Atlas of Leavitt Path Algebras of small graphs

TL;DR

This paper classifies all Leavitt path algebras from small graphs with up to three vertices satisfying Condition (Sing), providing a comprehensive 'atlas' of their isomorphism classes using various invariants.

Contribution

It offers a complete classification of Leavitt path algebras for small graphs under Condition (Sing), extending previous results to a broader class of graphs.

Findings

Complete classification of Leavitt path algebras for graphs with up to three vertices.

Identification of invariants such as K_0 group, socle, and loop counts for classification.

Recovery of previous classifications for purely infinite simple cases.

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. 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.