A Crash Course on Leavitt path algebras

Leavitt path algebras can be regarded as the algebraic counterparts of the graph C∗-algebras, the descendants from the algebras investigated by J. Cuntz in [11], which have been the focus of much attention from the analysts in the last two decades (see [13] for an overview of the subject). Moreover, Leavitt path algebras can also be viewed as a broad generalization of the algebras constructed by W. G. Leavitt in [12] to produce rings without the Invariant Basis Number property (i.e., whose modules have bases with different cardinals). The Leavitt path algebra LK(E) was introduced in 2004 in the papers [1] and [4]. LK(E) was first defined for a row-finite graph E (countable graph such that every vertex emits only a finite number of edges) and a field K. Although their history is very recent, a flurry of activity has followed since then. The main directions of research include: characterization of algebraic properties of a Leavitt path algebra LK(E) in terms of graph-theoretic properties of E; study of the modules over LK(E); computation of various substructures (such as the Jacobson radical, the center, the socle and the singular ideal)