Seminar: Representations of Lie groups - SS 17

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The representation theory of Lie groups is the fundamental tool to exploit continuous symmetries of mathematical problems. Consequently, it can be applied in virtually all areas of mathematics and is particularly useful in physics. For example,
• it is used to decompose partial differential equations into simpler equations on smaller spaces (the basic example is the diagonalization of linear PDOs via Fourier analysis),
• and it underlies the classification of elementary particles in physics.

The topic is accessed from the class of matrix Lie groups that contains many of the groups that are most relevant for applications. This approach avoids some technicalities of the general (manifold theoretic) case without oversimplifying the subject. In particular, a basic knowledge of linear algebra and analysis is sufficient for this approach.

The seminar covers some of following topics (depending on the number of participants):
• basic theory of Lie groups: Lie groups, Lie algebras, exponential mapping,
• basic representation theory: unitary representations, reducibility,
• the Baker-Campbell-Hausdorff formula,
• semisimple theory: roots, weights, Weyl groups,
• maybe compact groups.

Mostly, the content follows the textbook: 'Lie groups, Lie algebras, and representations' by Brian C. Hall, Springer, 2015 that can be accessed online at https://www.ub.tum.de/ebooks.

Every participant will give a one-hour presentation followed by 15-30 minutes of discussion. In addition, a summary of the content of the talk (about 5 pages) has to be written.

News

• 21.03.2016: Registration for the seminar is possible from 23.01.2017 in TUMonline.

Timetable

 Seminar Thursday, 16:00 - 17:30, MI 02.10.011

Course Material

 Week 1 (27.04.2017) Talk 1: Introduction; distribution of topics

 Week 2 (04.05.2017) Talk 2: Matrix Lie groups Johannes Baeumler   Zusammenfassung

 Week 3 (11.05.2017) Talk 3: The matrix exponential Isabel Wagner   Zusammenfassung

 Week 4 (18.05.2017) Talk 4: Lie algebras Maximilian Hess   Zusammenfassung

 Week 5 (25.05.2017) no seminar (Christi Himmelfahrt) -

 Week 6 (01.06.2017) Talk 6: Basic representation theory Jan Mueller   Zusammenfassung

 Week 7 (08.06.2017) Talk 7: The Baker-Campbell-Hausdorff formula Clara Bergner   Zusammenfassung

 Week 8 (15.06.2017) no seminar (Fronleichnam) -

 Week 9 (22.06.2017) Talk 9: The Representations of sl(3;\C) Rafael Bove Barrios   Zusammenfassung

 Week 10 (29.06.2017) Talk 10: Semisimple Lie algebras Nicolas Reiter   Zusammenfassung

 Week 11 (06.07.2017) Talk 11: Root systems David Hien   Zusammenfassung

 Week 12 (13.07.2017) Talk 12: Representations of semisimple Lie algebras Dimitri Dine

 Week 13 (20.07.2017) no seminar -

 Week 14 (27.07.2017) no seminar -

Literature

Representation theory
Hall   Lie groups, Lie algebras, and representations    Springer    2015