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some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl.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 {\displaystyle A} be an algebra over a field k {\displaystyle k} of characteristic zero and let g l ( A ) {\displaystyle {{\mathfrak {g}}l}(A)} be the algebra of infinite matrices over A {\displaystyle A} with only finitely many nonzero entries. Then the Lie algebra homology
H ⋅ ( g l ( A ) , k ) {\displaystyle H_{\cdot }({{\mathfrak {g}}l}(A),k)}
has a natural structure of a Hopf algebra. The space of its primitive elements of degree i {\displaystyle i} is denoted by K i + ( A ) {\displaystyle K_{i}^{+}(A)} and called the i {\displaystyle 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 ) . {\displaystyle HC_{i}(A)\cong K_{i+1}^{+}(A).}
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