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

Author(s)

Abrams, Gene ^[1] ; Nam, Tran Giang

Affiliation

^[1] Univ Colorado, Dept Math, Colorado Springs, CO 80918 - USA

• Document type: Journal article
• Source: Journal of Algebra; v. 547, p. 494-518, APR 1 2020.
• Web of Science Citations: 0

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. (C) 2019 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 18/06538-6 - Leavitt path algebras, Steinberg algebras and partial actions
• Grantee: Tran Giang Nam
• Support Opportunities: Scholarships in Brazil - Post-Doctoral