@phdthesis{Ulrich2016, author = {Micha{\"e}l Ulrich}, title = {Investigating noncommutative structures: quantum groups and dual groups in the context of quantum probability}, journal = {Erforschen nichtkommutativer Strukturen: Quantengruppen und Duale Gruppen im Rahmen der Quantenwharscheinlichkeitsrechnung}, url = {https://nbn-resolving.org/urn:nbn:de:gbv:9-002563-3}, year = {2016}, abstract = {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 product of the algebra with itself and verifying some coassociativity condition. It is also required that the compact quantum group verifies what is known as quantum cancellation property. It can be shown that (classical) compact groups are indeed a particular case of compact quantum groups. The area of compact quantum groups, and of quantum groups at large, is a fruitful area of research. Nevertheless, another generalization of group theory could be envisioned, namely by taking a comultiplication \&Delta taking values not in the tensor product but rather in the free product (in the category of unital *-algebras). This leads to the theory of dual groups in the sense of Voiculescu, also called H-algebras by Zhang. These objects have not been so thoroughly studied as their quantum counterparts. It is true that they are not so flexible and that we therefore do not know many examples of them and showing that some relations cannot exist in the dual group case because they do not pass the coproduct. Nevertheless, I have been interested during a great part of my PhD work by these objects and I have made some progress towards their understanding, especially regarding quantum L{\´e}vy processes defined on them and Haar states.}, language = {en} }