The classification question for Leavitt path algebras

TL;DR

This paper develops an algebraic version of the Gauge-Invariant Uniqueness Theorem and uses it to classify certain Leavitt path algebras via K-theory, also reestablishing known isomorphisms through graph algebra techniques.

Contribution

It introduces an algebraic Gauge-Invariant Uniqueness Theorem and applies it to classify Leavitt path algebras using K-theory and graph isomorphisms.

Findings

Classified Leavitt path algebras using K_0 groups and identity positions.

Established isomorphisms between matrix rings over classical Leavitt algebras.

Reproved known algebraic isomorphisms via graph algebra methods.

Abstract

We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the $K_0$ groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in $K_0$, classifies the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras.