Homework Set #3

1. Show that the normalizer of a Lie subalgebra K of a Lie algebra L is the maximal subalgebra of L containing K as an ideal.
2. Let p be a prime number, F a field of characteristic p, and V = F^p a vector space with basis e₁,...,e_p. Let x denote the linear transformation of V sending each e_i to ie_i, and let y denote the cyclic permutation e_i→e_i+1 (and e_p→e₁.) Show that the span of x and y gives a counterexample to Lie's theorem in characteristic p.
3. Show that the (5-dimensional) Lie algebra of 3×3 matrices with bottom row 0 and trace 0 is perfect but has a non-trivial solvable ideal.