K-theory of affine actions

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 $K_{TG}^{\mathrm{aff}}\left(X\right)\cong K{O}_G\left(X\right)$. Complexification produces analogous results involving $T_{\mathbb{C}}G$ and $K_G\left(X\right)$.

26 pages

References in corpus (2)

• Reduction of a symplectic-like Lie algebroid with momentum map and its application to fiberwise linear Poisson structures
• The Atiyah algebroid of the path fibration over a Lie group