Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

Quantum Markov semigroups (QMS), i.e. strongly continuous semigroups of unital completely positive maps, on compact quantum groups are studied. We show that translation invariant QMSs on the universal or reduced C⁎-algebra of a compact quantum group are in one-to-one correspondence with Levy processes on its ⁎-Hopf algebra. We use the theory of Levy processes on involutive bialgebras to characterize symmetry properties of the associated QMS. It turns out that the QMS is self-adjoint (resp. KMS-symmetric) if and only if the generating functional of the Levy process is invariant under the antipode (resp. the unitary antipode). Furthermore, we study Levy processes whose marginal states are invariant under the adjoint action. Finally, some related aspects of the potential theory as Dirichlet form, derivation and spectral triple are investigated. We discuss how the above results apply to compact groups, group C⁎-algebras of discrete groups, free orthogonal quantum groups On+ and twisted SUq(2) quantum group.