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

Aguiar, Marcelo; Lauve, Aaron

doi:10.2140/ant.2015.9.547

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

Marcelo Aguiar and Aaron Lauve

Vol. 9 (2015), No. 3, 547–583

DOI: 10.2140/ant.2015.9.547

Abstract

The Adams operators $Ψ_{n}$ 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 $Ψ_{n}$ 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 $n = - 1$ . 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.

Dedicated to the memory of Jean-Louis Loday.

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

Mathematical Subject Classification 2010

Primary: 16T05

Secondary: 16T30

Milestones

Received: 27 March 2014

Revised: 23 October 2014

Accepted: 1 December 2014

Published: 17 April 2015

Authors

Marcelo Aguiar
Department of Mathematics Cornell University Ithaca, NY 14853 United States

Aaron Lauve
Department of Mathematics and Statistics Loyola University Chicago Chicago, IL 60660 United States

Vol. 9, No. 3, 2015