http://gowers.wordpress.com/2013/03/25/an-experiment-concerning-mathematical-writing/

http://gowers.wordpress.com/2013/04/02/a-second-experiment-concerning-mathematical-writing/

It’s all very long, so you might not want to read all of it, but at least read through the first post, and maybe the first few paragraphs of the second post.

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I’ve attached a link to the pdf on wedge products and the determinant. It starts off fairly slowly, but about halfway through it gets more enlightening. It is very much all spelled out at a very approachable level.

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Don’t forget that next week is the problem session!

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I liked this TED talk by the Fields Medallist Cedric Villani.

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- 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 .

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- 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.

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https://www.dropbox.com/s/1f0zocn6e8bhm45/module.pdf

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The notes contain more detail on the standard representation of S3 and inter-relatedness (via characters) that arise between the different representations, their tensor products, direct sums and so on.

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