As usual you came through in spades, points 1-3 are proven.
Also you indicate here what is quite true, that we have been chewing over the same material----the exponential map, the logarithm of a matrix (which you defined earlier by a limit as I recall), the one parameter subgroup which is, by golly, a curve, and its derivative or tangent vector at the identity----in various different forms. At least I think we have been doing essentally that for a while. Maybe this theorem 3.18 of Hall can give us a place from which to move onwards.
Originally posted by Hurkyl
...
In the first identity in problem (3), notice that exp(tX) is a curve with tangent vector X @ t = 0, and φ*(X) is defined to be the tangent vector @ t = 0 to the image of exp(tX) under φ... that's precisely how we defined (*φ) in the geometric context!
There are still two parts to theorem 3.18 which I did not ask anyone to prove and I am going to nonchallantly leave them without proof. Anyone who wants can look it up in Hall.
The unproven parts are:
φ* exists and is a unique real linear map:
g -->
h,
and also that (φ o ψ)* = φ* o ψ*
The proof involves stuff we have already been doing lots of, you define phi-star in a by-now-very-familiar way by saying: "take X in
g that we want to define phi-star of, and make a one parameter subgroup exp(tX) which you can think of as a curve of matrices in G passing thru the identity matrix
and use phi to MAP THIS WHOLE CURVE into the matrix group H.
and since phi is a smooth group homomorphism the image is a nice smooth curve passing thru the identity in the matrix group H.
And then as destiny decrees you just look at the tangent vector of that curve up in the tangent space of matrices
h, and that is some matrix and you call THAT matrix = φ*(X)
then you have to check that this map is linear between the two vectorspaces (of matrices)
g -->
h, which just means trying it out with a scalar multiple rX and with a sum X+Y, and you have to check that it is the unique linear map that commutes with exponentiation namely
φ(exp(X)) = exp (φ*(X))
each of which little facts Brian Hall proves in one line on page 42 or 43 in case anyone wants to check up.
Now I think we can move on and see where this theorem and the discussion surrounding it have gotten us. In a way all the theorem does is work matrix multiplication into a familiar geometry picture
the geometry picture is two manifolds and a smooth map phi: M--->N that takes point x --->y
and we add just one thing to the picture namely that M and N are now matrix groups and x and y are the group identities (that is identity matrices) and phi is now a homomorphism----it preserves matrix multiplication.
this is just a tiny embellishment of the basic geometry picture and we want to know what happens with the lifted map of the Tangent spaces T
x ---> T
y
It is only natural to ask what happens when the smooth group homomorphism is lifted to the tangentspace level and the answer is this theorem which says that all is as well behaved as one could wish
not only is the thing linear and uniquely defined and consistent with the exponential map and one parameter subgroups (which are curves thru the identity) but we even get a bonus that the
map commutes with a certain "multiplication-like" operation upstairs called the [X,Y].
phi-star doesn't commute with ordinary matrix multiplication, it commutes with bracket. This is how god and nature tell us that we must endow the tangent space at the group identity with an algebraic structure involving the bracket.
We are predestined to do this because IT, the bracket, is what the lift of a group homomorphism preserves and it does not preserve anything else resembling multiplication.
And it is a linear map on tangent spaces so it preserves addition, so it is telling us what a Lie algebra is, namely vectorspace ops plus bracket----and whatever identities the bracket customarily obeys.
well that's one way to look at it. sorry if I have been long-winded.
now we can try a long jump to theorem 5.4 on page 68, which talks about Lie algebra representations, or else in a more relaxed frame of mind we can scope out some of the followup stuff that comes right after this theorem 3.18
oh, theorem 3.34 about the "complexification" of a real Lie algebra seems like a good thing to mention. sometimes we might need to drag in complex numbers to get some matrix diagonalized or solve some polynomial or for whatever reason and there is a regular proceedure for complexifying things when and if that is needed
well that is certainly enough said about theorem 3.18
(Hall's Theorem 3.18, restated with some more detail)
Let G and H be matrix Lie groups, with Lie algebras g and h. Let φ :G --> H be a Lie group homomorphism. Then there exists a unique real linear map φ*: g --> h,
such that for all X in g we have
φ(exp(X)) = exp (φ*(X)).
Moreover this unique real linear map φ* has certain properties:
1. For all X in g and all A in G,
φ*(AXA-1) = φ(A) φ*(X) φ(A-1)
2. For all X, Y in g,
φ*(XY - YX) = φ*(X) φ*(Y) - φ*(Y)φ*(X)
this is to say that φ* commutes with taking brackets or the "adjoint" map, whatever, namely
φ*([X,Y]) = [φ*(X), φ*(Y)]
3. The lifted map takes infinitesimal moves into the corresponding infinitesimals, namely,
φ*(X) = d/dt|t = 0 φ(exp(tX))
and FINALLY that the star operation is compatible with the composition of mappings
(φ o ψ)* = φ* o ψ*
)
