Constructing sequences of divided powers

Abstract

In my Sequences of divided powers in irreducible, cocommutative Hopf algebras, I demonstrated the existence of extensions of sequences of divided powers over arbitrary fields, if certain coheight conditions are met. Here, I show that if the characteristic of the field does not divide n, every sequence of divided powers of length n 1, in a cocommutative Hopf algebra, has an extension that can be written as a polynomial in the previous terms. (An algorithm for finding these polynomials is given, together with a list of some of them.) Furthermore, I show that if one uses this method successively for constructing a sequence of divided powers over a primitive, the only obstructions will occur at powers of the characteristic of the field. Some of the basic definitions of this paper are the following: (1) If H is a Hopf algebra and 0 $ g E H, then g is a grouplike if Ag = g 0 g. (2) If h c H, then h is a primitive if Ah = h 0 1 + 1 ? h. (3) A Hopf algebra will be called irreducible if every nontrivial subcoalgebra contains a fixed, nontrivial subcoalgebra, i.e., if H is irreducible, the identity is the unique grouplike. (4) An irreducible Hopf algebra will be called graded, if there exists a set of subspaces {Hj}j? O such that (a) H= G`0H,; (b) Ho= 1 k; (c) AHiC E=o Hj? Hi-; (d) HiHjC Hi+j (5) A sequence of divided powers Ox = 1, 1x, 2x, , x is a set of elements in a cocommutative, irreducible Hopf algebra such that Altx = 0 o t-iX, for all 0 < t < n. Received by the editors December 8, 1970. AMS 1970 subject classiflcations. Primary 16A24.