In mathematics, additive K-theory means some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.
Formulation
Following Boris Feigin and Boris Tsygan,[2] let be an algebra over a field
of characteristic zero and let
be the algebra of infinite matrices over
with only finitely many nonzero entries. Then the Lie algebra homology
has a natural structure of a Hopf algebra. The space of its primitive elements of degree is denoted by
and called the
-th additive K-functor of A.
The additive K-functors are related to cyclic homology groups by the isomorphism