The Adams operators on connected graded Hopf algebras

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 $H={⨁}_{m\in \mathbb{N}}{H}_m$ 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 ${\mathrm{\Psi }}_n{\mathrm{\mid }}_{H_m}$ 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 ${\mathrm{\Psi }}_n{\mathrm{\mid }}_{H_m}$, 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.