Sumários

25. Classification of finite dimensional irreducible representations of semisimple Lie algebras

18 Dezembro 2020, 11:00 Pedro Resende

Conclusion of the proof of the classification theorem. This lecture followed section 9.7 Brian Halls' book.


24. Verma modules

17 Dezembro 2020, 11:00 Pedro Resende

Verma modules. Irreducible representations obtained from Verma modules. This material followed sections 9.5 and 9.6 of Brian Hall's book.


23. Representations of semisimple Lie algebras

11 Dezembro 2020, 11:00 Pedro Resende

The tensor algebra of a vector space. The enveloping associative algebra of a Lie algebra. The Poincaré-Birkhoff-Witt theorem. Introduction to Verma modules. Exercise 7.13 of Brian Hall's book.


22. Representations of semisimple Lie algebras

10 Dezembro 2020, 11:00 Pedro Resende

Integral elements and dominant integral elements. Partial order on the set of integral elements. Basic properties. Integrality of the weights of the representations of semisimple Lie algebras, and invariance of the set of weights under the action of the Weyl group. Introduction to the theorem of the highest weight. This lecture was based on sections 8.7–8.8 and 9.1–9.2 of Brian Hall's book.


21. Semisimple Lie algebras and root systems

4 Dezembro 2020, 11:00 Pedro Resende

Exercises from chapter 7 of Brian Hall's book: 1, 2, 4, 6, 9, 12.