Problem set 3. Due Wednesday, October 14.
Mathematics 534, Term 1, 2015. Instructor: Reichstein.
Assume that the base field is the field of complex numbers C throughout.
1. Let L be a finite-dimensional Lie algebra. We showed in class
that the sum of two nilpotent ideals in L is again nilpotent. Use this
to prove that L has a unique maximal nilpotent ideal N (also known
as the nil-radical of L). That is, show that there exists an ideal N of
L such that
(i) N nilpotent, and
(ii) N contains every nilpotent ideal of L.
2. p. 20, Problem 1.
3. p. 21, Problem 5.
4. Let L be a finite-dimensional Lie algebra and S ⊂ L be a solvable
subalgebra of dimension d such that S 6= L. Show L has another
subalgebra S 0 such that S ⊂ S 0 and dim(S 0 ) = d + 1. Is S 0 necessarily
solvable?
5. (a) Let L be a solvable subalgebra of gln . Show that
n(n + 1)
dim(L) ≤
.
2
(b) Show that if L1 and L2 are both solvable subalgebras of gln
n(n + 1)
of dimension
, then L2 = gL1 g −1 for some invertible linear
2 n
transformation g : C → Cn .
(c) Give an example of solvable subalgebras L1 , L2 ⊂ gln such that
n(n + 1)
dim(L1 ) = dim(L2 ) =
− 1, and the assertion of part (b) fails.
2
−1
That is, L2 6= gL1 g for any g, as in part (b).
6. Use Cartan’s criterion to show that the algebra son of skewsymmetric n × n-matrices is not solvable for any n ≥ 3.