Investigating non commutative structures - quantum groups and dual groups in the context of quantum probability

Abstract : Noncommutative Mathematics are a very active domain. The idea underlying it is that instead of describing a space as a set of points, it is equivalent to describe it with the algebra of functions defined on said space. This algebra is commutative. Now we replace this algebra with an algebra that is not necessarily commutative any more and we want to interpret it as the algebra of functions defined on a « noncommutative space ». Quantum groups are an example of such a noncommutative generalization of the notion of group. They are C*-algebras equipped with a comultiplication that takes its values in the tensor product of the algebra with itself. Quantum groups are well-known and well studied. Nevertheless we can also define dual groups, which are similar to quantum groups, but the comultiplication takes now its values in the free product of the algebra with itself, instead of the tensor product. Though dual groups have been introduced in the 80s, they have not been much studied so far. The goal of this thesis is to study their properties, especially in the case of one particular dual group called the unitary dual group, by using methods from noncommutative probability (or quantum probability).
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Michael Ulrich. Investigating non commutative structures - quantum groups and dual groups in the context of quantum probability. Quantum Algebra [math.QA]. Université de Franche-Comté, 2016. English. ⟨NNT : 2016BESA2061⟩. ⟨tel-01652124v2⟩

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