On the \(\mathrm{mod}p\) Steenrod algebra and the Leibniz-Hopf algebra
For an odd prime \(p\), let \(\mathcal{A}_p\) denote the Bockstein-free part of the \(\mathrm{mod} \ p\) Steenrod algebra (which is generated as an associative algebra by the Steenrod powers \(\mathcal{P}^i\)). Using the Milnor coproduct definition, one gets a Hopf algebra structure for \(\mathcal{A}_p\), and its conjugation map is the subject of the present paper. The author considers the Leibnitz-Hopf algebra and its \(\mathrm{mod} \ p\) reduction, \(\mathcal{F}_p\), defined as the free associative \(\mathbb{Z}/p\) algebra on generators \(S^1, S^2, \ldots\), and carrying a natural Hopf algebra structure (using again a version of the Milnor coproduct). If each \(S^i\) is given the same degree as the corresponding \(\mathcal{P}^i\), by considering the Adem relations one gets a quotient map \(\pi : \mathcal{F}_p \rightarrow \mathcal{A}_p\). The point of this paper is to use this map, together with the known formula for the conjugation on \(\mathcal{F}_p\) (appearing here at the beginning of Section 3), to determine the conjugation of some admissible elements of \(\mathcal{A}_p\). The method is exemplified in Section 3, and is most successful for those that are images of elements of \(\mathcal{F}_p\) whose conjugation becomes simpler once one applies the Adem relations. The last section of this work gives a new, combinatorial proof of a result on the coarsening operation in the \(\mathrm{mod} \ 2\) dual Leibnitz-Hopf algebra, \(\mathcal{F}_2^*\), and presents an additional result on the conjugation invariants in \(\mathcal{F}_2^*\) using its algebra structure.
scientific article; zbMATH DE number 7220340
scientific article