Representation theory of compact Lie groups, spring 2014

Lecturer

Stefan Wagner

Scope

5 sp.

Type

Advanced studies

Prerequisites

The course is intended for advanced undergraduate students of mathematics and physics. Prior courses in linear algebra and analysis are required. Elementary knowledge of topology and differential geometry are useful but not assumed, and all the necessary theory will be developed during the course.

Description

To locate the theory of Lie groups within mathematics, one can say that Lie groups are groups with some additional structure that permits us to apply analytic techniques such as differentiation in a group theoretic context. Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. The aim of this course is to give an introduction to the representation theory of compact Lie groups including the construction of the Spin groups, Schur Orthogonality, the Peter–Weyl Theorem and the Plancherel Theorem. If time allows we will also discuss the link between Lie groups and Lie algebras and some Lie group cohomology.

Lectures

Weeks 3-9 and 11-18, Tuesday 14-16 in room C123.

Exams

Bibliography

The course will mainly follow

• Mark R. Sepanski, “Compact Lie groups ”, Graduate Texts in Mathematics 235, Springer, 2007.

Other useful references are

• J. A. de Azcrragaand and J. M. Izquierdo, “Lie Groups, Lie Algebras, Cohomology and some applications ”, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1998.
• J. Hilgert and K.-H. Neeb, “An Introduction to the Structure and Geomerty of Lie Groups and Lie Algebras”, Springer Monographs in Mathematics, 2012.
• K. H. Hofmann and S. A. Morris, “The Structure of Compact Groups ”, 2nd Revised and Augmented Edition, de Gruyter Studies in Mathematics 25, Cambridge University Press, 2006.

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