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{{Orphan|date=December 2012}}

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of [[algebraic K-theory]] in which, according to [[Spencer Bloch]], the [[general linear group]] ''GL'' has everywhere been replaced by its [[Lie algebra]] ''gl''.<ref>http://www.math.uchicago.edu/~bloch/addchow_rept.pdf</ref> It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.

==Formulation==

Following Feigin and Tsygan,<ref>B. Feigin, B. Tsygan. ''Additive K-theory'', LNM 1289, Springer</ref> let <math> A </math> be an algebra over a field <math> k </math> of [[characteristic zero]] and let <math> {\mathfrak gl}(A) </math> be the algebra of infinite matrices over <math>A</math> with only finitely many nonzero entries. Then the [[Lie algebra homology]]

:<math> H_\cdot ({\mathfrak gl}(A),k) \, </math> 

has a natural structure of a [[Hopf algebra]]. The space of its [[primitive element]]s of degree <math> i</math> is denoted by <math>K^+_i(A)</math> and called the <math>i</math>-th '''additive K-functor''' of&nbsp;''A''.

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

:<math> HC_i(A) \cong K^+_{i+1}(A). \, </math>

== References ==
{{Reflist}}

[[Category:K-theory]]


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