Primitive element (co-algebra)

In algebra, a primitive element of a co-algebra "C" (over an element "g") is an element "x" that satisfies
where formula_2 is the co-multiplication and "g" is an element of "C" that maps to the multiplicative identity 1 of the base field under the co-unit ("g" is called "group-like").
If "C" is a bi-algebra, i.e., a co-algebra that is also an algebra (with certain compatibility conditions satisfied), then one usually takes "g" to be 1, the multiplicative identity of "C". The bi-algebra "C" is said to be primitively generated if it is generated by primitive elements (as an algebra).
If "C" is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket formula_3 (graded commutator if "C" is graded).
If "A" is a connected graded cocommutative Hopf algebra over a field of characteristic zero, then the Milnor–Moore theorem states the universal enveloping algebra of the graded Lie algebra of primitive elements of "A" is isomorphic to "A". (This also holds under slightly weaker requirements.)