Sumários
10. Lie groups, Lie algebras and representations
29 outubro 2020, 11:00 • Pedro Resende
Exercises from chapters 4 and 5 of Brian Hall's book.
9. Representations of Lie groups and Lie algebras
23 outubro 2020, 11:00 • Pedro Resende
Real and complex finite dimensional representations. Basic definitions and examples: trivial, standard, and adjoint representations. Representations versus actions. Invariant subspaces, irreducible representations. Intertwiners (equivariant maps), isomorphic representations. The differential of a representation of matrix Lie groups. Functorial properties of the differential. Complexification versus irreducibility. Direct sums and tensor products of representations. Complete reducibility. Complete reducibility of invariant subspaces of completely reducible representations. Complete reducibility of unitary representations on Euclidean spaces. Schur's Lemma. This lecture followed the material in chapter 4 of Brian Hall's book.
8. Matrix Lie groups and Lie algebras
22 outubro 2020, 11:00 • Pedro Resende
Lie's second theorem for matrix groups (revision from previous lecture). An equivalence of categories between the category of simply connected matrix Lie groups and the category of their Lie algebras. Lie subgroups: connected Lie subgroups of matrix groups; Lie's third theorem. Universal covers. SL(2,R) as an example without a universal cover in the class of matrix Lie groups. General universal covers. Lie's first theorem. The equivalence of categories between the category of finite dimensional real Lie groups and the category of finite dimensional real Lie algebras. This lecture followed sections 5.8–5.10 of Brian Hall's book. See also these notes.
7. Matrix Lie groups and Lie algebras
16 outubro 2020, 11:00 • Pedro Resende
Global extension of local homomorphisms on simply connected matrix Lie groups. Lifting of Lie algebra homomorphisms for simply connected matrix Lie groups. Some functoriality properties of the liftings. Polar decomposition of matrices in SL(2;R) and SL(2;C). Simple connectedness of SL(2;C) and non-simple connectedness of SL(2;R). Lifting of Lie algebra homomorphisms from sl(2;R).
6. Matrix Lie Groups and Lie Algebras
15 outubro 2020, 11:00 • Pedro Resende
Exercises from Brian Hall's book: Chapter 3, ex. 6, 7, 10, 11, 15, 16, 17, 18, 19.