Math 261A, Lie Groups, Fall 2018

Instructor: David Nadler

Office Hours: by appointment, 815 Evans.

Lectures: Tuesdays and Thursdays 11am-12:30pm, 72 Evans.

Course number: 32532

Prerequisites: Math 214 or equivalent familiarity with differentiable manifolds.

Text: Varadarajan, Lie Groups, Lie Algebras, and Their Representations.

Other resources

Math 261A notes.

Some prior Math 261A classes:

Syllabus: Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits.

Homeworks: Unless otherwise noted, chapters and exercises are from the course text: Varadarajan, Lie Groups, Lie Algebras, and Their Representations.

  1. Week 1 (Thursday, August 23): Review of smooth manifolds. Ch 1: #4
  2. Week 2 (Tuesday, August 28 - Thursday, August 30): Examples of Lie groups/algebras, matrix groups/algebras, formal groups/enveloping algebras. Ch 2: #1, 2, 3, 4, 10, 25
  3. Week 3 (Tuesday, September 4 - Thursday, September 6): Homomorphisms, embeddings, coverings, Lie's Theorem. Ch 2: #5, 6, 7, 8, 27, 28, 39
  4. Week 4 (Tuesday, September 11 - Thursday, September 13): Group actions, representations. Ch 2: #11, 12, 13, 14, 38, 40
  5. Week 5 (Tuesday, September 18 - Thursday, September 20): Exponential map, Baker-Campbell-Hausdorff formula. Ch 2: #15, 16, 18, 19, 20, 21, 49, 50
  6. Week 6 (Tuesday, September 25 - Thursday, September 27): Universal enveloping algebras. Ch 3: #5, 8, 11, 14, 15, 16, 19
  7. Week 7 (Tuesday, October 2 - Thursday, October 4): Nilpotent, Solvable, Semisimple. Ch 3: #2, 3, 4, 17, 20, 21, 22, 23
  8. Week 8 (Tuesday, October 9 - Thursday, October 11): Semisimple and reductive Lie algebras. Ch 3: 26, 27, 28
  9. Week 9 (Tuesday, October 16 - Thursday, October 18): Cartan subalgebras. Ch 4:
  10. Week 10 (Tuesday, October 23 - Thursday, October 25): sl(2). Ch 4:
  11. Week 11 (Tuesday, October 30 - Thursday, November 1): Root systems. Ch 4:
  12. Week 12 (Tuesday, November 6 - Thursday, November 8): Simple complex Lie algebras/compact Lie groups. Ch 4:
  13. Week 13 (Tuesday, November 13 - Thursday, November 15): Finite-dimensional/algebraic representations. Ch 4:
  14. Week 14 (Tuesday, November 20): Character formulas. Ch 4:
  15. Week 15 (Tuesday, November 27 - Thursday, November 29): Chevalley Restriction Theorem, Harish Chandra center. Ch 4:

Evaluation: There will be weekly homework, a midterm and a final exam.

Midterm 1.

Midterm 1 with solutions.

Midterm 2.

Midterm 2 with solutions.

Final.