Math 629: Lie Groups
Problem Set #6 (and information about the exam on Monday)
0. Be sure you know the basic definitions (Lie group, Lie algebra, representation, etc), and basic theorems.
1. Describe why the Campbell-Baker-Hausdorff Theorem is useful or important. Be very specific that you are using the theorem in whatever form you prefer.
2. Find the dimension of U(n). Hint: you may use its Lie algebra. Make sure to justify why this hint is helpful if you use it.
3. Prove that Gl(n,C) is connected any way you know how.
4. Let Sl(n,C) act on C^n by matrix multiplication. Is this
representation faithful? Why or why not? Is it irreducible? Why or why
not?
5. Let SU(3) act on C^3 by matrix multiplication. Consider the group T
given by the set of all diagonal matrices in SU(3). Restrict the
representation of SU(3) to a representation of T. Is this
representation irreducible? Why or why not? If not, show a
decomposition into irreducible components.
6. #3 p. 121
7. In class we proved that if Pi is a representation of a matrix Lie
group G, there is a unique induced representation pi of Lie(G).
We also proved that, if Pi is irreducible, then pi is irreducible. Show
that the converse is also true: if pi is irreducible, then Pi is. This
is in the text if you need help with it.