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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.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 {\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).
References
- ^ Bloch, Spencer (2006-07-23). "Algebraic Cycles and Additive Chow Groups". Dept. of Mathematics, University of Chicago.
- ^ B. Feigin, B. Tsygan. Additive K-theory, LNM 1289, Springer