Volume 11, issue 3 (2011)

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A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

David Pengelley and Frank Williams

Algebraic & Geometric Topology 11 (2011) 1767–1780
Abstract

The hit problem for a cohomology module over the Steenrod algebra $\mathsc{A}$ asks for a minimal set of $\mathsc{A}$–generators for the module. In this paper we consider the symmetric algebras over the field ${\mathbb{F}}_{p}$, for $p$ an arbitrary prime, and treat the equivalent problem of determining the set of ${\mathsc{A}}^{\ast }$–primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo–Araki–May algebra $\mathsc{K}$, and consider the $\mathsc{K}$–module structure of the primitives, which form a sub $\mathsc{K}$–algebra of the dual of the infinite symmetric algebra. Our examples show that the $\mathsc{K}$–action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem.

Keywords
hit problem, symmetric algebra, Steenrod algebra, Kudo–Araki–May algebra, Dyer–Lashof algebra, $\mathit{BO}$, $\mathit{BU}$
Mathematical Subject Classification 2000
Primary: 16T05, 16T10, 16W22, 55R45, 55S10, 55S12
Secondary: 16W50, 55R40, 57T05, 57T10, 57T25