## New Modules from Old

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:

while

These are useful, and they’re only just the beginning.