mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, additive K-theory means some version of

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

in which, according to

Spencer Bloch Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Departm ...

, the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

''GL'' has everywhere been replaced by its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

''gl''. It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.

Formulation

Following

Boris Feigin Boris Lvovich Feigin (russian: Бори́с Льво́вич Фе́йгин) (born 20 November 1953) is a Russian mathematician. His research has spanned representation theory, mathematical physics, algebraic geometry, Lie groups and Lie algebra ...

and

Boris Tsygan Boris may refer to: People * Boris (given name), a male given name *:''See'': List of people with given name Boris * Boris (surname) * Boris I of Bulgaria (died 907), the first Christian ruler of the First Bulgarian Empire, canonized after his ...

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

A

be an algebra over a field

k

characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...

and let

(A)

be the algebra of infinite matrices over

A

with only finitely many nonzero entries. Then the

Lie algebra homology In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to prope ...

H_\cdot ((A),k)

has a natural structure of a

Hopf algebra Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedis ...

. 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 In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independent ...

groups by the isomorphism :

HC_i(A) \cong K^+_(A).

References

K-theory {{topology-stub