MAT 4144/5158 – Winter 2015
Assignment 4
Professor: Alistair Savage
Due date: March 25, 2015
Assignments are due by the end of department opening hours (approximately 4pm) on the due date. They should
be placed in the box for this course found in the department lobby. No assignments will be accepted after the
department receptionist has closed the box at the end of the day. Therefore, students should plan to submit their
assignments well before the end of the day. Assignments of more than one page that are not stapled will have
20% of the total available points deducted from their grade. See
http://mysite.science.uottawa.ca/asavag2/mat4144/assignments.html
for further details.
Questions not labeled by (MAT 4144) or (MAT 5158) are for both courses. Students registered in MAT 4144
may do the MAT 5158 problems for extra credit.
1. (5 points)
(a) We saw in class that if XY = Y X, then eX+Y = eX eY = eY eX . Find an example that shows that the
converse is not true. Hint: Think of the pure imaginary quaternions and the exponential map onto SU(2).
(b) Show that if etX esY = esY etX for all t, s ∈ R, then XY = Y X.
2. (4 points) Stillwell, Exercise 5.2.8 (this will involve solving Exercises 5.2.6 and 5.2.7 – you should include
these in your solution).
3. (2 points) Stillwell 7.2.1.
4. Prove the following:
(a) (3 points) sp(n, F ) = {X ∈ M2n (F ) | JX T J = X}, where F = R or C, and J = J2n is the matrix used in
the definition of the symplectic group.
(b) (3 points) The Lie algebra of the Heisenberg group is {X ∈ M3 (R) | X is strictly upper triangular}.
5. Prove the following isomorphisms.
∼ so(n, C).
(a) (MAT 4144) (3 points) so(n)C =
∼ sp(n, C).
(b) (3 points) sp(n, R)C =
∼ sp(n, C). Here you should use the realization of sp(n) in terms of complex matrices.
(c) (3 points) sp(n)C =
That is, take sp(n) = sp(n, C) ∩ u(2n).
6. (MAT 5158) (4 points) Let V be a finite-dimensional real vector space, considered as a group under vector
addition. Prove that V is a Lie group. That is, verify directly that the axioms of an (abstract) Lie group are
satisfied. Do not use any results about matrix Lie groups.