John Stillwell - Naive Lie Theory.pdf - Index of

5.3 The tangent space of U(n), SU(n), Sp(n) 99
5.2.2 Deduce from Exercise 5.2.1 that the tangent space of SO(3) is a real vector
space of dimension 3.
5.2.3 Check that [I,J]=K, [J,K]=I, and[K,I]=J. (This shows, among other
things, that the 3 × 3 real skew-symmetric matrices are closed under the
Lie bracket operation.)
5.2.4 Deduce from Exercises 5.2.2 and 5.2.3 that the tangent space of SO(3) under
the Lie bracket is isomorphic to R 3 under the cross product operation.
5.2.5 Prove directly that the n × n skew-symmetric matrices are closed under the
Lie bracket, using X T = −X and Y T = −Y.
The argument above shows that exponentiation sends each skew-symmetric
X to an orthogonal e X , but it is not clear that each orthogonal matrix is obtainable
in this way. Here is an argument for the case n = 3.
⎛
5.2.6 Find the exponential of the matrix B = ⎝ 0 −θ 0 ⎞
θ 0 0⎠.
0 0 0
5.2.7 Show that Ae B A T = e ABAT for any orthogonal matrix A.
5.2.8 Deduce from Exercises 5.2.6 and 5.2.7 that each matrix in SO(3) equals e X
for some skew-symmetric X.
5.3 The tangent space of U(n), SU(n), Sp(n)
We know from Sections 3.3 and 3.4 that U(n) and Sp(n), respectively, are
the groups of n×n complex and quaternion matrices A satisfying AA T = 1.
This equation enables us to find their tangent spaces by essentially the same
steps we used to find the tangent space of SO(n) in the last two sections.
The outcome is also the same, except that, instead of skew-symmetric matrices,
we get skew-Hermitian matrices. As we saw in Section 5.1, these
matrices X satisfy X + X T = 0.
Tangent space of U(n) and Sp(n). The tangent space of U(n) consists of
all the n × n complex matrices satisfying X + X T = 0. The tangent space
of Sp(n) consists of all n×n quaternion matrices X satisfying X +X T = 0,
where X denotes the quaternion conjugate of X.
Proof. From Section 5.1 we know that the tangent vectors at 1 to a space
of matrices satisfying AA T = 1 are matrices X satisfying X + X T = 0.