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

2022/01/09 by Roozbeh Hazrat, Hazrat, Roozbeh, Kulumani M. Rangaswamy +1
Mathematics · Computer Science · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Algebra and Logic

paper · pdf · doi:10.48550/arxiv.2201.03113

Abstract

Two unanswered questions in the heart of the theory of Leavitt path algebras are whether Grothendieck group K₀ is a complete invariant for the class of unital purely infinite simple algebras and, a weaker question, whether L₂ (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 L₂ ≅ L_2-. Along the way, we give new easy to construct algebras having stable free but not free modules.

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