View source for Primitive element (co-algebra)

{{one source |date=May 2024}}
In algebra, a '''primitive element''' of a [[co-algebra]] ''C'' (over an element ''g'') is an element ''x'' that satisfies
:<math>\mu(x) = x \otimes g + g \otimes x</math>
where <math>\mu</math> is the [[co-multiplication]] and ''g'' is an element of ''C'' that maps to the [[1#In Mathematics|multiplicative identity]] 1 of the [[Ground field|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 <math>[x, y] = xy - yx</math> ([[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.)

== References ==
*http://www.encyclopediaofmath.org/index.php/Primitive_element_in_a_co-algebra

[[Category:Coalgebras]]


{{algebra-stub}}