Center for Ring Theory Colloquium

ABSTRACT: We characterize the minimal left ideals of a Leavitt path algebra as those which are isomorphic to principal left ideals generated by line points; that is, by vertices whose trees contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line point vertices. This characterization allows us to compute the socle of certain algebras that arise as the Leavitt path algebra of a row-finite graph. A complete description of the socle of a Leavitt path algebra is given: it is a locally matricial algebra. In the more general case of arbitrary graphs we use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it in this context as well.