K-theory for Leavitt path algebras: Computation and classification

We show that the long exact sequence for K-groups of Leavitt path algebras deduced by Ara, Brustenga, and Cortiñas 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 K0K0-group and K1K1-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 K0K0-group and the K6K6-group does suffice to classify these Leavitt path algebras up to Morita equivalence.