3Lie algebras
III Symmetries, Fields and Particles
3.2 Differentiation
We are eventually going to get a Lie algebra from a Lie group. This is obtained
by looking at the tangent vectors at the identity. When we have homomorphisms
f
:
G → H
of Lie groups, they are in particular smooth, and taking the derivative
will give us a map from tangent vectors in
G
to tangent vectors in
H
, which
in turn restricts to a map of their Lie algebras. So we need to understand how
differentiation works.
Before that, we need to understand how tangent vectors work. This is
completely general and can be done for manifolds which are not necessarily Lie
groups. Let
M
be a smooth manifold of dimension
D
and
p ∈ M
a point. We
want to formulate a notion of a “tangent vector” at the point
p
. We know how
we can do this if the space is
R
n
— a tangent vector is just any vector in
R
n
.
By definition of a manifold, near a point p, the manifold looks just like R
n
. So
we can just pretend it is R
n
, and use tangent vectors in R
n
.
However, this definition of a tangent vector requires us to pick a particular
coordinate chart. It would be nice to have a more “intrinsic” notion of vectors.
Recall that in
R
n
, if we have a function
f
:
R
n
→ R
and a tangent vector
v
at
p
, then we can ask for the directional derivative of
f
along
v
. We have a
correspondence
v ←→
∂
∂v
.
This directional derivative takes in a function and returns its derivative at a
point, and is sort-of an “intrinsic” notion. Thus, instead of talking about
v
, we
will talk about the associated directional derivative
∂
∂v
.
It turns out the characterizing property of this directional derivative is the
product rule:
∂
∂v
(fg) = f(p)
∂
∂v
g + g(p)
∂
∂v
f.
So a “directional derivative” is a linear map from the space of smooth functions
M → R to R that satisfies the Leibnitz rule.
Definition
(Tangent vector)
.
Let
M
be a manifold and write
C
∞
(
M
) for the
vector space of smooth functions on
M
. For
p ∈ M
, a tangent vector is a linear
map v : C
∞
(M) → R such that for any f, g ∈ C
∞
(M), we have
v(fg) = f(p)v(g) + v(f)g(p).
It is clear that this forms a vector space, and we write
T
p
M
for the vector space
of tangent vectors at p.
Now of course one would be worried that this definition is too inclusive, in
that we might have included things that are not genuinely directional derivatives.
Fortunately, this is not the case, as the following proposition tells us.
In the case where
M
is a submanifold of
R
n
, we can identify the tangent
space with an actual linear subspace of
R
n
. This is easily visualized when
M
is a
surface in
R
3
, where the tangent vectors consists of the vectors in
R
3
“parallel to”
the surface at the point, and in general, a “direction” in
M
is also a “direction”
in
R
n
, and tangent vectors of
R
n
can be easily identified with elements of
R
n
in
the usual way.
This will be useful when we study matrix Lie groups, because this means the
tangent space will consist of matrices again.
Proposition.
Let
M
be a manifold with local coordinates
{x
i
}
i=1,··· ,D
for some
region U ⊆ M containing p. Then T
p
M has basis
∂
∂x
j
j=1,··· ,D
.
In particular, dim T
p
M = dim M.
This result on the dimension is extremely useful. Usually, we can manage to
find a bunch of things that we know lie in the tangent space, and to show that
we have found all of them, we simply count the dimensions.
One way we can obtain tangent vectors is by differentiating a curve.
Definition
(Smooth curve)
.
A smooth curve is a smooth map
γ
:
R → M
.
More generally, a curve is a C
1
function R → M.
Since we only want the first derivative, being C
1
is good enough.
There are two ways we can try to define the derivative of the curve at time
t
= 0
∈ R
. Using the definition of a tangent vector, to specify
˙γ
(0) is to tell how
we can differentiate a function
f
:
M → R
at
p
=
γ
(0) in the direction of
˙γ
(0).
This is easy. We define
˙γ(0)(f) =
d
dt
f(γ(t)) ∈ R.
If this seems too abstract, we can also do it in local coordinates.
We introduce some coordinates
{x
i
}
near
p ∈ M
. We then refer to
γ
by
coordinates (at least near p), by
γ : t ∈ R 7→ {x
i
(t) ∈ R : i = 1, ··· , D}.
By the smoothness condition, we know
x
i
(
t
) is differentiable, with
x
i
(0) = 0.
Then the tangent vector of the curve γ at p is
v
γ
= ˙x
i
(0)
∂
∂x
i
∈ T
p
(M), ˙x
i
(t) =
dx
i
dt
.
It follows from the chain rule that this exactly the same thing as what we
described before.
More generally, we can define the derivative of a map between manifolds.
Definition
(Derivative)
.
Let
f
:
M → N
be a map between manifolds. The
derivative of f at p ∈ M is the linear map
Df
p
: T
p
M → T
f(p)
N
given by the formula
(Df
p
)(v)(g) = v(g ◦ f)
for v ∈ T
p
M and g ∈ C
∞
(N).
This will be useful later when we want to get a map of Lie algebras from a
map of Lie groups.