  The paper revisits concretely the algebraic K-theory in the light of the
global program of Langlands by taking into account the new algebraic
interpretation of homotopy viewed as deformation(s) of Galois representations
given by compactified algebraic groups. More concretely, we introduce higher
algebraic bilinear K-theories referring to homotopy and cohomotopy and related
to the reducible bilinear global program of Langlands as well as mixed higher
bilinear KK-theories related to dynamical geometric bilinear global program of
Langlands.
