18.755 tenth and last problems, due Monday, December 7, 2015
I reserve the right to add problems to this list until Wednesday, December 2. If I do that,
I will tell you in class December 2 and 4.
0 1
, and Σr to be the 2r × 2r matrix with r diagonal blocks σ and
1. Define σ =
1 0
all other entries zero. Finally define

Ip
J(p, q, r) =  0
0
0
−Iq
0

0
0 ,
Σr
an integer matrix of size n = p + q + 2r with square equal to the identity. Prove that if there
is an integer matrix g of determinant ±1 such that
gJ(p, q, r)g−1 = J(p′ , q ′ , r ′ ),
then p = p′ , q = q ′ , and r = r ′ . (Hint: in the correspondence between tori and lattices
discussed in class, the matrix J(p, q, r) defines an automorphism j(p, q, r) of U (1)n .)
2. How many different compact connected Lie groups have Lie algebra su(n) ⊕ R?
3. Let X ∗ be the lattice Z4 , with dual lattice X∗ = Z4 as usual. The root datum for
(U (4), U (1)4 ) is
R0 = {ei − ej |1 ≤ i 6= j ≤ 4},
R0∨ = {ei − ej |1 ≤ i 6= j ≤ 4}.
Find all root data (X ∗ , R, X∗ , R∨ ) so that R ⊃ R0 and R∨ ⊃ R0∨ .