One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry

TL;DR

This paper establishes a one-to-one correspondence between generating functionals and cocycles on quantum groups with symmetry, leading to new insights into quantum Lévy processes and the Haagerup property.

Contribution

It proves that all cocycles on Hopf *-algebras with symmetry originate from generating functionals, extending previous results and applying to quantum Lévy processes and quantum group properties.

Findings

Quantum Lévy processes decompose into Gaussian and non-Gaussian parts.

The Haagerup property is characterized by the existence of proper cocycles.

All cocycles arise from generating functionals under symmetry assumptions.

Abstract

We prove that under a symmetry assumption all cocycles on Hopf *-algebras arise from generating functionals. This extends earlier results of R.Vergnioux and D.Kyed and has two quantum group applications: all quantum L\'evy processes with symmetric generating functionals decompose into a maximal Gaussian and purely non-Gaussian part and the Haagerup property for discrete quantum groups is characterized by the existence of an arbitrary proper cocycle.