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 09/16:
  • isomorphism between so_3(C) and sl_2(C);
  • group isomorphism between SU_2 and S^3 with the quaternion structure (bijection: 10 points, group homomorphism: 10 points; total: 20 points; see page 6/Example 2.5 (5));
  • optional: page 22/2.8, 2.9, 2.10.
• Due on 09/23:
  • Show that SU_2 is contained in Sp_2(C), so it is isomorphic to Sp(2).
  • The 5 properties of characters.
• Due on 09/30:
  • 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).
• Due on 10/7 [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 10/14 page 107/5.7: the statement needs a correction, namely A is upper triangular with identical entries on the diagonal; part 2 of Theorem 5.40.
• Due on 10/21 page 107/5.1 (5 points for (1) and 10 for (2)), 5.2.
• Due on 10/28
  • 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 11/7 p.130/6.5 (6 pts Cartan subalgebra, 9 pts the 4 root spaces).
• Due on 11/18
  • 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. Is the reciprocal true?
  • page 160/7.3.
• Due on 11/28 page 160/7.2 (10+10 points), 7.5 (for type D, just determine the fundamental weights, and give an efficient description of the lattices P and Q); page 161-162/7.11(2), 7.14, 7.17 (15 pts).
• Due on 12/12
  • decompose the tensor product of V_{L_1-2L_3} and V_{L_1} for sl_3 (10 points); you need to show that dimension of the weight space L_1-L_3 in V_{2(L_1-L_3)} is 2 (only exhibit 2 linearly independent vectors, as in class: 10 points), and then you can use the fact that the 0-weight space has dimension 3.
  • page 162/7.15 (hint: consider a subrepresentation and act on a basis by s_i, conclude that the root system is not irreducible).
  • page 194/8.6 (15 pts, hint: use the adjoint representation and its root space decomposition).