One-to-one correspondence between generating functionals and cocycles on   quantum groups in presence of symmetry
  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. 
Biswarup Das, Uwe Franz, Anna Kula and Adam Skalski
math.QA math.OA