Problem 3:
By using the fact that any unitary matrix can be diagonalized by conjugation
by a unitary matrix, show that all unitary matrices can be written aseX, for
Xa skew-adjoint matrix inu(n).
By contrast, show that
A=
(
− 1 1
0 − 1
)
is in the groupSL(2,C), but is not of the formeXfor anyX∈sl(2,C) (this
Lie algebra is all 2 by 2 matrices with trace zero).
Hint: For 2 by 2 matricesX, one can show (this is the Cayley-Hamilton
theorem: matricesXsatisfy their own characteristic equation det(λ 1 −X) = 0,
and for 2 by 2 matrices, this equation isλ^2 −tr(X)λ+ det(X) = 0)
X^2 −tr(X)X+ det(X) 1 = 0
ForX∈sl(2,C),tr(X) = 0, so hereX^2 =−det(X) 1. Use this to show that
eX= cos(
√
det(X)) 1 +
sin(
√
det(X))
√
det(X)
X
Try to use this foreX=Aand derive a contradiction (taking the trace of the
equation, what is cos(
√
det(X))?)
Problem 4:
- Show thatMis an orthogonal matrix iff its rows are orthonormal vectors
for the standard inner product (this is also true for the columns). - Show thatMis a unitary matrix iff its columns are orthonormal vectors
for the standard Hermitian inner product (this is also true for the rows).
B.3 Chapters 5 to
Problem 1:
On the Lie algebrasg=su(2) andg=so(3) one can define the Killing form
K(·,·) by
(X,Y)∈g×g→K(X,Y) =tr(XY)
- For both Lie algebras, show that this gives a bilinear, symmetric form,
negative definite, with the basis vectorsXjin one case andljin the other
providing an orthogonal basis if one usesK(·,·) as an inner product. - Another possible way to define the Killing form is as
K′(X,Y) =tr(ad(X)◦ad(Y))
Here the Lie algebra adjoint representation (ad,g) gives for eachX∈ga
linear map
ad(X) :R^3 →R^3