The Adams operators on connected graded Hopf algebras

Abstract

The Adams operators on a Hopf algebra H are the convolution powers of the identity map of H. They are also called Hopf powers or Sweedler powers. It is a natural family of operators on H that contains the antipode. We study the linear properties of the Adams operators when $H={⨁}_{m\in N}{H}_m$ is connected graded. The main result is that for any of such H, there exist a PBW type homogeneous basis and a natural total order on it such that the restrictions ${\Psi }_n{|}_{H_m}$ of the Adams operators are simultaneously upper triangularizable with respect to this ordered basis. Moreover, the diagonal coefficients are determined in terms of n and a combinatorial number assigned to the basis elements. As an immediate consequence, we obtain a complete description of the characteristic polynomial of ${\Psi }_n{|}_{H_m}$, both on eigenvalues and their multiplicities, when H is locally finite and the base field is of characteristic zero. It recovers the main result of the paper [2] by Aguiar and Lauve, where the approach is different from ours.