# Hopf Algebras And Cohomology Operations

1992

Dissertation

## Degree Name

Doctor of Philosophy

## Abstract

The main purpose of this thesis is to study product structures of Hopf algebras, in particular for the cases of the Steenrod algebra and the Brown-Peterson algebra. In Chapter 1, we are given a Hopf algebra with basic operations {dollar}\{lcub}D\sp{lcub}R{rcub} \vert R :{dollar} exponential sequences{dollar}\{rcub}{dollar}, the coalgebra structure being the Cartan Formula. The main theorem is a formula expressing the products {dollar}D\sp{lcub}R{rcub}\cdot D\sp{lcub}S{rcub}{dollar} of basic operations using a star operation of scalar parameters. In Chapter 2, the Milnor Product Formula for the Steenrod algebra is shown to be a consequence of the main theorem in Chapter 1. The weighted symmetrical polynomials {dollar}\sigma\sb1{dollar}, {dollar}\sigma\sb2\...{dollar}, {dollar}\sigma\sb{lcub}n{rcub}{dollar} are introduced to formulate iterated products of Steenrod operations in terms of the Milnor basis. Simple proofs of the Bullet-MacDonald version of the Adem Relations, the Peterson Formula and Davis's results are at hand. Moreover, a closed formula for calculating the anti-automorphism of the Steenrod algebra is given which is much simpler than the one given by Milnor. There is also a theorem showing a close relation between the composition of Steenrod operations and the Dickson and Mui Invariants. Finally, all results of Monks on the nilpotence of Sq{dollar}\sp{lcub}2r\sb1+1,\...,2r\sb{lcub}n{rcub}+1{rcub}{dollar} are re-captured with an explanation of the existence of a gap between the upper and lower bounds given by Monks. In Chapter 3, the Quillen Product Formula for the Brown-Peterson algebra and a product formula for the Landweber-Novikov algebra are obtained. A complete formula for calculating the anti-automorphism of the Brown-Peterson algebra is given paralleling the one for the Steenrod algebra. Based on a closed formula developed in the thesis, a method is fully described for calculating the composition of Brown-Peterson operations. This method is simpler and more systematic than Zahler's method. It gives the complete result of any composition and involves no inductive steps. The introduction of a new set of rational generators {dollar}\{lcub}w\sb{lcub}i{rcub}\{rcub}{dollar} and a new set of rational operations {dollar}\{lcub}{lcub}\bf q{rcub}\sb{lcub}R{rcub}\{rcub}{dollar} proves useful in drawing corollaries.

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