The graded Grothendieck group and the classification of Leavitt path algebras

TL;DR

This paper demonstrates that the graded K_0-group fully classifies Leavitt path algebras, similar to how K-theory classifies AF C*-algebras, with proofs for specific graph classes.

Contribution

It establishes the graded K_0-group as a complete invariant for Leavitt path algebras, extending classification results to certain graph types.

Findings

Graded K_0-group classifies Leavitt path algebras for acyclic graphs.

Complete classification achieved for comet and polycephaly graphs.

Parallel to Elliott's classification of AF C*-algebras.

Abstract

This paper is an attempt to show that, parallel to Elliott's classification of AF $C^*$-algebras by means of $K$-theory, the graded $K_0$-group classifies Leavitt path algebras completely. In this direction, we prove this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and comet and polycephaly graphs (graphs which each head is connected to a cycle or a collection of loops).