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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.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

[math]\displaystyle{ H_\cdot ({\mathfrak gl}(A),k) }[/math]

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

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

[math]\displaystyle{ HC_i(A) \cong K^+_{i+1}(A). }[/math]

## References

1. ↑ http://www.math.uchicago.edu/~bloch/addchow_rept.pdf Template:Bare URL PDF
2. ↑ B. Feigin, B. Tsygan. Additive K-theory, LNM 1289, Springer

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