John Stillwell - Naive Lie Theory.pdf - Index of

3.6 Maximal tori in SO(n), U(n), SU(n), Sp(n) 63
• as a group of 2k × 2k real matrices
R θ1 ,θ 2 ,...,θ k
=
⎛
⎞
cos θ 1 −sinθ 1
sinθ 1 cosθ 1 cosθ 2 −sinθ 2
sin θ 2 cosθ 2 ,
. .. ⎜
⎟
⎝
cosθ k −sinθ k
⎠
sinθ k cos θ k
where all the blank entries are zero,
• as a group of k × k unitary matrices
⎛
Z θ1 ,θ 2 ,...,θ k
= ⎜
⎝
e iθ 1
e iθ 2
⎞
. ..
⎟
⎠ ,
e iθ k
where all the blank entries are zero and e iθ = cos θ + isinθ,
• as a group of k × k symplectic matrices
⎛
Q θ1 ,θ 2 ,...,θ k
= ⎜
⎝
e iθ 1
e iθ 2
⎞
. ..
⎟
⎠ ,
e iθ k
where all the blank entries are zero and e iθ = cos θ + isinθ. (This
generalization of the exponential function is justified in the next
chapter. In the meantime, e iθ may be taken as an abbreviation for
cosθ + isinθ.)
We can also represent T k by larger matrices obtained by “padding” the
above matrices with an extra row and column, both consisting of zeros
except for a 1 at the bottom right-hand corner (as we did to produce the
matrices R ′ θ
in SO(3) in the previous section). Using this idea, we find the
following tori in the groups SO(2m),SO(2m+1),U(n),SU(n),andSp(n).