# Moscow Mathematical Journal

Volume 19, Issue 3, July–September 2019 pp. 523–548.

Classification of Leavitt Path Algebras with Two Vertices

**Authors**:
Müge Kanuni (1), Dolores Martín Barquero (2), Cándido Martín González (3), and Mercedes Siles Molina (3)

**Author institution:**(1) Department of Mathematics, Düzce University, Konuralp 81620 Düzce, Turkey

(2) Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Universidad de Málaga, 29071 Málaga, Spain

(3) Departamento de Álgebra Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n. 29071 Málaga, Spain

**Summary: **

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.

2010 Math. Subj. Class. Primary: 16D70; Secondary: 16D25, 16E20, 16D30.

**Keywords:**Leavitt path algebra, IBN property, type, socle, extreme cycle,

*K*

_{0}.

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