Hopf Algebras And Cohomology Operations

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.