Math 820 - Lie Algebras and Their Representations: Syllabus and Homework

Syllabus

Textbook for assignments: A. Kirillov Jr, An Introduction to Lie Groups and Lie Algebras.

All homework problems are worth 10 points unless otherwise specified.

• Due on 09/19: isomorphism between so(3,C) and sl(2,C).
• Due on 09/29:
  • group isomorphism between SU(2) and S^3 with the quaternion structure (bijection: 10 points, group homomorphism: 10 points; total: 20 points)
  • p.81/4.4 parts 2 and 3 (10 points for each question; total: 20 points)
  • decompose the tensor product of Sym^a(V) and Sym^b(V) into irreducibles, for sl_2=sl_2(V)
  • optional: p.22/2.7
• Due on 10/03 [I,J] is an ideal if I,J are. Optional: f^{-1}(I) is an ideal if f is a map of Lie algebras and I is an ideal.
• Due on 10/10 p.106-107/5.1 (5+10 pts), 5.2, 5.7. Optional: the orthogonal complement of an ideal (with respect to a symmetric and invariant bilinear form) is an ideal.
• Due on 10/17
  • Rederive the Casimir element of sl_2(C) starting from definition.
  • p.130/6.1
  • p.131/6.6 (1),(2),(3) (10+2+10 pts).
• Due on 10/24 p.130/6.5 (6 pts Cartan subalgebra, 9 pts the 4 root spaces)
• Due on 10/31
  • Calculate explicitly the root strings for sl_n, and the integers h_alpha(beta).
  • If the inner product of two roots is strictly positive, show that their difference is a root. You are required to use a different proof than the one by reduction to rank 2 root systems, which you can find in textbooks. Is the reciprocal true?
• Due on 11/7 page 160/7.2 (10+10 points), 7.3, 7.5 (for type D, just determine the fundamental weights, and give an efficient description of the lattices P and Q).
• Due on 11/14 page 161-162/7.11(2), 7.14, 7.17 (15 pts).
• Due on 11/21
  • decompose the tensor product of V_{L_1-2L_3} and V_{L_1} for sl_3
  • finish the calculation of the weight space of weight L_1 for V_{2L_1-L_3} for sl_3, by addressing the other two ways to reach this weight space from the highest weight
  • problems 162/7.15, 194/8.6 (15 pts).
• Due on 12/1 Prove that "rho=1/2 sum_{alpha in R_+} alpha" is equal to "sum_i omega_i" (omega_i are the fundamental weights, dual to alpha_j^v). Hint: prove first that s_i permutes the positive roots with the exception of alpha_i, which is (clearly) sent to -alpha_i.
• Due on 12/8 Derive the rule for multiplying an arbitrary Schur polynomial by a Schur polynomial indexed by a row partition from the Littlewood-Richardson rule stated in class.