Lie Groups and Lie Algebras — 1st Semester 2008/2009
AnnoucementsThe final grades are available. SyllabusLie Groups: examples; SU(2), SO(3), SL(2); homogeneous spaces; some theorems about matrices; Lie theory; representation theory; compact groups and integration; maximal compact subgroups; the Peter-Weyl theorem; functions on R^n and S^(n-1); induced representations; the complexification of a compact group; the unitary and simmetric groups; the Borel-Weyl theorem; representations of non-compact groups; representations of SL(2); the Heisenberg group. Lie Algebras: basic concepts; representations and modules; special kinds of Lie algebras; the Lie algebras sl(n,C); Cartan subalgebras; the Cartan decomposition; the Killing form; the Weyl group; Dynkin diagrams; the universal enveloping algebra; Verma modules; finite dimensional irreducible modules; Weyl's character and dimension formulae; fundamental representations. Applications: quantum mechanics, particle physics, general relativity (if there is time and/or interest). BibliographyCarter, Segal e McDonald, Lectures on Lie Groups and Lie Algebras, Cambridge University Press (1995) Warner, Foundations of Differential Manifolds and Lie Groups, Springer (1971) Brocker e tom Dieck, representations of Compact Lie Groups, Springer (1985) Humphreys, Introduction to Lie Algebras and Representation Theory, Springer (1972) Grading PolicyTests: There will be two tests each counting 35% towards the grade (dates to be arranged). You will be able to make up for one of these tests the week after classes end. Homework: There will be weekly problem sets making up 30% of the grade. Late homework will not be accepted. Homework
For more exercises check the course webpages from previous years (in Portuguese): TestsLinks |