Lévy Processes on Quantum Hypergroups

Abstract

Quantum hypergroups are non-commutative versions of hypergroups and were introduced by Yu.A. Chapovsky and L.I. Vainerman. Assuming that the quantum hypergroup satisfies a certain positivity condition (Schoenberg's correspondence), we show that Lévy processes, like in the quantum group case, are given by solutions of quantum stochastic differential equations in the sense of R.L. Hudson and K.R. Parthasarathy. We prove that quantum hypergroups of double coset type satisfy Schoenberg's correspondence. As an example we discuss the quantum hypergroup U<2>//U<1> with U<n> the non-commutative analogue of the coefficient algebra of the unitary group.