Additive K-theory

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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl. 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, let $$ A $$ be an algebra over a field $$ k $$ of characteristic zero and let $$ {\mathfrak gl}(A) $$ be the algebra of infinite matrices over $$A$$ with only finitely many nonzero entries. Then the Lie algebra homology


 * $$ H_\cdot ({\mathfrak gl}(A),k) $$

has a natural structure of a Hopf algebra. The space of its primitive elements of degree $$ i$$ is denoted by $$K^+_i(A)$$ and called the $$i$$-th additive K-functor of A.

The additive K-functors are related to cyclic homology groups by the isomorphism


 * $$ HC_i(A) \cong K^+_{i+1}(A). $$