### MAT563 - Groupes compacts et groupes de Lie (2022-2023)

This course aims to study two classes of groups and
certain aspects of their representation theory.
In the first part, we study compact groups.
In addition to finite groups, this class of groups contains
projective limits of finite groups
(which can appear as a Galois group of infinite extension),
and groups of matrices like O(n) or SU(n).

We generalize representation theory from finite groups
to these groups,by imposing topological conditions
(continuity) so that their study remains possible
and relevant. The main result of this part is
the Peter-Weyl theorem, that we
approach via the Fourier transform.

In the second part of the course, we study linear groups,
that is to say groups realized as groups of matrices.
The main tool here is differential calculus,
which allows in particular to introduce the Lie algebra of
a linear group.
We introduce vocabulary and some results concerning
Lie algebras, independent of group theory.

We study the correspondence between groups and Lie algebra,
connectivity problems and covering of groups.

The finite-dimensional representations of these groups
are studied via the action of their Lie algebra.

We then use the results obtained in the first two parts
to study in depth the representation theory
of certain compact linear Lie groups
(first SU(2) and SO(3), then SU(n)).

We show tthat every compact group is the projective
limit of compact linear groups.

Groups and the theory of their representations
constitute a central field of mathematics,
both through the numerous and very rich applications
(physics, arithmetic, etc.), but also through
the diversity of tools necessary for their study
(algebra, topology, analysis functional,
category theory).

Prerequisites: representation theory of
finite groups (for example MAT556),
rudiments of general topology and functional analysis.

Indicative course plan:

1- Topological groups, examples (profinite groups,
matrix groups, etc.), and generalities.

2- Locally compact and compact groups. Haar measure,
spaces L^p(G)

3 - Representations of compact groups.
Fourier transform.

4- Representations of compact groups, continued.
Peter-Weyl theorem. Character theory.

5- Linear groups. Exponential applications. Lie Algebra

6- Lie correspondence

7- Homomorphisms and coverings.
Finite dimensional representations

8- SU(2), SO(3), spherical harmonics

9- SU(n) and the highest weight theorem.

Langage: French, (english on demand by non french speaking students)

Credits ECTS : 5