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In algebra, a primitive element of a co-algebra C is an element x that satisfies
where μ {\displaystyle \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.
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.
If C is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket = x y − y x {\displaystyle =xy-yx} .
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