Recall the construction of an associative algebra over a field We have a module homomorphism such that
If the algebra is unital, then we also a canonical map with defined as the identity of and corresponding assumption that
on (with canonical identifications ).
We can dually define a –coalgebra as a -module together with a comultiplication map such that
We also call it unital if there is a map such that
on (with the canonical identification ).
Definition 1. A –bialgebra is a unital -algebra and unital -coalgebra such that
Definition 2. An element of a coalgebra is –primitive if Elements are primitive if they are 1-primitive.
If is the embedding into the enveloping algebra, then the map gives a -bialgebra structure.
Definition 3. Let be a set and be the free algebra over We define the free Lie –algebra over as the quotient where is the ideal generated by elements of the form and We will denote this Lie algebra by and its product by
The enveloping algebra of the free Lie algebra ends up being the free associative algebra over
 Bourbaki, Nicholas. Lie Groups and Lie Algebras, Chapters 1-3. Springer-Verlag. 1971.