# Primitive element in a co-algebra

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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)
How to Cite This Entry:
Primitive element in a co-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_element_in_a_co-algebra&oldid=50993
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article