## 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.

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## 3 Comments »

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[…] 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 |