There are a few standard techniques we can use to generate new modules for a Lie algebra from old ones. We’ve seen direct sums already, but here are a few more.
One way is to start with a module and then consider its dual space . I say that this can be made into an -module by setting
for all , , and . Bilinearity should be clear, so we just check the defining property of a module. That is, we take two Lie algebra elements and check
so for all , as desired.
Another way is to start with modules and and form their tensor product . Now we define a module structure on this space by
We check the defining property again. Calculate:
These are useful, and they’re only just the beginning.