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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.^[1] It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.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

${\displaystyle H_{\cdot }({{\mathfrak {g}}l}(A),k)}$

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

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

${\displaystyle HC_{i}(A)\cong K_{i+1}^{+}(A).}$

References