Higher conjugation cohomology in commutative Hopf algebras (Q2716974)

For \(A\) a graded, commutative Hopf algebra, an action of the symmetric group \(\Sigma_n\) on \(A^{\otimes n-1}\) is studied. This action, first introduced by \textit{S. Whitehouse} [Symmetric group actions on tensor products of Hopf algebroids, Université d'Artois preprint 99-5], is a generalization of the action of \(\Sigma_2\) on \(A\) given by \(\mu\circ(1\otimes\chi)\circ\Delta=\eta\circ\varepsilon\), where \(\mu\), \(\chi\), \(\Delta\), \(\eta\), and \(\varepsilon\) are the multiplication, antipode, comultiplication, unit, and counit respectively. This paper proves that, under certain situations (such as \(A\) being graded, connected, commutative, and biassociative) the cohomology ring \(H^*(\Sigma_n;A^{\otimes n-1})\) is independent of \(\Delta\) provided than \(n\) and \((n-2)!\) are invertible in the base ring, in spite of the fact that the action of \(\Sigma_n\) uses \(\Delta\) explicitly. The entire paper is devoted to proving this theorem, which it does by defining a bijection \(f\) between \(\widetilde A^{\otimes n-1}\) and \(A^{\otimes n-1}\), where \(\widetilde A\) is a tensor product of monogenic Hopf algebras which is isomorphic to \(A\), and showing it commutes with the \(\Sigma_n\)-action. Of course, this result facilitates the calculation of \(H^*(\Sigma_n,A^{\otimes n-1})\) by enabling us to use a simpler coproduct.