SYMMETRIC GROUP ACTIONS ON TENSOR PRODUCTS OF HOPF ALGEBROIDS

Abstract

We describe an action of the symmetric group Σ _n on A ^{⊗ n − 1}, the n − 1-fold tensor product of A over K, for (K,A) a Hopf algebroid. This arises in a natural way in stable homotopy theory: when A = E _* E, the ‘co-operations’ in the cohomology theory associated to a suitable ring spectrum E, this action is induced from the natural action on the n-fold smash product E ⁽ⁿ⁾. The case n = 2 is classical: the switch action of Σ₂ on E ∧ E induces the canonical conjugation of E _* E. Therefore we may think of the symmetric group actions as ‘higher order conjugation maps’.

ACKNOWLEDGMENTS

I would like to thank Haynes Miller for helpful comments. I acknowledge the support of a TMR grant from the European Union, held at the Laboratoire d'Analyse, Géometrie et Applications (UMR 7539 au CNRS), Université Paris-Nord.