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

Abstract

We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra LK(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 LK(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 LK(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.