The local-triviality dimension of actions of compact quantum groups

TL;DR

This paper introduces the local-triviality dimension for actions of compact quantum groups on unital C*-algebras, linking it to classical bundle triviality and applying it to extend the Borsuk-Ulam theorem in noncommutative geometry.

Contribution

It defines a new invariant called local-triviality dimension for quantum group actions and demonstrates its implications for freeness and noncommutative Borsuk-Ulam conjectures.

Findings

Actions with finite local-triviality dimension are free.

The notion generalizes classical local triviality to the quantum setting.

Application to noncommutative Borsuk-Ulam conjecture under certain subgroup conditions.

Abstract

We define the local-triviality dimension for actions of compact quantum groups on unital C*-algebras. The resulting compact quantum principal bundle is said to be locally trivial when this dimension is finite. For commutative C*-algebras, this notion recovers the standard definition of local triviality of compact principal bundles. We prove that actions with finite local-triviality dimension are automatically free. Then we apply this new notion to prove the noncommutative Borsuk-Ulam-type conjecture under the assumption that a compact quantum group admits a non-trivial classical subgroup whose induced action has finite local-triviality dimension. This is a noncommutative extension of the Borsuk-Ulam-type theorem for locally trivial principal bundles.