Seminar "Functional analysis and noncommutative geometry". Speaker: Riya Dogra

The talk is based on a survey paper by T. Banica, J. Bichon, and B. Collins and will focus on some of the recent work on quantum permutation groups focusing mainly on some concepts needed to build the subject.
We shall start by introducing the concept of a C*-algebra and some fundamental results about the spectrum of an element in that algebra.
Next we shall extend the theory to de fine a finite dimensional Hopf algebra leading to a result about commutative and cocommutative group algebras generated by a finite group. Using these concepts, we will try to see the construction of orthogonal, unitary and symmetric quantum groups.
We also prove that the universal algebras A_o(n), A_u(n), and A_s(n) are finitely generated Hopf algebras.
Finally, we look at an introduction and state a result about quantum permutation groups.