Recall a representation of an algebra on an -module is an algebra morphism When we have a Lie algebra a representation of it on an -module is an algebra morphism
Hence in particular we have We correspondingly call a –module. A representation is faithful if it is injective. It is simple/irreducible if its -module is simple.
Let and be representations of Lie algebras and on and respectively. Then is a representation of on defined by
where are the canonical inclusions of in and in Hence is both a -module and a -module.
Definition 1. is solvable if its derived series for some It is nilpotent if its lower central series for some The radical of is the largest solvable ideal in We say is semisimple if it has no nonzero abelian ideals.
Hence it follows that the center of a nilpotent Lie algebra is nontrivial. Also note that a nilpotent Lie algebra is solvable. is the smallest ideal such that has radical
Theorem 2 (Engel). Let be a vector space and be a finite dimensional subalgebra of . If then there exists a with such that for all
It follows that a Lie algebra is nilpotent iff is nilpotent for all
Note that defines a symmetric bilinear form on called the Killing form of The Killing form also satisfies
Proposition 3. is solvable iff
Proposition 4. is semisimple iff iff is nondegenerate.
Theorem 5 (Weyl). Every finite dimensional representation of a semisimple Lie algebra is completely reducible (i.e. the -module is semisimple).
A Lie algebra is simple if its only ideals are and and noncommutative.
Proposition 6. is semisimple iff it is a product of simple algebras.
A Lie algebra is reductive if its adjoint representation is semisimple. That is, it is reductive if it can be written where is semisimple and is abelian (since here we want as a module to be semisimple–so that as an algebra, it has an an abelian term). This also turns out to be equivalent to being semisimple.
Definition 7. The nilpotent radical of is the intersection of the kernels of all finite dimensional simple representations of
For an element in the nilpotent radical, it follows that is nilpotent. Now for any nilpotent endomorphism of an algebra, we have that
has a finite number of nonzero terms, and is hence well-defined. If we can write with semisimple and nilpotent, then we will call a Levi subalgebra of
Theorem 8 (Levi-Malcev). Every Lie algebra has a Levi subalgebra (that is, it can be written as above). Moreover, every other Levi subalgebra is the image of under for some in the nilpotent radical of
It turns out that a subalgebra of is a Levi subalgebra iff it is a maximal semisimple subalgebra.
Theorem 9 (Ado). Let be the largest nilpotent ideal in Then admits a finite dimensional faithful representation such that every element of is nilpotent.
 Bourbaki, Nicholas. Lie Groups and Lie Algebras, Chapters 1-3. Springer-Verlag. 1971.