The structure of free irreducible, cocommutative Hopf algebras

Abstract

In [9] M. Sweedler showed that the coalgebra structure of cocommutative, irreducible Hopf algebras over perfect fields is the coproduct of “sequence of divided power” coalgebras (i.e., coalgebras whose bases consist of a sequence of divided powers). In this paper it is shown that, over perfect fields of characteristic p > 0 and under certain rather weak conditions, the objects in this category and in the category of commutative, cocommutative, irreducible Hopf algebras which are free over cocommutative, pointed, irreducible coalgebras are coproducts of certain elementary Hopf algebras. (B. Ditters in [2] has independently proved this theorem in some special cases.) These elementary Hopf algebras will have evident analogies to sequence of divided power coalgebras and, as shown in the appendix, are isomorphic to Witt Hopf algebras in the commutative case. The second major theorem of this paper is that if $j is a cocommutative, irreducible Hopf algebra and F(g) the free Hopf algebra of 4j, then the Hopf kernel of F(e) -+ 4j is also a coproduct of the above mentioned elementary Hopf algebras. Section 1 consists of a listing of definitions and basic facts needed in the remainder of the paper. Section 2 culminates in the structure theorem for free cocommutative, irreducible Hopf algebras. For this proof we need to know that these Hopf algebras contain primitives and sequences of divided powers of a certain explicit form. This is done in Propositions 2.5 and 2.10, respectively. The proof of the former strongly uses Sweedler’s structure theorem, by saying in effect, for that theorem to be true, the desired primitives must exist. Section 3 includes a demonstration that the Hopf kernel of F(G) + Jj is