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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group "GL" has everywhere been replaced by its Lie algebra "gl".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 formula_1 be an algebra over a field formula_2 of characteristic zero and let formula_3 be the algebra of infinite matrices over formula_4 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 formula_6 is denoted by formula_7 and called the formula_8-th additive K-functor of "A".
The additive K-functors are related to cyclic homology groups by the isomorphism