An averaging trick for smooth actions of compact quantum groups on manifolds

TL;DR

The paper demonstrates a method to modify smooth actions of compact quantum groups on manifolds to preserve a Riemannian structure, enabling better geometric understanding of such actions.

Contribution

It introduces an averaging technique that ensures quantum group actions preserve a Riemannian metric on manifolds, extending classical symmetry concepts to quantum symmetries.

Findings

Existence of a Riemannian structure preserved by quantum group actions

Averaging trick applicable to smooth quantum symmetries

Extension of classical symmetry preservation to quantum setting

Abstract

We prove that, given any smooth action of a compact quantum group (in the sense of \cite{rigidity}) on a compact smooth manifold satisfying some more natural conditions, one can get a Riemannian structure on the manifold for which the corresponding $C^\infty(M)$-valued inner product on the space of one-forms is preserved by the action.