Atlas of Leavitt path algebras of small graphs

Abstract

The aim of this work is the description of the isomorphism classes of all Leavitt path algebras coming from graphs satisfying Condition (Sing) with up to three vertices. In particular, this classification recovers the one achieved by Abrams et al. [1] in the case of graphs whose Leavitt path algebras are purely infinite simple. The description of the isomorphism classes is given in terms of a series of invariants including the K 0 group, the socle, the number of loops with no exits and the number of hereditary and saturated subsets of the graph.2000 Mathematics Subject Classification. Primary 16D70.

Classification of Leavitt Path Algebras with Two Vertices

“…The classification problem of Leavitt path algebras (up to isomorphisms) has been present in the literature since the pioneering works [1] and [2]. The study of the classification of Leavitt path algebras associated to small graphs was started in [6], where the authors considered graphs with at most 3 vertices satisfying Condition (Sing), i.e, there is at most one edge between two vertices. This work can be also of interest, not only for people studying Leavitt path algebras, but also for a broader audience; concretely, for those working on graph C ˚algebras (as these are the analytic cousins of Leavitt path algebras).…”

“…It was shown in [1,Theorem 2.3] that every shift of a graph E produces an epimorphism between the corresponding Leavitt path algebras over a field K, which is an isomorphism provided the graph E satisfies Condition (L) or the field K is infinite. This result can be extended to arbitrary fields, and the condition can be eliminated, as the second, third and fourth author mentioned in [6] (see page 583) and proved in a condensed way. Here we include a more detailed proof.…”

“…Using Theorem 2.1 we can affirm that if there exists m P N, m ą 1 such that (1) is satisfied, then pm, mq´p1, 1q is in the Z-span of the relations given in (6), that is, there exist k 1 , k 2 P Z such that…”