Higher algebraic K-theories related to the global program of Langlands

TL;DR

This paper explores advanced algebraic K-theories within the Langlands program, introducing new bilinear and KK-theories linked to Galois representations and geometric structures.

Contribution

It introduces higher algebraic bilinear K-theories and mixed bilinear KK-theories connected to the global Langlands program, offering new algebraic interpretations of homotopy.

Findings

Development of higher algebraic bilinear K-theories.

Introduction of mixed higher bilinear KK-theories.

Connections established between K-theories and Langlands program structures.

Abstract

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.