Brownian motion on involutive braided spaces


  • Uwe Franz Département de mathématiques de Besançon, Université de Bourgogne Franche-Comté, 16, route de Gray, 25 030 Besançon cedex, France
  • Michael Schürmann Institut für Mathematik und Informatik, Universität Greifswald, Walther-Rathenau-Straße 47, 17489 Greifswald, Germany
  • Monika Varšo Institut für Mathematik und Informatik, Universität Greifswald, Walther-Rathenau-Straße 47, 17489 Greifswald, Germany



We study (quantum) stochastic processes with independent and stationary increments (i.e., Lévy processes), and in particular Brownian motions in braided monoidal categories. The notion of increments is based on a bialgebra or Hopf algebra structure, and positivity is taken w.r.t. an involution. We show that involutive bialgebras and Hopf algebras in the Yetter–Drinfeld categories of a quasi- or coquasi-triangular $\ast$-bialgebra admit a symmetrization (or bosonization) and that their Lévy processes are in one-to-one correspondence with a certain class of Lévy processes on their symmetrization. We classify Lévy processes with quadratic generators, i.e., Brownian motions, on several braided Hopf-$\ast$-algebras that are generated by their primitive elements (also called braided $\ast$-spaces), and on the braided $\mathrm{SU}(2)$-quantum groups.


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