Additive K theory - Alchetron, The Free Social Encyclopedia

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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl. 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, let A be an algebra over a field k of characteristic zero and let g l ( A ) be the algebra of infinite matrices over A with only finitely many nonzero entries. Then the Lie algebra homology

H ⋅ ( g l ( 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

H C i ( A ) ≅ K i + 1 + ( A ) .

References

Additive K-theory Wikipedia

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