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.
[…] are a few constructions we can make, starting with the ones from last time and applying them in certain special […]
Pingback by More New Modules from Old « The Unapologetic Mathematician | September 21, 2012 |
Shouldn’t the eighth line be the negative of what it is now?
Comment by Tom Gregory | December 27, 2014 |
Yes, thanks; fixed.
Comment by John Armstrong | December 28, 2014 |