K-theory for Leavitt path algebras: computation and classification

TL;DR

This paper extends the computation of algebraic K-groups for Leavitt path algebras to countable graphs with infinite emitters and explores their classification using K-theory, especially over number fields.

Contribution

It provides explicit formulas for higher algebraic K-groups of Leavitt path algebras over various fields and establishes new classification results using K_0 and K_6 groups.

Findings

Extended the long exact sequence for K-groups to countable graphs with infinite emitters.

Computed explicit formulas for higher algebraic K-groups over finite and algebraically closed fields.

Proved that K_0 and K_6 groups suffice for classification over number fields.

Abstract

We show that the long exact sequence for K-groups of Leavitt path algebras deduced by Ara, Brustenga, and Cortinas extends to Leavitt path algebras of countable graphs with infinite emitters in the obvious way. Using this long exact sequence, we compute explicit formulas for the higher algebraic K-groups of Leavitt path algebras over certain fields, including all finite fields and all algebraically closed fields. We also examine classification of Leavitt path algebras using K-theory. It is known that the K_0-group and K_1-group do not suffice to classify purely infinite simple unital Leavitt path algebras of infinite graphs up to Morita equivalence when the underlying field is the rational numbers. We prove for these Leavitt path algebras, if the underlying field is a number field (which includes the case when the field is the rational numbers), then the pair consisting of the K_0-group and the K_6-group does suffice to classify these Leavitt path algebras up to Morita equivalence.