MAT 4144/5158 – Winter 2015
Assignment 2
Professor: Alistair Savage
Due date: February 11, 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. (MAT 4144) (3 points) Show that Sp(1, R) = SL(2, R), Sp(1, C) = SL(2, C), and Sp(1) = SU(2).
2. (3 points) Stillwell, Exercise 3.4.3. You may assume the results of Exercises 3.4.1 and 3.4.2 (you should
work these out as an exercise, but do not need to include them in your assignment). This question amounts to
proving the statement Sp(n) = Sp(n, C) ∩ U(2n) made in class.
3. This question shows that the Euclidean group is isomorphic to a matrix Lie group.
(a) (3 points) Show that for x1 , x2 ∈ Rn and R1 , R2 ∈ O(n), we have
(Tx1 R1 )(Tx2 R2 ) = Tx1 +R1 x2 (R1 R2 )
n
and that for x ∈ R and R ∈ O(n), we have
(Tx R)−1 = T−R−1 x, R−1 .
This shows that E(n) is a semidirect product of the group of translations and the group O(n). More
precisely, E(n) is isomorphic to the group consisting of pairs (x, R) for x ∈ Rn and R ∈ O(n) with
multiplication
(x1 , R1 )(x2 , R2 ) = (x1 + R1 x2 , R1 R2 ).
(b) (3 points) Show that the map E(n) → GL(n + 1, R) given by


x1

.. 

R
. 
Tx R 7→ 


xn 
0 ··· 0 1
is an injective group homomorphism and conclude that E(n) is a linear Lie group.
4. (3 points) Prove that if G is a matrix Lie group, then the component of G containing the identity is a
subgroup of G (see Exercises 3.2.1–3.2.3 of Stillwell).
5. (MAT 4144) (2 points) Prove that SL(n, C) is path-connected. Hint: Modify one of the proofs given in
class that GL(n, C) is path-connected.
2
6. (MAT 5158)
(a) (3 points) Using the polar decomposition of SL(n, R) given in class and the path-connectedness of SO(n),
show that SL(n, R) is path-connected. Hint: Recall that if P is a real, symmetric matrix, then there exists a
real, orthogonal matrix R1 such that P = R1 DR1−1 , where D is diagonal.
(b) (2 points) Let GL(n, R)+ be the subgroup of GL(n, R) consisting of those matrices with positive determinant. Prove that GL(n, R)+ is path-connected. Hint: Use part (a).
Bonus Question. Background: Suppose E is a complex vector space. A sesquilinear form on E is a
bi-additive function ϕ : E × E → C such that
ϕ(αx, βy) = ᾱβϕ(x, y)
for all x, y ∈ E, α, β ∈ C.
A sesquilinear form ϕ is Hermitian if
ϕ(x, y) = ϕ(y, x)
for all x, y ∈ E.
By picking a suitable basis, a non-degenerate Hermitian form can be written in the form
(1)
ϕ(x, y) = ±x̄1 y1 ± x̄2 y2 ± · · · ± x̄n yn ,
where (x1 , x2 , . . . , xn ) and (y1 , y2 , . . . , yn ) are the coordinates of x and y in this basis. If p is the number of +
signs in (1) and q is the number of − signs, then we say the signature of ϕ is (p, q). This signature is independent
of the basis.
Let ϕ be a non-degenerate Hermitian form on Cn with signature (p, q). Then we define
U(p, q) = {A ∈ GL(n, C) | ϕ(Ax, Ay) = ϕ(x, y) ∀ x, y ∈ Cn },
SU(p, q) = {A ∈ U(p, q) | det A = 1}.
(Our notation is justified by the fact that, up to isomorphism, these matrix Lie groups depend only on the
signature of ϕ.)
Problem: Show that Sp(n, R) ∼
= Sp(n, C) ∩ SU(n, n).