# 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 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, 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 (graded commutator if *C* is graded).

If *A* is a connected graded cocommutative Hopf algebra over a field of characteristic zero, then the Milnorâ€“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.)