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!

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