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Hopf algebras with one grouplike element

1967; American Mathematical Society; Volume: 127; Issue: 3 Linguagem: Inglês

10.1090/s0002-9947-1967-0210748-5

1088-6850

Moss E. Sweedler,

Finite Group Theory Research

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 77 ® Y(G) (as a coalgebra) where Y(G) is the group algebra of G the group of grouplike elements-elements of K where dg=g ® g -and 77 is the unique maximal sub-Hopf algebra of K containing one grouplike element, namely 1.If the characteristic of the field is zero then 77 is isomorphic to the universal enveloping algebra of its primitive elements-elements where dx=l ®x+x ® 1-which form a Lie algebra.These results of Kostant prompt the present study of 77 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 77 to be primitively generated is that for all a' e 77' (the dual to 77 which has a natural algebra structure) where =0 then a'p=0, p the characteristic of the field.When the field is perfect 77 modulo the left ideal generated by the primitives (the ideal is actually twosided) with its vector space structure altered is isomorphic to a sub-Hopf algebra of 77.The main results come from the study of divided powers.°x, xx,..., *x is a sequence of divided powers if for « = 0,..., t, cf(nx) = 2"=o *x ® n_ix; in characteristic zero if x is primitive, letting ix=xi/¡!gives an infinite sequence of divided powers.We prove a generalization of the Birkhoff-Witt theorem in which divided powers replace ordinary powers.The results obtained here on Hopf algebras and divided powers are used in an extension of Galois theory to include all finite normal field extensions.A Hopf algebra replaces the Galois group.The Hopf algebra is the group algebra of the Galois group in case the field extension is separable.If the extension is purely inseparable the Hopf algebra has only one grouplike element.These results will appear in a subsequent paper.In the area of algebraic groups divided powers are of interest since certain infinite sequences of divided powers correspond to oneparameter subgroups.