Course material from previous semesters

Course material for current semester (Spring, 2008)

Announcements

Office Hours:
New Monday office hour: 2:00-3:00 (to avoid conflict with the Faculty Research Perspecitves talks)
Other office hours: MW 4:00-5:00 (and usually Th 2:00-3:00)

Reference books:
The books by Rossmann and Goodman-Wallach have been put on reserve in the math library.

Homework Assignment #1:

Changes and Hints for Exercises 1.1.5

• Exercise #4:
  In part (a), also prove that B is nondegenerate
  In part (b), show the signature is (1,2)
  In part (c), show that the map is a homomorphism into SO(V,B) with kernel +/- I. (You don't have to prove surjectivity; for F = C this will be proved in the lectures later, and for F = R it is not true since SO(1,2) has two connected components.)
• Exercise #7:
  Here \wedge^2 g is the linear transformation on \bigwedge F^4 that sends u \wedge v to gu \wedge gv. Show that the map e_i \wedge e_j \to e_{ij} - e_{ji} gives a linear isomorphism between \bigwedge F^4 and the space of 4 \times 4 skew-symmetric matrices, and that \wedge^2 g becomes the linear transformation A \to gAg^t for A a skew-symmetric matrix.
• Exercise #8:
  The symplectic subgroup of SL(4, F) should be labeled Sp(2, F), in the notation of the book. Use the isomorphism from Exercise #7 to prove the first part of (a).
• Exercise #9:
  Use the map e_i \otimes e_j \to e_{ij} to give an isomorphism between F^2 \otimes F^2 and M_2(F). Then the action of (a,b) \in G on M_2(F) is x \to axb^t and the bilinear form is the same as in Exercise #4.
• For all these exercises, use triangular form (Schur or Jordan) to show that linear transformations have determinant 1 and to show that the kernel of a homomorphism is +/- I.

Homework Assignment #2:

• There is a typo in the definition of the differential of a Lie group homomorphism given in Proposition 1.3.10. Change d\pi to d\phi.
• Exercise 1.3.7 #2 refers to GL(n, R), not SL(n, C).
• Here is the new Exercises 1.3.7, #7.

Homework Assignment #3:

• Exercises 1.5.4 #1 should read: Let (\pi, V) be a rational representation.

Homework Assignment #4: (due March 26)

• The list of graded homework problems has been changed (March 4). See the revised homework exercises.
• In Exercises 1.6.4, #2, the hint should say: Take H = g diag[H_1, ..., H_k] g^{-1}.
• Here is the new Exercises 1.7.3, #5.

Homework Assignments #5, #6, and #7:

• Here are the homework exercises for the rest of the term (posted March 20).

Homework Assignment #5: (due April 9)

• In Exercises 2.1.3 #4, the hint should say: Show the map
  X ---> A + \sqrt{-1} Bs_l
  gives a Lie algebra isomorphism from \frak k to gl(l, C).

Text: Selected chapters from the 2nd edition of Representations and Invariants of the Classical Groups by Roe Goodman and Nolan R. Wallach (to appear). These chapters will be available for downloading from this page during the first and second week of the term. The third printing (2003) of the revised first edition of this book is available from Cambridge University Press (ISBN 0-521-66348-2) and other booksellers.

Description and Prerequisites: This course will be an introduction to Lie groups and algebraic groups. The prerequisites are real analysis, linear algebra, and elementary topology. Students who took Prof. Buch's course 640:556 Representation Theory and are interested in learning more Lie theory will find this course provides a natural continuation and alternate points of view. However, no prior knowledge of Lie algebras, Lie groups, or representation theory will be assumed.

Course Outline:

1. The classical linear groups (real and complex forms)
2. Closed subgroups of GL(n) as Lie groups
3. Linear algebraic groups and rational representations
4. Structure of complex classical groups: maximal torus, roots, adjoint representation, Weyl group
5. Highest weight theory for representations of semisimple Lie algebras
6. Complete reducibility of representations of semisimple Lie algebras and classical groups

Grading: There are graded   homework exercises that are due every two weeks during the term.
(revised March 20)

Lectures: Here is a detailed syllabus
(revised April 17)

Course Notes:

• Chapter 1: Lie Groups and Algebraic Groups
    (pdf format)   (revised 3/24/08)
• Chapter 2: Structure of Classical Groups
    (pdf format)   (revised 4/10/08)
• Chapter 3: Highest Weight Theory
    (pdf format)   (revised 5/06/08)
• Appendix A: Algebraic Geometry
    (pdf format)   (revised 1/10/08)
• Appendix B: Linear and Multilinear Algebra
    (pdf format)   (revised 1/10/08)
• Appendix C: Associative Algebras and Lie Algebras
    (pdf format)   (revised 1/10/08)
• Appendix D: Manifolds and Lie Groups
    (pdf format)   (revised 1/10/08)