Smooth actions of compact quantum groups on compact smooth manifolds

TL;DR

This paper defines and analyzes smooth actions of compact quantum groups on compact smooth manifolds, establishing their injectivity and conditions for lifting actions to bimodule morphisms that preserve Riemannian structures.

Contribution

It introduces a formal definition of smooth actions of CQGs on manifolds and characterizes when these actions can be lifted to bimodule morphisms preserving geometric structures.

Findings

Smooth actions of CQGs are always injective.

A necessary and sufficient condition for lifting actions to bimodule morphisms is identified.

Preservation of Riemannian inner products is equivalent to the lift condition.

Abstract

Definition of a smooth action of a CQG on a compact, smooth manifold is given and studied. It is shown that a smooth action is always injective. Furthermore A necessary and sufficient condition for a lift of the smooth action as a bimodule morphism on the bimodule of one forms has been deduced and it is also shown to be equivalent to the condition of preserving some Riemannian inner product on the manifold.