19165

WiSe 13/14: Lie groups and Lie algebras: Geometry, Representations, Combinatorics

Raman Sanyal;Christian Stump

Hinweise für Studierende

Prerequisits:
A solid background in analysis and (linear) algebra. Knowledge in differential geometry/topology or representation theory is helpful but not required.

Kommentar

Lie groups and Lie algebras are ubiquitous in mathematics. Their study combines in an elegant way ideas and notions from algebra, geometry, and combinatorics. Lie groups make an appearance as symmetries of mathematical objects such as physical systems, differential equations, geometric objects, etc. They are best understood in terms of associated Lie algebras and their representations.

In this seminar, we will approach Lie groups and their Lie algebras through their geometry, their representations, and their combinatorics. To emphasize the hands-on character of the seminar and to keep the prerequisits to a minimum, we will follow the book by B. Hall supplemented by the more general and more abstract perspective of Bröcker-tom Dieck, Fulton-Harris, and Humphreys. The main advantage of the approach by Hall is the focus on matrix Lie groups, i.e., closed subgroups of the general linear group. This way familiarity with differentiable manifolds is reduced to a solid background in analysis and linear algebra.

In the first half, we will cover the basic theory of (matrix) Lie groups and Lie algebras including the Baker-Campbell-Hausdorff formula. Fundamental notions are supplemented by a wealth of examples. In the second half we will focus on semisimple Lie algebras, including their Weyl groups and root systems, and their representations.

The goal of the seminar is a coherent picture of Lie theory along the lines of the book by Brian Hall. To this end, participants are expected to actively participate throughout. Schließen

Literaturhinweise

T. Bröcker, T. tom Dieck - Representations of Compact Lie Groups
B. Hall - Lie gropus, Lie algebras, and representations: An elementary introduction
W. Fulton, J. Harris - Representation Theory: A first course
J.E. Humphreys - Introduction to Lie Algebras and Representation Theory Schließen