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,vy 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.