# Lie groups and Lie algebras: Geometry, Algebra, and Combinatorics

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Double feature: Wednesday 19. Feb. 10:30-12:00 and 13:00-14:30

The seminar takes place Wednesday 10:15 - 11:45 at A2 (Villa) seminar room. Please see below for the seminar description, literature, and what to expect. We as well started a discussion page for registered users where we can discuss problems from the book or elsewhere or questions.

# Schedule

The winter semester 2013/14 consists of 16 weeks. The last two presentations are likely to take place during the semester break.

1. Introductory meeting 30-45min(Christian/Raman, 2013-10-16)
• remaining topics
2. Matrix Lie groups (Moritz S., 2013-10-16)
3. Properties of matrix Lie groups (Moritz S./Michael, 2013-10-23)
4. Lie groups as smooth manifolds with a group structure, based on «Reference(“FH91”,”Chapter 7”)» (Thomas, 2013-10-30)
5. The exponential mapping, part I (Anna-Lena, 2013-11-06)
6. The exponential mapping, part II (Miriam, 2013-11-13)
7. The Baker-Campbell-Hausdorff formula, part I (Moritz F., 2013-11-20)
8. The Baker-Campbell-Hausdorff formula, part II (Lucia, 2013-11-27)
9. Introduction to representation theory, based on «Reference(“FH91”,”Chapter 1-4”)» and «Reference(“Ha03”,”Chapter 7”)» (Olaf, 2013-12-04)
10. The representations of $\mathbf{SU}(3)$, part I (Heuna, 2013-12-11)
11. The representations of $\mathbf{SU}(3)$, part II (Lars, 2013-12-18)
12. Semisimple Lie algebras, part I (Albert, 2014-01-08)
13. Semisimple Lie algebras, part II (Florian, 2014-01-15)
14. The semisimple Lie algebra $sl(n,\mathbb{C})$ (Leif, 2014-01-22)
15. Representations of complex semisimple Lie algebras, part I (Leif / Katharina, 2014-01-29)
16. Representations of complex semisimple Lie algebras, part II (Katharina / Sarah, 2014-02-05)
17. Representations of complex semisimple Lie algebras, part III (Sarah, 2014-02-12)
18. More on roots and weights, part I (Tobias, 2014-02-19, 10:30-12:00)
19. More on roots and weights, part II (Jean-Philippe, 2014-02-19, 13:00-14:30)

# Seminar description

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 «Reference(“Ha03”,”B. Hall”, “Lie groups, Lie algebras, and representations: An elementary introduction”,”Springer (2003)”)» supplemented by the more general and more abstract perspective of Bröcker-tom Dieck «Reference(“BtD85”,”T. Bröcker and T.tom Dieck”,”Representations of Compact Lie Groups”,”Springer (1985)”)» , Fulton-Harris «Reference(“FH91”,”W. Fulton and J. Harris”,”Representation theory: A first course”,”Springer (1991)”)» , and Humphreys «Reference(“Hum72”,”J. Humphreys”,”Introduction to Lie Algebras and Representation Theory”,”Springer (1972)”)» . 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 a solid background in analysis and linear algebra are the only prerequisits.

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.

# Prerequisits

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

«Reference»

# What to expect

• Active participation expected

• Talks are 50-70 min + questions and discussion

• Filling the wiki with content

• We hope to get a successively filling of the wiki with an annotated outline of the book

• We want to discuss your talk ** one week** before along an outline of about 2 pages. Please setup the outline in the Wiki before we meet

• We print the outline (as a handout) on the day of the presentation

• The outline should be expanded with more details from your talk. The hard limit is 5 pages.

• Deadline for the final annotated outline is 7 days after the talk

• Blackboard recommended, overhead and projector available

• The Brian Hall book on sale on Springer’s webpage
• Hall Lie groups and Lie algebras
• arXiv version
• Fulton-Harris Representation theory