Leavitt Path Algebras
Gene Abrams, Pere Ara, Mercedes Siles Molina Leavitt path algebras June 14, 2016 Springer vi Preface The great challenge in writing a book about a topic of ongoing mathematical research interest lies in determining who and what. Who are the readers for whom the book is intended? What pieces of the research should be included? The topic of Leavitt path algebras presents both of these challenges, in the extreme. Indeed, much of the beauty inherent in this topic stems from the fact that it may be approached from many different directions, and on many different levels. The topic encompasses classical ring theory at its finest. While at first glance these Leavitt path algebras may seem somewhat exotic, in fact many standard, well-understood algebras arise in this context: matrix rings and Laurent polynomial rings, to name just two. Many of the fundamental, classical ring-theoretic concepts have been and continue to be explored here, including the ideal structure, Z-grading, and structure of finitely generated projective modules, to name just a few. The topic continues a long tradition of associating an algebra with an appropriate combinatorial structure (here, a directed graph), the subsequent goal being to establish relationships between the algebra and the associated structures. In this particular setting, the topic allows for (and is enhanced by) visual, pictorial representation via directed graphs. Many readers are no doubt familiar with the by-now classical way of associating an algebra over a field with a directed graph, the standard path algebra. The construction of the Leavitt path algebra provides another such connection.
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