Automated Proving

Here’s links to Tim Gowers’ posts on his automated theorem prover:

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.

Exercises From Cartan Subalgebras and Root Systems

  1. Why are diagonal matrices their own normaliser?
  2. How you do define determinant for a general linear operator? (i.e without defining a basis)
  3. Why is the exponential map well defined for matrices? exp: gl_{n} \rightarrow GL(g) Where the exponential map takes a matrix to its exponential Taylor series.
  4. Consider gl_{2} \& gl_{3}, what is \mathrm{ad}(x), and what is e^{\mathrm{ad}(x)} ?
  5. Show the rank of  gl_{n} is n, where rank is the dimension of a Cartan Subalgebra.
  6. Show that the regular elements in gl_{n} are diagonal matrices with no repeated entries.
  7. If we define the map -1 : V \rightarrow V which takes a root \alpha \rightarrow -\alpha, For which root systems is this in the Weyl Group?
  8. (Very optional) Why is the Dynkin diagram for E_{8} not allowed to have a connection on the central root?

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

Exercises Week 5

Complex Lie Algebras in Dimensions 1, 2, 3:

  1. Let L_2 be the unique non-abelian Lie algebra of dimension 2. What is the realisation of L_2 as a Lie subalgebra of a matrix algebra?
  2. Suppose we have a Lie algebra of dimension 3 and rank 1. Show that we can choose a basis \{X, Y, Z\} such that [X, Y] = [X, Z] = 0. We know that the resulting Lie algebra has [Y, Z] = X (WLOG). Show this is isomorphic to the Lie algebra of strictly upper-triangular 3 \times 3 matrices. Write an explicit isomorphism.
  3. Let \mathfrak{g}_\alpha be the dimension 3 Lie algebra with [X, Y] = Y, [X, Z] = \alpha Z, [Y, Z] = 0. Show that \mathfrak{g}_\alpha \simeq \mathfrak{g}_{\alpha'} iff \alpha = \alpha' or \alpha = \frac{1}{\alpha'}. What is the realisation of this as a Lie subalgebra of a matrix algebra?
  4. What is the realisation of the dimension 3 Lie algebra with [X, Y] = Y, [X, Z] = Y + Z, [Y, Z] = 0? Show that this is not isomorphic to the above.

Rep Theory of \mathfrak{sl}_2(\mathbb{C}):

  1. Construct one infinite-dimensional irreducible representation of \mathfrak{sl}_2(\mathbb{C}).
  2. Show that the standard representation, with H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, X= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, is isomorphic to V(1).
  3.  Show that \textrm{Sym}^n S \simeq V(n).