Title:
Higher Conjugation Cohomology in Commutative Hopf Algebras
Authors:
M. D. Crossley and Sarah Whitehouse
AMS Classification Numbers:
16W30, 57T05, 20C30, 20J06, 55S25
Addresses:
Max-Planck-Institut fuer Mathematik,
P.O. Box 7280,
D-53072 Bonn,
Germany.
Departement de Mathematiques,
Universite d'Artois - Pole de Lens,
Rue Jean Souvraz,
S. P. 18 - 63207 Lens,
France.
Email addresses:
crossley\@member.ams.org
whitehouse\@euler.univ-artois.fr
Abstract Text:
The dual Steenrod algebra can be expressed as the homotopy of a
smash product of two copies of the Eilenberg-MacLane spectrum,
and the conjugation arises by permutation of the two factors.
This can be generalized to an action of the symmetric group
$\Sigma_n$ acting on an n-1-fold tensor product of copies of the
dual Steenrod algebra; this action was described by the second
author in [6], purely in terms of the Hopf algebra structure.
So, formally, one has a similar action for any commutative Hopf
algebra. In this paper, we study the cohomology ring
$H^*(\Sigma_n; A^{\otimes n-1})$, where $A$ is a graded
commutative Hopf algebra.
We show that for a certain class of Hopf algebras the cohomology
ring is independent of the coproduct provided $n$ and $(n-2)!$
are invertible in the ground ring.
Then, by choosing a sufficiently simple coproduct, we are able to
deduce significant information about the $\Sigma_n$ invariants of
$A^{\otimes n-1}$, including dimensions and algebra structure.
In particular, we give a complete solution to the "conjugation
invariants" problem for the mod $p$ dual Steenrod algebra when
$p>2$.