Corners of Leavitt path algebras of finite graphs are Leavitt path algebras

2019/02/10 by Gene Abrams, Abrams, Gene, Tran Giang Nam +1
Mathematics · #16S99 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1902.03641

openalex publication_date 2019/02/10 · openalex created_date 2019/02/21 · openalex updated_date 2026/07/28

Abstract

We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra L_K(E) of a finite graph E with coefficients in a field K as a direct sum of matrix rings over K, direct sum with a corner of the Leavitt path algebra L_K(F) of a graph F for which every regular vertex is the base of a loop. Moreover, in this case one may transform the graph E into the graph F via some step-by-step procedure, using the "source elimination" and "collapsing" processes. We use this to establish the main result of the article, that every nonzero corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. Indeed, we prove a more general result, to wit, that the endomorphism ring of any nonzero finitely generated projective L_K(E)-module is isomorphic to the Leavitt path algebra of a graph explicitly constructed from E. Consequently, this yields in particular that every unital K-algebra which is Morita equivalent to a Leavitt path algebra is indeed isomorphic to a Leavitt path algebra.

Citations

• The classification question for Leavitt path algebras

Related

• Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C^*-algebras
• A survey of some of the recent developments in Leavitt path algebras
• The Leavitt path algebra of a graph
• Realization of graded matrix algebras as Leavitt path algebras
• Classification of Leavitt path algebras with two vertices