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Additive K-theory

From Wikipedia, the free 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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.

Formulation

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Following Boris Feigin and Boris Tsygan,[2] let be an algebra over a field of characteristic zero and let be the algebra of infinite matrices over with only finitely many nonzero entries. Then the Lie algebra homology

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

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

References

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  1. ^ Bloch, Spencer (2006-07-23). "Algebraic Cycles and Additive Chow Groups" (PDF). Dept. of Mathematics, University of Chicago.
  2. ^ B. Feigin, B. Tsygan. Additive K-theory, LNM 1289, Springer