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Seminar on Lie algebras

Dynkin diagrams of simple Lie algebras
Original file by Tomruen, created under CC BY-SA 3.0 License.

Florent Schaffhauser
Heidelberg University
Winter Semester 2023-2024

Organisation

The purpose of this seminar is to cover the basics of Lie theory, with a view towards understanding the structure and classification of semisimple Lie algebras.

  • Target audience: Bachelor and Master students.
  • Language of instruction: English.
  • Time and place: Block Seminar, 15-16 December 2023.
  • Zulip channel:     (please contact me by email to register).
  • Registration: on Müsli.
  • Introductory meeting: 11 October 2023.

Program

Topics to be covered in the seminar:

  1. Nilpotent and solvable Lie algebras
  2. Simple and semisimple Lie algebras
  3. Cartan subalgebras
  4. Representations of the Lie algebra sl(2; C)
  5. Root systems and root space decompositions
  6. Cartan matrices and Dynkin diagrams
  7. The Jacobson-Morozov theorem and parabolic subalgebras
  8. Nilpotent orbits and Kostant’s theorem
The A2 root system Original file by Mark L MacDonald, Smithers888 and Maksim, under CC BY-SA 3.0 License.

References

  1. David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras,  Van Nostrand Reinhold (1993).
  2. James E. Humphreys, Introduction to Lie algebras and representation theory,  GTM Vol. 9.,  Springer-Verlag (1972).
  3. Jean-Pierre Serre, Complex semisimple Lie algebras,  Springer-Verlag (1987).  Translated from the French by G. A. Jones.
The G2 root system
Original file by Mathphysman, under CC BY-SA 4.0 License.

Schedule

# Date Topic Speaker Slides Notes
0 06/12 Lie groups and Lie algebras Florent Schaffhauser
1 15/12 Nilpotent and solvable Lie algebras Richard Pospich
2 15/12 Simple and semisimple Lie algebras Arian Gjini
3 15/12 Reductive Lie algebras, Cartan subalgebras Feline Bailer
4 15/12 Representations of the Lie algebra \(\mathfrak{sl}( 2; \mathbb{C} )\) Aljoscha Helm
5 16/12 Root systems and root space decompositions Till Janke
6 16/12 Cartan matrices and Dynkin diagrams Philipp Müller
7 16/12 Classification of Dynkin diagrams Emanuel Roth
8 16/12 Nilpotent orbits Marcel Eichberg
9 16/12 Nilpotent orbits in the classical Lie algebras Liva Deiler