K-theory of affine actions

TL;DR

This paper studies affine actions of the tangent group on vector bundles over a G-manifold, establishing categorical equivalences, monadic adjunctions, and isomorphisms of Grothendieck groups relating to equivariant K-theory.

Contribution

It introduces the concept of affine actions of the tangent group, proves categorical equivalences, and relates their Grothendieck groups to classical equivariant K-theory.

Findings

Category of affine tangent group actions is equivalent to a slice category of G-equivariant vector bundles.

A monadic adjunction connects affine tangent group actions to G-equivariant vector bundles.

Grothendieck groups of affine actions are isomorphic to classical equivariant K-theory groups.

Abstract

For a Lie group $G$ and a vector bundle $E$ we study those actions of the Lie group $TG$ on $E$ for which the action map $TG\times E \to E$ is a morphism of vector bundles, and call those \emph{affine actions}. We prove that the category $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ of such actions over a fixed $G$-manifold $X$ is equivalent to a certain slice category $\mathfrak g_X \backslash \mathrm{Vect}_G\left(X\right)$. We show that there is a monadic adjunction relating $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ to $\mathrm{Vect}_G\left(X\right)$, and the right adjoint of this adjunction induces an isomorphism of Grothendieck groups $K_{TG}^{\mathrm{aff}}\left(X\right) \cong KO_G\left(X\right)$. Complexification produces analogous results involving $T_\mathbb C G$ and $K_G\left(X\right)$.