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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.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

$${\displaystyle H_{\cdot }({{\mathfrak {g}}l}(A),k)}$$

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

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

$${\displaystyle HC_{i}(A)\cong K_{i+1}^{+}(A).}$$

References

B. Feigin, B. Tsygan. Additive K-theory, LNM 1289, Springer