From Mitchell’s talk on -modules:
- If is a division algebra, then check that every -module is of the form , and as a corollary, all -modules are of the form .
- Prove the following lemma: If is a field, then is a (finitely generated) -module, then is a (finite dimensional) vector space over .
- Is the ring semisimple? (we did this one in class)
- Prove the following theorem of Wedderburn: If is algebraically closed, then any semisimple algebra over deomposes as . As a corollary, this holds for . What does represent? What does this mean in the context of representations?
From my talk on representations of group products, and induced representations:
- Let be representations. Consider the representation defined appropriately. Is it interesting?
- Show that the character of is .
- We have shown that if are irreducible, is irreducible. Show then that in fact every irreducible representation of is of this form.
- Verify and experiment with the examples in my notes (to be posted soon-ish).
From Reuben’s talk on Lie algebras:
- Show there is a bijection between representations of , and left -modules.
- Show that under the cross product can be realised (by an isomorphism) as , the Lie subalgebra of containing matrices satisfying .
- Suppose there exist such that and are groups, and . What can we say about ? Does it necessarily hold that ?
- Find an example of an indecomposable -module which is not irreducible (i.e. contains nontrivial proper submodules).
- Prove the following lemmas: For a Lie algebra , and , ; and for , . (It follows directly that nilpotent Lie algebras are solvable.)
- Show that Lie algebras consisting of upper triangular matrices are solvable, and strictly upper triangular matrices are nilpotent.
- Show that every irreducible representation of a solvable Lie algebra is one-dimensional.
And finally, from Chris’ talk on semisimple Lie algebras:
- For , show that the radical is given by the centre, the subalgebra of nonzero scalar matrices, of the form . Show also that .
- Prove the following direction of the Cartan-Killing criterion: If is a semisimple Lie algebra, then its Killing form is nondegenerate.
- Prove that , , are simple Lie algebras.