arXiv Analytics

Sign in

arXiv:1710.04642 [math.DG]AbstractReferencesReviewsResources

K-theory of affine actions

James Waldron

Published 2017-10-12Version 1

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 $K_G\left(X\right)$.

Related articles: Most relevant | Search more
arXiv:1503.06566 [math.DG] (Published 2015-03-23)
Tulczyjew's Triplet for Lie Groups II: Dynamics
arXiv:math/0301002 [math.DG] (Published 2003-01-01, updated 2003-01-25)
The uncertainty principle for operators determined by Lie groups
arXiv:1509.04414 [math.DG] (Published 2015-09-15)
Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces