Complex Lie Algebras in Dimensions 1, 2, 3:
- Let be the unique non-abelian Lie algebra of dimension 2. What is the realisation of as a Lie subalgebra of a matrix algebra?
- Suppose we have a Lie algebra of dimension 3 and rank 1. Show that we can choose a basis such that . We know that the resulting Lie algebra has (WLOG). Show this is isomorphic to the Lie algebra of strictly upper-triangular matrices. Write an explicit isomorphism.
- Let be the dimension 3 Lie algebra with , , . Show that iff or . What is the realisation of this as a Lie subalgebra of a matrix algebra?
- What is the realisation of the dimension 3 Lie algebra with , , ? Show that this is not isomorphic to the above.
Rep Theory of :
- Construct one infinite-dimensional irreducible representation of .
- Show that the standard representation, with , , , is isomorphic to .
- Show that .
Here are notes which correspond to things that I said a couple of weeks ago, including some exercises. If anything needs rewording/etc, let me know.
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.
Here’s a list of some questions which had been left unanswered from the week 1 meeting:
- Consider representations on vector spaces where the base field is not (e.g. vector spaces over ). What can go wrong?
- If you choose two different characters (degree 1 representations) of a group, why aren’t they isomorphic?
- Let be a finite group. How many different degree 1 representations does have?
- Consider a group which acts on a set , and show that this induces a corresponding action on (the set of all functions mapping from into some other set, say ). Show that the regular representation of is a particular case of this.
- Theorem 1 of Serre states: Let be a representation, and let be a subspace of which is stable under ; then there is a complement space which is also stable.
Write a proof of this theorem in your own words. Highlight where we use the following facts: (1) is finite; (2) is finite-dimensional; (3) is over the field .
For each of the three conditions, find a counter-example to show that the theorem doesn’t hold if the condition is dropped.
(This will also be a midterm exam question.)
- If is a group, why is (the free vector space on ) the same as the regular representation of ?
- For vector spaces , does it hold that ?
- If and are finite dimensional vector spaces, prove that . Show that this does not hold in general when are infinite-dimensional.
- For vector spaces , prove that .
be a finite dimensional vector space, and
denote its dual and
denote its double dual. One knows that
is isomorphic to
. However, the isomorphism
is much better than the isomorphism
!! Here is one reason:
; i.e., we have a representation
. Show that
. Let us denote these representations by
. Show that the representations
are always isomorphic, where as, it could happen that
are not isomorphic!!
One says that the isomorphism
, where as the isomorphism
is not (it depends on a choice of basis).