On conjugation invariants in the dual Steenrod algebra

Abstract

The dual Steenrod algebra {\scr A}\sb * at the prime 2 is agraded connected Hopf algebra and, as such, supports a canonicalconjugation χ. This paper studies the subalgebra{\scr A}\sp χ\sb * of elements in {\scr A}\sb * which areinvariant under χ, that is, χ(a)=a. Despite theclassical nature of the problem, it has received almost noattention in the literature, and only arises here as an outgrowthof work of the second author and A. Robinson on E\sb \inftyring spectra. Much more space has been devoted to discussinginvariants of the conjugation of the Steenrod algebra itself;see, in particular, A. M. Gallant [Proc. Amer. Math. Soc. 76(1979), no. 1, 161 166; MR 81a:55029], and various papers by J.Silverman.\par The main result of this paper is a completecalculation of the conjugation invariants with ξ\sb 1inverted, where ξ\sb 1 is the non zero class in degree 1. Theresult is simply stated: {\scr A}\sp χ\sb *[ξ\sp {1}\sb 1]={\bf F}\sb 2[ξ\sp {\pm1}\sb 1,ε\sb 2,ε\sb 3,\cdots],where ε\sb i is an easily described class of degree2\sp i+2. Furthermore, bounds on the dimension, by degree, of{\scr A}\sp χ\sb * itself are given, and the paper closeswith a specific conjecture which would complete the calculation