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

- (),
- and

**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

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