite graphs by Serre property

Abstract

Two unanswered questions in the heart of the theory of Leavitt path algebras are whether Grothendieck group $K_0$ is a complete invariant for the class of unital purely infinite simple algebras and, a weaker question, whether $L_2$ (the Leavitt path algebra associated to a vertex with two loops) and its Cuntz splice algebra $L_{2-}$ are isomorphic. The positive answer to the first question implies the latter. In this short paper, we raise and investigate another question, the so-called Serre's conjecture, which sits in between of the above two questions: The positive answer to the classification question implies Serre's conjecture which in turn implies $L_2 \cong L_{2-}$. Along the way, we give new easy to construct algebras having stable free but not free modules.