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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.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 𝔤l(A) be the algebra of infinite matrices over A with only finitely many nonzero entries. Then the Lie algebra homology

H⋅(𝔤l(A),k)

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

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

HCi(A)≅Ki+1+(A).

## References

1. ↑ Bloch, Spencer (2006-07-23). Algebraic Cycles and Additive Chow Groups. Dept. of Mathematics, University of Chicago. http://www.math.uchicago.edu/~bloch/addchow_rept.pdf.
2. ↑ B. Feigin, B. Tsygan. Additive K-theory, LNM 1289, Springer

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