Math 621 - 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:
  • isomorphisms between so_3(C), (su_2)_C (i.e., the complexification of su_2), 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 10/3:
  • Show that SU_2 is contained in Sp_2(C), so it is isomorphic to Sp(2).
  • p.81/4.4 all parts (10 points for each question; total: 30 points)
• Due on 10/10: decompose the tensor product of Sym^a(V) and Sym^b(V) into irreducibles, for sl_2=sl_2(V).
• Due on 10/22:
  • [I,J] is an ideal if I,J are ideals.
  • Optional: f^{-1}(I) is an ideal if f is a map of Lie algebras and I is an ideal.
  • page 107/5.7: the statement needs a correction, namely A is upper triangular with identical entries on the diagonal.
• Due on 10/31:
  • part 2 of Theorem 5.40;
  • page 107/5.1 (1);
  • page 107/5.2 -- enough if you compare (E_ij, E_ji) using the two bilinear forms.
• Due on 11/7
  • Rederive the Casimir element of sl_2(C) starting from definition;
  • page 130/6.1.
• Due on 11/26:
  • page 130-131/6.5 (1),(2),(3),(4) (10+2+10+10 pts);
  • page 130/6.4 (10 pts Cartan subalgebra, 10 pts the 4 root spaces);
  • page 160/7.2 (10+10 points).
• Due on 12/17 at noon in my mailbox, based on the example shown in class:
  • show the weight diagram of the irreducible representation of sl_3 with highest weight L_1-2L_3;
  • use an explicit calculation to show that the dimension of a particular weight space is 2, rather than 3 (see also Fulton-Harris, bottom of page 181);
  • exhibit the semistandard Young tableaux corresponding to all the weight spaces.