Banach manifolds of spectrally small quantum-group representations

TL;DR

This paper demonstrates that finite-spectrum representations of compact quantum groups form Banach analytic manifolds with rich geometric structures, extending classical representation theory to the quantum setting.

Contribution

It establishes the differential-geometric structure of finite-spectrum quantum group representations, generalizing classical results to the quantum context and identifying conditions for norm-continuity.

Findings

Representations form Banach analytic manifolds.

Orbit maps are locally analytically split.

Finite-spectrum unitary representations are norm-continuous.

Abstract

We prove that finite-spectrum representations of compact quantum groups either in unital $C^*$-algebras $A$ or on Banach spaces $E$ exhibit the same Banach-space-modeled differential-geometric structure as their classical analogues: (a) they are Banach analytic manifolds; (b) locally homogeneous under conjugation by the pertinent Banach Lie group $U(A)$ or $GL(E)$; (c) with orbit maps fibering principally; (d) and hence with said orbit maps admitting local analytic splitting. We also identify the finite-spectrum unitary representations as precisely those that are norm-continuous in the appropriate sense when the compact quantum group has at least one classical point, again generalizing the classical parallel present in various forms in work of Kallman, Shtern and the author.