Meetings take place in Howard House (first term: Wednesday at 2pm; second term: Tuesday at 10 am).

We are using the following material:

• [1]   Representations of Semisimple Lie groups, A.W. Knapp and P.E. Trapa (lecture notes available here).
• [2]   Lie Groups Beyond an Introduction - Second Edition, A.W. Knapp, Birkhäuser, 2005.
• [3]   Representation Theory of Semisimple Lie Groups - An Overview Based on Examples, A.W. Knapp, Princeton University Press, 1986.

Meetings

1. Wednesday 22nd November 2017: Finite-dimensional representations of SL(2,R), SL(2,C), and their Lie algebras.
2. Wednesday 29th November 2017: Solvable, nilpotent and semisimple Lie algebras. Lie's and Engel's Theorems, Cartan's Criterion for semisimplicity.
3. Wednesday 13th December 2017: Root-space decomposition for sl(n,C). The Lie algebra sl(n,C) is simple.
4. Tuesday 30th January 2018: Generalised weight-space decomposition for a representation of a nilpotent Lie algebra. Existence of Cartan subalgebras.
5. Tuesday 6th February 2018: Root-space decomposition for complex semisimple Lie algebras. Split real forms and compact real forms.
6. Tuesday 13th February 2018: Cartan involutions and Cartan decomposition. Iwasawa decomposition. Examples.
7. Tuesday 27th February 2018: The universal enveloping algebra and its universal property. Poincare'-Birkhoff-Witt Theorem. Casimir element.
8. Thursday 8th March 2018: Schur's Lemma. Characters and orthogonality relations. Peter-Weyl Theorem.

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