6Representation of Lie algebras

III Symmetries, Fields and Particles

6.1 Weights
Let
ρ
be a representation of
g
on
V
. Then it is completely determined by the
images
H
i
7→ ρ(H
i
) gl(V )
E
α
7→ ρ(E
α
) gl(V ).
Again, we know that
[ρ(H
i
), ρ(H
j
)] = ρ([H
i
, H
j
]) = 0.
So all
ρ
(
H
i
) commute, and by linear algebra, we know they have a common
eigenvector v
λ
. Again, for each H h, we know
ρ(H)v
λ
= λ(H)v
λ
for some λ(H), and λ lives in h
.
Definition
(Weight of representation)
.
Let
ρ
:
g gl
(
V
) be a representation
of
g
. Then if
v
λ
V
is an eigenvector of
ρ
(
H
) for all
H h
, we say
λ h
is a
weight of ρ, where
ρ(H)v
λ
= λ(H)v
λ
for all H h.
The weight set S
ρ
of ρ is the set of all weights.
For a weight
λ
, we write
V
λ
for the subspace that consists of vectors
v
such
that
ρ(H)v = λ(H)v.
Note that the weights can have multiplicity, i.e.
V
λ
need not be 1-dimensional.
We write
m
λ
= dim V
λ
1
for the multiplicity of the weight.
Example. By definition, we have
[H
i
, E
α
] = α
i
E
α
So we have
ad
H
i
E
α
= α
i
E
α
.
In terms of the adjoint representation, this says
ρ
adj
(H
i
)E
α
= α
i
E
α
.
So the roots α are the weights of the adjoint representation.
Recall that for
su
C
(2), we know that the weights are always integers. This
will be the case in general as well.
Given any representation
ρ
, we know it has at least one weight
λ
with an
eigenvector
v V
λ
. We’ll see what happens when we apply the step operators
ρ(E
α
) to it, for α Φ. We have
ρ(H
i
)ρ(E
α
)v = ρ(E
α
)ρ(H
i
)v + [ρ(H
i
), ρ(E
α
)]v.
We know that
[ρ(H
i
), ρ(E
α
)] = ρ([H
i
, E
α
]) = α
i
ρ(E
α
).
So we know that if v V
λ
, then
ρ(H
i
)ρ(E
α
)v = (λ
i
+ α
i
)ρ(E
α
)v.
So the weight of the vector has been shifted by
α
. Thus, for all vectors
v V
λ
,
we find that
ρ(E
α
)v V
λ+α
.
However, we do not know a priori if
V
λ+α
is a thing at all. If
V
λ+α
=
{
0
}
, i.e.
λ + α is not a weight, then we know that ρ(E
α
)v = 0.
So the Cartan elements
H
i
preserve the weights, and the step operators
E
α
increment the weights by α.
Consider the action of our favorite
su
(2)
α
subalgebra on the representation
space. In other words, we consider the action of
{ρ
(
h
α
)
, ρ
(
e
α
)
, ρ
(
e
α
)
}
on
V
.
Then
V
becomes the representation space for some representation
ρ
α
for
su
(2)
α
.
Now we can use what we know about the representations of
su
(2) to get
something interesting about V . Recall that we defined
h
α
=
2
(α, α)
H
α
=
2
(α, α)
(κ
1
)
ij
α
j
H
i
.
So for any v V
λ
, after some lines of algebra, we find that
ρ(h
α
)(v) =
2(α, λ)
(α, α)
v.
So we know that we must have
2(α, λ)
(α, α)
Z
for all
λ S
ρ
and
α
Φ. So in particular, we know that (
α
(i)
, λ
)
R
for all
simple roots
α
(i)
. In particular, it follows that
λ h
R
(if we write
λ
as a sum
of the
α
(i)
, then the coefficients are the unique solution to a real system of
equations, hence are real).
Again, from these calculations, we see that
L
λS
ρ
V
λ
is a subrepresentation
of
V
, so by irreducibility, everything is a linear combination of these simultaneous
eigenvectors, i.e. we must have
V =
M
λS
ρ
V
λ
.
In particular, this means all ρ(H) are diagonalizable for H h.