Classification of Leavitt path algebras with two vertices

TL;DR

This paper classifies Leavitt path algebras with up to two vertices using invariants like K-theory and socle, revealing that certain ideals are invariant under isomorphism and that shift moves preserve algebra structure.

Contribution

It provides a complete classification of small Leavitt path algebras and proves invariance of specific ideals and isomorphisms under graph transformations.

Findings

Ideal generated by extreme cycles is invariant under isomorphism.

Shift move produces an isomorphism for row-finite graphs.

Classification based on invariants like K_0 and socle.

Abstract

We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the $K_0$ group, $\det(N'_E)$ (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated by vertices in extreme cycles. The starting point is a simple linear algebraic result that determines when a Leavitt path algebra is IBN. An interesting result that we have found is that the ideal generated by extreme cycles is invariant under any isomorphism (for Leavitt path algebras whose associated graph is finite). We also give a more specific proof of the fact that the shift move produces an isomorphism when applied to any row-finite graph, independently of the field we are considering.