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.