is NOT a space of realnbynmatrices. It is the space of skew-Hermitian
matrices, which in general are complex. While the matrices are complex, only
real linear combinations of skew-Hermitian matrices are skew-Hermitian (recall
that multiplication byichanges a skew-Hermitian matrix into a Hermitian
matrix). Within this space of skew-Hermitian complex matrices, if one looks at
the subspace of real matrices one gets the sub-Lie algebraso(n) of antisymmetric
matrices (the Lie algebra ofSO(n)⊂U(n)).
Any complex matrixZ∈M(n,C) can be written as a sum of
Z=
1
2
(Z+Z†) +
1
2
(Z−Z†)
where the first term is self-adjoint, the second skew-Hermitian. This second
term can also be written asitimes a self-adjoint matrix
1
2(Z−Z†) =i(
1
2 i(Z−Z†)
)
so we see that we can get all ofM(n,C) by taking all complex linear combina-
tions of self-adjoint matrices.
There is an identity relating the determinant and the trace of a matrix
det(eX) =etrace(X)which can be proved by conjugating the matrix to upper-triangular form and
using the fact that the trace and the determinant of a matrix are conjugation
invariant. Since the determinant of anSU(n) matrix is 1, this shows that the
Lie algebrasu(n) ofSU(n) will consist of matrices that are not only skew-
Hermitian, but also of trace zero. So in this casesu(n) is again a real vector
space, with the trace zero condition a single linear condition giving a vector
space of real dimensionn^2 −1.
One can show thatU(n) andu(n) matrices can be diagonalized by conjuga-
tion by a unitary matrix and thus show that anyU(n) matrix can be written as
an exponential of something in the Lie algebra. The corresponding theorem is
also true forSO(n) but requires looking at diagonalization into 2 by 2 blocks. It
is not true forO(n) (you can’t reach the disconnected component of the identity
by exponentiation). It also turns out to not be true for the groupsSL(n,R)
andSL(n,C) forn≥2 (while the groups are connected, they have elements
that are not exponentials of any matrix insl(n,R) orsl(2,C) respectively).
5.3 A summary
Before turning to Lie algebra representations, we’ll summarize here the classes of
Lie groups and Lie algebras that we have discussed and that we will be studying
specific examples of in later chapters:
- The general linear groupsGL(n,R) andGL(n,C) are the groups of all
invertible matrices, with real or complex entries respectively. Their Lie