Definition 1. A Lie algebra is an -algebra such that and
Hence provided is not a ring of characteristic Also, if is an -algebra, then we can define a Lie algebra whose product is defined by
Hence in particular, simply doubles the product of Recall that a derivation on an algebra satisfies
For let us define the adjoint of as defined by
Proposition 2. For is a derivation on
Proof. By tensorial properties of algebras we have
Proposition 3. Let be a commutative Lie algebra over a ring with Then If then is commutative, regardless of the ring.
Proof. If is commutative, then The second claim is trivial.
Let be a left ideal in and Then and hence So is also a right ideal. Hence we simply consider ideals of Lie algebras without reference to leftness or rightness.
Definition 4. Let be a submodule of such that for all derivations on then is called a characteristic ideal of
It’s easy to see that a characteristic ideal is actually an ideal in since we have
Proposition 5. Let be an ideal of and be a characteristic ideal of then is an ideal of
Proof. If is a derivation on then since is a characteristic ideal. But multiplication by is a derivation on Hence for
For two ideals we have that and are ideals in and that and In the case of an -algebra , we will let denote the submodule generated by elements This is in fact an ideal in a Lie algebra for if is a derivation, then
Hence it is stable under adjoint action.
Definition 6. The ideal is called the derived ideal of Inductively we can define
The sequence is called the derived series of Let us define and
The sequence is called the lower central series of
Definition 7. Let be a Lie algebra, be its tensor algebra (with interpreted as a module), and be the ideal generated by elements of the form Then is called the enveloping algebra of
Theorem 8. The enveloping algebra satisfies the universal property that if is an algebra homomorphism for some associative unital algebra and is the canonical inclusion, then there exists a unique morphism and such that
Proposition 9. If and are Lie algebras, then
Let us define with defined as above, and It follows that is a filtered algebra, which we will call the associated filtered algebra to
Theorem 10 (Poincare-Birkhoff-Witt) Let be the associated filtered algebra to and be the symmetric algebra of as a module. Then if is free,
It follows that if is free, then the canonical homomorphism is injective.
 Bourbaki, Nicholas. Lie Groups and Lie Algebras, Chapters 1-3. Springer-Verlag. 1971.