It is shown how the relations of the renormalized squared white noise defined by Accardi, Lu, and Volovich [ALV99] can be realized as factorizable current representations or Lévy processes on the real Lie algebra sl 2. This allows to obtain its Itô table, which turns out to be infinite-dimensional. The linear white noise without or with number operator is shown to be a Lévy process on the Heisenberg-Weyl Lie algebra or the oscillator Lie algebra. Furthermore, a joint realization of the linear and quadratic white noise relations is constructed, but it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator. Classical Lévy processes are shown to arise as components of Lévy process on real Lie algebras and their distributions are characterized.

Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed.Comment: 28 pages; v3 omits the former lemma 2.1 due to a gap in the proof. This does not affect any other results. The paper will appear in the Journal of Algebr

Abstract. Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced C * -algebra of a compact quantum group which are translation invariant (w.r.t. to the coproduct) are in one-to-one correspondence with Lévy processes on its * -Hopf algebra. We use the theory of Lévy processes on involutive bialgebras to characterize symmetry properties of the associated quantum Markov semigroup. It turns out that the quantum Markov semigroup is GNS-symmetric (resp. KMS-symmetric) if and only if the generating functional of the Lévy process is invariant under the antipode (resp. the unitary antipode). Furthermore, we study Lévy processes whose marginal states are invariant under the adjoint action. In particular, we give a complete description of generating functionals on the free orthogonal quantum group O + n that are invariant under the adjoint action. Finally, some aspects of the potential theory are investigated. We describe how the Dirichlet form and a derivation can be recovered from a quantum Markov semigroup and its Lévy process and we show how, under the assumption of GNS-symmetry and using the associated Schürmann triple, this gives rise to spectral triples. We discuss in details how the above results apply to compact groups, group C * -algebras of countable discrete groups, free orthogonal quantum groups O + n and the twisted SU q (2) quantum group.

The authors propose piecewise deterministic Markov processes as an alternative approach to model gene regulatory networks. A hybrid simulation algorithm is presented and discussed, and several standard regulatory modules are analysed by numerical means. It is shown that despite of the partial simplification of the mesoscopic nature of regulatory networks such processes are suitable to reveal the intrinsic noise effects because of the low copy numbers of genes.

Abstract. We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π : A → M n (C) is a finite dimensional representation of a Hopf C * -algebra, we prove that the idempotent state associated to its Hopf image A ′ must be the convolution Cesàro limit of the linear functional ϕ = tr • π. We discuss then some consequences of this result, notably to inner linearity questions.