Higher Conjugation Cohomology in Commutative Hopf Algebras

We study the action of the symmetric group Σn on a tensor product of n − 1 copies of a commutative Hopf algebra A, defined by the second author [8]. We show that for ‘nice’ Hopf algebras, the cohomology algebra H∗(Σn;A⊗n−1) is independent of the coproduct if n.(n − 2)! is invertible in the ground ring. Let A be a graded, connected, unital, counital, associative, coassociative Hopf algebra. In section 8 of [6] it was shown how A has a ‘conjugation’ or ‘antipode’ χ satisfying the equality μ ◦ (1⊗ χ) ◦∆ = η ◦ , where μ and ∆ are the product and coproduct and η and are the unit and counit/augmentation. In particular, χ(1) = 1 and, for x of positive degree, χ(x) = −x+ ∑