#### Vol. 301, No. 2, 2019

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K-theory of affine actions

### James Waldron

Vol. 301 (2019), No. 2, 639–666
DOI: 10.2140/pjm.2019.301.639
##### 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 $TG×E\to E$ is a morphism of vector bundles, and call those affine actions. We prove that the category ${Vect}_{TG}^{\phantom{\rule{0.3em}{0ex}}aff}\left(X\right)$ of such actions over a fixed $G$-manifold $X$ is equivalent to a certain slice category ${\mathfrak{g}}_{X}\setminus {Vect}_{G}\left(X\right)$. We show that there is a monadic adjunction relating ${Vect}_{TG}^{\phantom{\rule{0.3em}{0ex}}}$

##### Keywords
vector bundles, equivariant K-theory, differential geometry
Primary: 19E99