Sumários
Maximal tori in compact groups.
20 dezembro 2018, 09:30 • Gustavo Granja
Definition and examples of the maximal torus and the Weyl group. The maximal torus theorem and consequences. Relation between the root space decomposition for compact groups and semisimple Lie algebras. Representations of compact Lie groups vs representations of complex semisimple algebras. Examples.
Non-abelian Fourier analysis
19 dezembro 2018, 09:30 • Gustavo Granja
Proof of Peter-Weyl. Decomposition of the regular representation L^2(G) as a G-bimodule in terms of the irreducible representations of G.
Characters of representations of compact groups.
13 dezembro 2018, 09:30 • Gustavo Granja
Constructions with representations. Representations are characterized by their characters. L^2 orthogonality for characters. Statement of the Peter-Weyl theorem and proof that it is equivalent to the existence of a faithful representation for every compact Lie group.
Chevalley basis. Classification of representations.
12 dezembro 2018, 09:30 • Gustavo Granja
Proof of existence of a Chevalley basis for a semisimple Lie algebra. The compact form of a complex semi-simple Lie algebra. Classification of irreducible representations by highest weights with respect to a base for the root system.
Classification of root systems.
6 dezembro 2018, 09:30 • Gustavo Granja
Description of the properties and classification of root systems. Serre's relations for a semisimple algebra.