View source for Primitive element in a co-algebra

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Let  $  ( C, \mu, \epsilon ) $
be a [[Co-algebra|co-algebra]] over  $  k $. 
An element  $  x \in C $
is called group like if  $  \mu ( x ) = g \otimes g $. 
An element  $  x \in C $
is called primitive over the group-like element  $  g $
if  $  \mu ( x ) = g \otimes x + x \otimes g $, 
[[#References|[a2]]], p. 199. Let  $  ( B,m,e, \mu, \epsilon ) $
be a bi-algebra (see [[Hopf algebra|Hopf algebra]]) and let  $  P ( B ) $
be the set of primitive elements over the group-like element  $  1 \in B $
of  $  B $, 
considered as a co-algebra. Then  $  P ( B ) $
becomes a [[Lie algebra|Lie algebra]] under the commutator bracket

$$ 
[ x,y ] = xy = yx,
$$

(using the multiplication of  $  B $). 
This is the Lie algebra of primitive elements.

For  $  k $
a [[Field|field]] of characteristic zero, the functors  $  L \mapsto U ( L ) $, 
the [[Universal enveloping algebra|universal enveloping algebra]] of the Lie algebra  $  L $, 
and  $  H \mapsto P ( H ) $, 
where  $  H $
is a Hopf algebra (or bi-algebra) over  $  k $, 
establish an equivalence between the category of Lie algebras and the category of co-commutative irreducible bi-algebras (such bi-algebras are automatically Hopf algebras).

In particular,  $  P ( U ( L ) ) \simeq L $, 
$  U ( P ( H ) ) \simeq H $
for such a bi-algebra (Hopf algebra) [[#References|[a2]]], [[#References|[a1]]]; for the graded version of this correspondence, see [[Hopf algebra|Hopf algebra]] and the references quoted there. See also [[Lie polynomial|Lie polynomial]] for the concrete case that  $  L $
is a free Lie algebra (cf. [[Lie algebra, free|Lie algebra, free]]) on a set  $  X $
and  $  U ( L ) = { \mathop{\rm Ass} } ( X ) $
is the free associative algebra over  $  X $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Abe,   "Hopf algebras" , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.E. Sweedler,   "Hopf algebras" , Benjamin  (1963)</TD></TR></table>