CLASSIFICATION OF LEAVITT PATH ALGEBRAS USING ALGEBRAIC K-THEORY

We show that the long exact sequence for the $K$-theory of Leavitt path algebras over row-finite graphs, discovered by Ara, Brustenga, and Corti{~n}as, extends to Leavitt path algebras of arbitrary countable graphs. Using this long exact sequence, we compute explicit formulas for the higher $K$-groups of Leavitt path algebras in several situations, including all of the $K$-groups of Leavitt path algebras over finite fields and algebraically closed fields. We then focus on the classification up to Morita equivalence of purely infinite simple unital Leavitt path algebras over countably infinite graphs. The $K0$-group and the $K1$-group are not sufficient to classify these Leavitt path algebras when the underying field is the rational numbers. We prove that when the underlying field is a number field (which includes the rational numbers), then these Leavitt path algebras are classified up to Morita equivalence by the $K0$-group and the $K6$-group.