Problem 1: Let g be a finite dimensional Lie algebra over the complex numbers. With OR
we will denote the same Lie algebra over the real numbers: as sets 9 and 9R are the same,
but the first one is vector space over C and the second is a vector space over R.
a) Explain what is the difference between definition of ideals in 9 and ideals in 9Ri
b) Prove that if 9R is simple Lie algebra then g is also simple;
c) Describe all 9 which are simple, but OR is not simple.
Problem 3: Let 9 be a finite dimensional Lie algebra. For any two subspaces U,V of
9 with [U,V] will denote the subspace of 9 spanned by all elements u. with u € U and
e € V. Define the following decreasing chains of subspaces of gr
and
(0) g.
The Lie algebra 9 is called nilpotent if g' = 0 for some i; and 9 is called solvable if = 0
for some i.
a) Prove that if U and V are ideals in g then (U.V] is also an ideal;
b) Prove that all go and are ideals;
c) Show that every nilpotent Lie algebra is solvable;
d) Prove that every Lie algebra of dimension less or equal to 2 is solvable. Are all such
algebras nilpotent?
e) Give an example of a nilpotent Lie algebra which is not abelian.

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