Ph.D.-thesis ‘Locally compact quantum groups’

Thesis consultant: Alfons Van Daele
Public defence on March 23, 2001 in the K.U.Leuven. the people from the jury.

Download . The slides of my thesis defence can also be found:

The idea of quantum groups goes back towards the sixties, when G.I. Kac attempted to provide one framework for study regarding both groups as well as their dual objects. In early eighties quantum groups made an appearance inside a different setting once the Leningrad school discovered quantizations of Lie algebras. These quantized universal enveloping algebras are types of Hopf algebras. The idea of Hopf algebras is entirely algebraic and therefore not sufficient for any study of all of the facets of quantum groups. The very first topological method of quantum groups was the idea of Kac algebras, which enables study regarding in your area compact groups, their duals and many other interesting examples. Nonetheless the idea of Kac algebras isn’t general enough to incorporate the above mentioned quantized Lie algebras or even the compact quantum groups.

Along with J. Kustermans and building around the work of countless mathematicians we’ve given a meaning of in your area compact quantum groups which provides coverage for all Kac algebras and all sorts of other known examples. The very first chapter of the thesis is dedicated to this definition and also to study regarding in your area compact quantum groups generally.

Ordinary groups are frequently based on their action on the space, think for example from the number of affine transformations from the plane. So it’s natural to create quantum groups act upon quantum spaces, which is completed in the 2nd chapter of the thesis. We read the canonical implementation of those actions with a unitary representation from the quantum group.

We cope with the key work of M. Enock and R. Nest, who demonstrated that quantum symmetry groups appear naturally in study regarding subfactors.

Already within the sixties an organized method to construct types of finite quantum groups was created by G.I. Kac. Along with L. Vainerman we demonstrated that Kac’ cocycle bicrossed product construction still experiences within our topological setting which approach we take to obtain new types of in your area compact quantum groups. Simultaneously, we create a general theory of extensions, so we prove a 1-to-one correspondence between so-known as cleft extensions and also the cocycle bicrossed products. All of this is worked within the 3rd chapter. These results parallel important algebraic developments produced by S. Majid.

Ph.D.-thesis ‘Locally compact quantum groups’

Thesis consultant: Alfons Van Daele
Public defence on March 23, 2001 in the K.U.Leuven. the people from the jury.

Download . The slides of my thesis defence can also be found:

The idea of quantum groups goes back towards the sixties, when G.I. Kac attempted to provide one framework for study regarding both groups as well as their dual objects. In early eighties quantum groups made an appearance inside a different setting once the Leningrad school discovered quantizations of Lie algebras. These quantized universal enveloping algebras are types of Hopf algebras. The idea of Hopf algebras is entirely algebraic and therefore not sufficient for any study of all of the facets of quantum groups. The very first topological method of quantum groups was the idea of Kac algebras, which enables study regarding in your area compact groups, their duals and many other interesting examples.

Nonetheless the idea of Kac algebras isn’t general enough to incorporate the above mentioned quantized Lie algebras or even the compact quantum groups.

Along with J. Kustermans and building around the work of countless mathematicians we’ve given a meaning of in your area compact quantum groups which provides coverage for all Kac algebras and all sorts of other known examples. The very first chapter of the thesis is dedicated to this definition and also to study regarding in your area compact quantum groups generally.

Ordinary groups are frequently based on their action on the space, think for example from the number of affine transformations from the plane. So it’s natural to create quantum groups act upon quantum spaces, which is completed in the 2nd chapter of the thesis. We read the canonical implementation of those actions with a unitary representation from the quantum group. We cope with the key work of M. Enock and R. Nest, who demonstrated that quantum symmetry groups appear naturally in study regarding subfactors.

Already within the sixties an organized method to construct types of finite quantum groups was created by G.I. Kac. Along with L. Vainerman we demonstrated that Kac’ cocycle bicrossed product construction still experiences within our topological setting which approach we take to obtain new types of in your area compact quantum groups. Simultaneously, we create a general theory of extensions, so we prove a 1-to-one correspondence between so-known as cleft extensions and also the cocycle bicrossed products. All of this is worked within the 3rd chapter. These results parallel important algebraic developments produced by S. Majid.