Automated Proving

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

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/

http://gowers.wordpress.com/2013/04/14/answers-results-of-polls-and-a-brief-description-of-the-program/

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).

Exercises Weeks 3 & 4

From Mitchell’s talk on \mathbb{C}[G]-modules:

  1. If D is a division algebra, then check that every D-module is of the form D^r, and as a corollary, all M_n(D)-modules are of the form \underbrace{D^r \oplus \dots \oplus D^r}_{n}.
  2. Prove the following lemma: If F is a field, then V is a (finitely generated) F[G]-module, then V is a (finite dimensional) vector space over F.
  3. Is the ring \mathbb{Z} semisimple? (we did this one in class)
  4. Prove the following theorem of Wedderburn: If F is algebraically closed, then any semisimple algebra over F deomposes as \bigoplus_{i=1}^r M_{n_i}(F). As a corollary, this holds for F[G]. What does r represent? What does this mean in the context of representations?

From my talk on representations of group products, and induced representations:

  1. Let \rho_1: G_1 \to GL(V_1), \rho_2: G_2 \to GL(V_2) be representations. Consider the representation \rho_1 \oplus \rho_2 : G_1 \times G_2 \to GL(V_1 \oplus V_2) defined appropriately. Is it interesting?
  2. Show that the character of \rho_1 \otimes \rho_2 is \chi(g_1,g_2)=\chi_1(g_1) \chi_2(g_2).
  3. We have shown that if \rho_1, \rho_2 are irreducible, \rho_1 \otimes \rho_2 is irreducible. Show then that in fact every irreducible representation of G_1 \times G_2 is of this form.
  4. Verify and experiment with the examples in my notes (to be posted soon-ish).

From Reuben’s talk on Lie algebras:

  1. Show there is a bijection between representations of L, and left L-modules.
  2. Show that \mathbb{R}^3 under the cross product can be realised (by an isomorphism) as \mathfrak{so}_2(\mathbb{R}), the Lie subalgebra of \mathfrak{gl}_2(\mathbb{R}) containing matrices satisfying X+X^T=0.
  3. Suppose there exist G, \circ_1, \circ_2 such that (G, \circ_1) and (G, \circ_2) are groups, and (g \circ_1 h) \circ_2 k = g \circ_1 (h \circ_2 k). What can we say about G? Does it necessarily hold that \circ_1 = \circ_2?
  4. Find an example of an indecomposable L-module which is not irreducible (i.e. contains nontrivial proper submodules).
  5. Prove the following lemmas: For a Lie algebra L, and m, n \geq 1, [L^m, L^n] \subseteq L^{m+n}; and for n \geq 0, L^{(n)} \subseteq L^{2^n}. (It follows directly that nilpotent Lie algebras are solvable.)
  6. Show that Lie algebras consisting of upper triangular matrices are solvable, and strictly upper triangular matrices are nilpotent.
  7. Show that every irreducible representation of a solvable Lie algebra is one-dimensional.

And finally, from Chris’ talk on semisimple Lie algebras:

  1. For \mathfrak{g} = \mathfrak{gl}_n, show that the radical is given by the centre, the subalgebra of nonzero scalar matrices, of the form z I_n, z \in F^*. Show also that \mathfrak{g}/\textrm{Rad}(\mathfrak{g}) \simeq \mathfrak{sl}_n.
  2. Prove the following direction of the Cartan-Killing criterion: If \mathfrak{g} is a semisimple Lie algebra, then its Killing form is nondegenerate.
  3. Prove that \mathfrak{sl}_n, \mathfrak{so}_n, \mathfrak{sp}_{2n} are simple Lie algebras.