Sumários
20. Semisimple Lie algebras and root systems
3 dezembro 2020, 11:00 • Pedro Resende
Duals of root systems. Bases and Weyl chambers. Action of the Weyl group on the set of open Weyl chambers: basic properties, transitivity and freeness. Reflections associated to the positive simple roots as generators of the Weyl group. Dynkin diagrams. Classification of the irreducible root systems and of the complex simple Lie algebras. (Cf. sections 8.3–8.6 and 8.9–8.11 of Brian Hall's book.)
19. Semisimple Lie algebras and root systems
27 novembro 2020, 11:00 • Pedro Resende
The root systems of the classical Lie algebras. Simple Lie algebras versus complexification of Lie algebras. Theorem: a nonabelian Lie algebra of a compact matrix Lie group is simple if and only if its complexification is simple as a complex Lie algebra. A semisimple Lie algebra is simple if and only if its root system is irreducible. This lecture followed Brian Hall's book, sections 7.6 and 7.7.
18. Semisimple Lie algebras and root systems
26 novembro 2020, 11:00 • Pedro Resende
More properties of the roots of semisimple Lie algebras: the only multiples of a root a that are roots are a and – a; the root spaces have dimension one. Weyl group of a semisimple algebra. Invariance of the set of roots under the action of the Weyl group. Root systems. The Weyl group of a root system. Direct sum of roots systems, irreducible root systems. General facts about angles between roots and root norms. Classification of the rank-two root systems. This lecture was based on sections 7.3–7.5 and 8.1–8.2 of Brian Hall's book.
17. Representations of Lie groups and Lie algebras
20 novembro 2020, 11:00 • Pedro Resende
Exercises from chapter 6 of Brian Hall's book: 1, 2, 3, 6, 7, 10, 12.
16. Semisimple Lie algebras
19 novembro 2020, 11:00 • Pedro Resende
Cartan subalgebras of semisimple Lie algebras. Basic properties and examples. Roots and root spaces relative to a Cartan subalgebra. The subalgebra isomorphic to sl(2;C) associated to each root. This lecture was based on Brian Hall, sections 7.2–7.3.