Deformaties van spectrale tripletten en hun kwantumisometriegroepen via monoïdale equivalenties

摘要

In this thesis, we develop a deformation procedure for spectral triples. The initial data for the deformation are:1) a compact spectral triple i.e. a noncommutative generalisation of a compact manifold;2) a compact quantum group G acting algebraically and by orientation preserving isometries on the spectral triple (i.e. a noncommutative generalisation of a compact group acting isometrically on a compact manifold);3) a unitary fiber functor on G, or equivalently, a monoidal equivalence between G and a second compact quantum group i.e. an equivalence between the representation categories of the two compact quantum groups seen as monoidal categories.This procedure is proven to be a generalisation of the cocycle deformation which Goswami and Joardar introduced in 2014. Moreover, it is a proper generalisation: we construct an example of our method which is not a deformation à la Goswami-Joardar. This example is found by deforming a spectral triple on a suitable deformation of the sphere using a monoidal equivalence between a q-deformation of SU(2) and another well chosen orthogonal quantum group.Finally, we prove that this deformation method is compatible with the notion of quantum isometry group: the quantum isometry group of a deformed spectral triple is a suitable deformation of the quantum isometry group of the initial quantum isometry group.