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 r**eductive** 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.

[1] Bourbaki, Nicholas. *Lie Groups and Lie Algebras, Chapters 1-3*. Springer-Verlag. 1971.