Investigating noncommutative structures : quantum groups and dual groups in the context of quantum probability The history of Mathematics has been lead in part by the desire for generalization: once an object was given and had been understood there was the desire to find a more general version of it to fit it into a broader framework. Noncommutative Mathematics fits into this description as its interests are objects analoguous to vector spaces or probability spaces etc. but without the commonsense interpretation that those latter objects possess. Indeed a space can be described by its points but also and equivalently by the set of functions on this space. This set is actually a commutative algebra sometimes equipped with some more structure: * algebra C* algebra von Neumann algebras Hopf algebras etc. The idea that lies at the basis of noncommutative Mathematics is to replace such algebras by algebras that are not necessarily commutative any more and to interpret them as algebras of functions on noncommutative spaces . Of course these spaces do not exist independently from their defining algebras but facts show that a lot of the results holding in (classical) probability or (classical) group theory can be extended to their noncommutative counterparts or find therein powerful analogues. The extensions of group theory into the realm of noncommutative Mathematics has long been studied and has yielded the various quantum groups. The easiest version of them the compact quantum groups consist of C* algebras equipped with a * homomorphism &Delta with values in the tensor ... Quantengruppe Zufallsmatrizen Lévy prozess Quantenwahrscheinlichkeitstheorie Random matrices Lévy processes quantum groups quantum probability free probability