We use our understanding of the representation theory of
A real form of a complex Lie algebra
In lectures, we didn’t give this definition, just doing the special case of
For dimension reasons, necessary and sufficient conditions are that
The Lie algebra
The Lie algebra
so
Explicitly, we may write
and
in
Recall that
Let
Since the defining equation
and so
Let
There is a one-to-one correspondence between representations of
Given a
It is easy to see that this preserves the Lie bracket. The proposition follows (the final statement is left as an exercise). ∎
As a corollary we immediately obtain
The representation theories of
The claims about Lie algebras follow from the above discussion. Every (finite-dimensional) irreducible representation
of
Every finite-dimensional complex-linear representation of
Let
Then
as representations of
The argument in this theorem is called Weyl’s unitary trick. For a similar application of this idea, see the proof of Proposition 6.2.