# 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!