#### Document Type

Article

#### Publication Date

2015

#### Publication Title

Algebra and Number Theory

#### Volume

9

#### Issue

3

#### Pages

547-583

#### Publisher Name

Mathematical Science Publishers

#### Abstract

The Adams operators ‰n on a Hopf algebra H are the convolution powers of the identity of H. They are also called Hopf powers or Sweedler powers. We study the Adams operators when H is graded connected. The main result is a complete description of the characteristic polynomial — both eigenvalues and their multiplicities — for the action of the operator ‰n on each homogeneous component of H. The eigenvalues are powers of n. The multiplicities are independent of n, and in fact only depend on the dimension sequence of H. These results apply in particular to the antipode of H, as the case n D 1. We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In the case where H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and for q-Hopf algebras.

#### Recommended Citation

Aguiar, Marcelo and Lauve, Aaron. The Characteristic Polynomial of the Adams Operators on Graded Connected Hopf Algebras. Algebra and Number Theory, 9, 3: 547-583, 2015. Retrieved from Loyola eCommons, Mathematics and Statistics: Faculty Publications and Other Works, http://dx.doi.org/10.2140/ant.2015.9.547

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

#### Copyright Statement

First published in ‘Algebra and Number Theory" in Volume 9, Issue 3, 2015, published by Mathematical Sciences Publishers. © Mathematical Science Publishers, 2015.

## Comments

Author posting. © Mathematical Science Publishers, 2015. This article is posted here by permission of MSP for personal use, not for redistribution. This article was published in Algebra and Number Theory, Volume 9, Issue 3, 2015. http://dx.doi.org/10.2140/ant.2015.9.547