On conjugation invariants in the dual Steenrod algebra

Abstract

Abstract. We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra, A * , with a view to determining the subspace, A χ * , of elements invariant under χ. We give bounds on the dimension of this subspace for each degree and show that, after inverting ξ 1 , it becomes polynomial on a natural set of generators. Finally we note that, without inverting ξ 1 , A χ * is far from being polynomial.

Higher Conjugation Cohomology in Commutative Hopf Algebras

“…We note that if p > 2 then Theorem 1.2 satisfactorily solves the 'conjugation invariants' problem for the mod p dual Steenrod algebra, in marked contrast to the partial solution [3] available when p = 2.…”

“…For example, we need to know the Σ n -invariants, since these form H 0 . However, in [3] we saw how complicated this calculation could be when we attempted it for n = 2 and for A an object familiar to algebraic topologists: the mod 2 dual Steenrod algebra.…”

“…be invertible is rather curious. Some such condition is certainly required; [3] illustrates this in the case where n = 2 and similar examples can be constructed for larger n showing that n, at least, must be invertible. We know of no examples where n is invertible and the conclusion of the theorem does not hold but the condition that n(n − 2)!…”

A note on conjugation invariants in the dual Leibniz-Hopf algebra

“…In earlier papers [4,5], conjugation map was interested because of its link with the Steenrod algebra, and earlier works of Crossley and Whitehouse [6,7]. More precisely, in [5] the conjugation invariants which form a submodule, Ker(χ−1), where 1 denotes the identity homomorphism, were determined both for F * and F * ⊗ Z/p, for any odd prime p. In the proof of [5, Theorem 2.5], it was given that: Ker(χ−1) = Im(χ+1) in F * , and a spanning set for Im(χ+1) was introduced by a matrix representation of χ + 1.…”

On the mod p Steenrod algebra and the Leibniz-Hopf algebra

“…In [40,41] the author used the π * to introduce an alternative view of the Adem relations for any prime number. In [40], motivated by the work of Crossley and Whitehouse [11], the author attempts to solve conjugation invariant problem in the mod 2 dual Steenrod algebra by using the homomorphism π * . Conjugation invariants in the mod p dual Steenrod algebra, A * p , are determined in [12].…”