Algebra & Number Theory

The characteristic polynomial of the Adams operators on graded connected Hopf algebras

Marcelo Aguiar and Aaron Lauve

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Abstract

The Adams operators Ψn 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 Ψn 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 = 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.

Article information

Source
Algebra Number Theory, Volume 9, Number 3 (2015), 547-583.

Dates
Received: 27 March 2014
Revised: 23 October 2014
Accepted: 1 December 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842311

Digital Object Identifier
doi:10.2140/ant.2015.9.547

Mathematical Reviews number (MathSciNet)
MR3340544

Zentralblatt MATH identifier
1330.16020

Subjects
Primary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
Secondary: 16T30: Connections with combinatorics

Keywords
Adams operator characteristic operation convolution power Hopf power antipode trace graded connected Hopf algebra Hopf monoid in species $q$-Hopf algebra Schur indicator Eulerian idempotent

Citation

Aguiar, Marcelo; Lauve, Aaron. The characteristic polynomial of the Adams operators on graded connected Hopf algebras. Algebra Number Theory 9 (2015), no. 3, 547--583. doi:10.2140/ant.2015.9.547. https://projecteuclid.org/euclid.ant/1510842311


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