Hopf algebras with one grouplike element

Abstract

Introduction. We are interested in the coalgebra structure of cocommutative Hopf algebras. Over an algebraically closed field a cocommutative Hopf algebra K with antipode is of the form H 0 r(G) (as a coalgebra) where r(G) is the group algebra of G the group of grouplike elements-elements of K where dg =g 0 g -and H is the unique maximal sub-Hopf algebra of K containing one grouplike element, namely 1. If the characteristic of the field is zero then H is isomorphic to the universal enveloping algebra of its primitive elements-elements where dx= 1 0 x + x 0 1-which form a Lie algebra. These results of Kostant prompt the present study of H when the characteristic is not zero. We do not insist the field be algebraically closed but merely that the unique simple subcoalgebra of our Hopf algebra is the 1-dimensional space spanned by the unit. In this case the subalgebra generated by the primitive elements is a restricted universal enveloping algebra but not necessarily the entire Hopf algebra. A necessary and sufficient condition for H to be primitively generated is that for all a' E H' (the dual to H which has a natural algebra structure) where = 0 then a'P= 0, p the characteristic of the field. When the field is perfect H modulo the left ideal generated by the primitives (the ideal is actually two