Primitive element in a co-algebra

Let $ ( C, \mu, \epsilon ) $ be a 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 $, [a2], p. 199. Let $ ( B,m,e, \mu, \epsilon ) $ be a bi-algebra (see 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 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 of characteristic zero, the functors $ L \mapsto U ( L ) $, the 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) [a2], [a1]; for the graded version of this correspondence, see Hopf algebra and the references quoted there. See also Lie polynomial for the concrete case that $ L $ is a free Lie algebra (cf. Lie algebra, free) on a set $ X $ and $ U ( L ) = { \mathop{\rm Ass} } ( X ) $ is the free associative algebra over $ X $.

References

[a1]	E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977)
[a2]	M.E. Sweedler, "Hopf algebras" , Benjamin (1963)