Homework Set #10

1. Give a necessary condition on an element h of a maximal toral subalgebra H of a semisimple Lie algebra L, in terms of the root system Φ of L with respect to H, for H to be the centralizer of h in L.
2. The rank of a semisimple Lie algebra L is the minimal dimension of the centralizer of any element h of L. Prove that the dimension of every maximal toral subalgebra of L is greater than or equal to the rank of L.
3. If L is a semisimple Lie algebra and H is a maximal toral subalgebra, we say that x∈L is a root element if x∈L_α for some root α. Prove that the minimum number of root elements necessary to generate L as a Lie algebra is at least 1+dim(H) and at most 2dim(H).
4. Prove that L is a semisimple Lie algebra with maximal toral subalgebras H₁ and H₂ giving rise to root systems Φ₁ and Φ₂, then Φ₁ is irreducible if and only if Φ₂ is.