Semi-simple and Affine Lie Algebras (GQT)

The theory of Lie groups was initiated by the Norwegian mathematician Sophus Lie (1842 - 1892) with the purpose of analyzing differential equations in the presence of ymmetries. Much about a  Lie group can be understood from its linearization at the identity, the so called Lie algebra.

In the course we will systematically develop the structure theory of these Lie algebras. In particular we will study the semisimple algebras. Over the field of complex numbers these are precisely the complexified Lie algebras of the compact Lie groups with finite center.

The structure of semisimple Lie algebras can be understood
in terms of so called root systems and the associated reflection (or Weyl) groups. We will discuss the classification of these algebras in terms of the so-called Dynkin diagrams. Important (i.e. in quantum physics) is the representation theory of semisimple algebras. We will discuss the classification of irreducible representations in terms of weight theory. The beautiful character and dimension formulas of Weyl will be discussed.

Towards the end of the course we will construct the (untwisted) affine Lie algebras. They appear in physics, e.g. in Wess-Zumino-Witten models, under the name current algebras. These are central extensions of a loop algebra, i.e.,  the Lie algebra of polynomial maps from the circle to a simple finite dimensional Lie algebra. We will show that a loop algebra admits a presentation which resembles
the presentation with Serre relations in the finite dimensional case. The presentation will be generalized to obtain Kac-Moody Lie algebras. If time  permits we will address the representation theory of these.