For the quiz of Oct.17.

L always denotes a finite dimensional Lie algebra over C.

1) For any ideal I of L define I^{(n)} for n=-1,0,1,2,... by
I^{(-1)}=L, I^{(0)}=I, I^{(n)}=[I^{(n-1)},I^{(n-1)}] for n=1,2,3,...
Let I,J be ideals of L. Show (by induction) that for N=-1,0,1,2,... we have
(I+J)^{(N)}\subset \sum_{M,P in [-1,N]; M+P+1=N}(I^{(M)}\cap J^{(P)})
Deduce that if I,J are solvable then I+J is solvable.

2) If d,d' are derivations of L then dd'-d'd is a derivation of L.

3) The Lie algebra of 2 by 2 matrices with entries in C with trace zero
(under the bracket AB-BA) is simple.

4) Assume that L is solvable. Assume known that there exist ideals 
L_0\subset L_1\subset...\subset L_n=L of L, dim L_i=i. Assume known 
Engel's theorem. Shows that [L,L] is a nilpotent Lie algebra.

5) Let V be a finite dimensional vector space. Let x\in End(V). Let
A\subset B be subspaces of V such that x(B)\subset A. Show that the
semisimple part of x maps B into A.

6) Let I be an ideal of L. Show that the Killing form of I is the 
restriction to I of the Killing form of L.

7) Show that the radical of the Killing form of L is a solvable ideal. 
(You may assume known Cartan's theorem.)

8) Show that if the Killing form of L is nondegenerate then L is
semisimple. (You may use that if J is a solvable ideal of L then
[J,J] is nilpotent.)

9) Let M,N be L-modules. Then the rule (xf)(v)=x(f(v))-f(xv) where 
x\in L, f\in Hom(M,N), v\in M makes Hom(M,N) into an L-module.

10) Let x be a derivation of L. Let s\in End(L) be the semisimple
part of x\in End(L). Show that s is a derivation of L.