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

Annoucements

The final grades are available.

Summaries

Grades

Syllabus

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).

Bibliography

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.

Homework

• Week 1 (due on September 30)
• Week 2 (due on October 7)
• Week 3 (due on October 14)
• Week 4 (due on October 21)
• Week 5 (due on October 29)
• Week 6 (due on November 5)
• Week 7 (due on November 19)
• Week 8 (due on November 26)
• Week 9 (due on December 3)
• Week 10 (due on December 10)

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

• 2007/2008
• 2005/2006
• 2001/2002

Tests

• Test 1
• Test 2
• Make Up Test

Links

• Mathematicians Biographies
• Eric Weinstein's World of Mathematics
• Planetmath