Primitive element (co-algebra)

In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies

\mu(x) = x \otimes g + g \otimes x

where \mu 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). C is said to be primitively generated if it is generated by primitive elements.

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.

If C is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket [x, y] = xy - yx (graded commutator if C is graded.)

If A is a (certain graded) 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.

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