E8 roots

Lie Groups and Lie Algebras — 1st Semester 2008/2009

Lecturer: José Natário
Email: jnatar@math.tecnico.ulisboa.pt
Office: Mathematics Building, 4th floor, room 4.13
Schedule: Tuesdays from 15:30 to 17:00 and Wednesdays from 12:30 to 14:00, both in room P9
Office Hours: Mondays from 14:00 to 15:00. If you need to talk to me at some other time please email me.


The final grades are available.




Lie 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).


Carter, 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 Policy

Tests: 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.


For more exercises check the course webpages from previous years (in Portuguese):



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