Seminar on Lie groups, Lie algebras and their representations

Lie groups are important to describe symmetries, both in mathematics and in applications (physics, chemistry, engineering,. . . ). The classical Lie groups are for example the orthogonal groups O(n), the unitary groups U(n), but mathematicians and physicists are also fascinated by more exotic examples such as the symmetry group of the octonions which is discussed a lot in modern mathematical physics. Many of these Lie groups can be represented as subgroups of Gl(k,R) for some sufficiently large k, but there are also Lie groups which cannot. Lie groups are manifolds G together with a multiplication μ : G×G→ G which is a smooth map, such that (G,μ) is a group.1 Lie groups and their representation is a mighty theory which allows effect calculations both for problems inside mathematics and also for applications outside.