Recall that in the representation theory of , one considered an element
and its action on a representation
. We looked for its largest eigenvalue and the corresponding highest weight vector.
There is something along the same lines to be done here for arbitrary semisimple Lie algebras, though it is much more complicated (and interesting). I’m only going to scratch the surface today.
Let be a semisimple Lie algebra and
a Cartan subalgebra. Then
is to play the role of
in
; the
matrices in
are now replaced by the root space decomposition
We know that acts on a representation
of
by commuting semisimple transformations, so we can write
where . These are called the weight spaces, and the
are called weights.
Now
by an analog of the “fundamental calculation,” proved as follows. Let . Then
So .
To talk about “highest weights,” we need an ordering on the roots . We can do this by choosing a suitable real functional on the vector space
which is irrational with respect to the lattice generated by
. In particular, with respect to such a functional, we can talk about positive roots
and negative roots
, i.e. those that are mapped to positive (resp. negative) values under this functional.
So, let’s say that a highest weight vector in some not-necessarily-finite-dimensional representation is one that is annihilated by
for
. One way to get a highest weight vector in a finite-dimensional representation is to choose any nonzero vector in
, where
maximizes the linear functional on
among the weights. It turns out that we can study the simple representations of
—and that is enough to study all representations of
, by complete reducibility—by using these highest weight vectors. In particular, there is a unique one in every simple
-module.
First, we define the Borel subalgebra as
(Borel subalgebra means maximal solvable.) It is clear that this is solvable, because the commutator . Here
is fact a nilpotent subalgebra, and
. One should think of
as being something like diagonal matrices (indeed, if we are working with
, this is indeed a Cartan subalgebra), and
something like upper-traingular matrices.
We can also define an opposite Borel subalgebra
and a corresponding nilpotent .
Proposition 1 If
is a highest weight vector, then
is a
-submodule of
, and it is the smallest one containing
.
In detail, if is a basis for
, then
is spanned by vectors of the form
(Here the may be zero.)
The reason for this proposition is that the Poincare-Birkhoff-Witt theorem implies that, as subrings of ,
and since is obviously a
-subrepresentation and
annihilates
, the result is clear.
Note that if has weight
and
(where
), we have
which is a lower weight vector.
Now, we are interested in the case where the highest weight vector generates all of ; in this case we will call
a highest weight module, though it may not be finite-dimensional. Let the weight be
. Of course, any simple finite-dimensional representation is a highest weight module. However, it is also possible to construct infinite-dimensional ones (as we will see). We will discuss some of the properties of highest weight modules.
First, note that the highest weight vector is necessarily unique. As we saw, when we multiply it by
, its weight goes down. So
. Moreover, I claim that for all
. This is because there are only finitely many ways the weights of the
‘s in (*) can add up to
by positivity. Thus, we do not necessarily have finite-dimensionality, but the spaces in question are not too bad either.
Proposition 2
Ifis a subrepresentation, then
This is proved by a familiar inductive trick. Suppose we have where
for
a nonrendant collection of weights. Suppose
is the smallest natural number with such a contradiction possible; then each
, and clearly
. Then we may choose
to take different values on all the
; in particular
But this is an expression of the same form resolving into its weight components, so by the inductive hypothesis
for
belongs to
, contradiction.
So we have a few tools for dealing with highest weight modules. Next, we’re going to specialize this to the case of and then use that to get a few more results about the root space decomposition.
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