Additive K-theory explained

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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.^[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.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

A be an algebra over a field k of characteristic zero and let { ak gl}(A) be the algebra of infinite matrices over A with only finitely many nonzero entries. Then the Lie algebra homologyH _{⋅} ({ ak 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).

Notes and References

1. Web site: Bloch . Spencer . 2006-07-23 . Algebraic Cycles and Additive Chow Groups . Dept. of Mathematics, University of Chicago.
2. B. Feigin, B. Tsygan. Additive K-theory, LNM 1289, Springer