John Stillwell - Naive Lie Theory.pdf - Index of

98 5 The tangent space
as in ordinary calculus. (This can be checked by differentiating the series
for e tX .) It follows that A(t) has the tangent vector A ′ (0) =X at 1, and
therefore each X such that X +X T = 0 occurs as a tangent vector to SO(n)
at 1, as required.
□
As mentioned in the previous section, a matrix X such that X +X T = 0
is called skew-symmetric. Important examples are the 3 × 3 skewsymmetric
matrices, which have the form
⎛
0 −x
⎞
−y
X = ⎝x 0 −z⎠.
y z 0
Notice that sums and scalar multiples of these skew-symmetric matrices
are again skew-symmetric, so the 3 × 3 skew-symmetric matrices form a
vector space. This space has dimension 3, as we would expect, since it is
the tangent space to the 3-dimensional space SO(3). Less obviously, the
skew-symmetric matrices are closed under the Lie bracket operation
[X 1 ,X 2 ]=X 1 X 2 − X 2 X 1 .
Later we will see that the tangent space of any Lie group G is a vector space
closed under the Lie bracket, and that the Lie bracket reflects the conjugate
g 1 g 2 g −1
1
of g 2 by g −1
1
∈ G. This is why the tangent space is so important
in the investigation of Lie groups: it “linearizes” them without obliterating
much of their structure.
Exercises
According to the theorem above, the tangent space of SO(3) consists of 3 × 3 real
matrices X such that X = −X T . The following exercises study this space and the
Lie bracket operation on it.
5.2.1 Explain why each element of the tangent space of SO(3) has the form
⎛ ⎞
0 −x −y
X = ⎝x 0 −z⎠ = xI + yJ + zK,
y z 0
where
⎛
I = ⎝ 0 −1 0 ⎞ ⎛
1 0 0⎠, J = ⎝ 0 0 −1 ⎞ ⎛
0 0 0 ⎠, K = ⎝ 0 0 0 ⎞
0 0 −1⎠.
0 0 0 1 0 0
0 1 0