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Primitive element in a co-algebra

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Let 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