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2015 The characteristic polynomial of the Adams operators on graded connected Hopf algebras
Marcelo Aguiar, Aaron Lauve
Algebra Number Theory 9(3): 547-583 (2015). DOI: 10.2140/ant.2015.9.547

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.

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Marcelo Aguiar. Aaron Lauve. "The characteristic polynomial of the Adams operators on graded connected Hopf algebras." Algebra Number Theory 9 (3) 547 - 583, 2015. https://doi.org/10.2140/ant.2015.9.547

Information

Received: 27 March 2014; Revised: 23 October 2014; Accepted: 1 December 2014; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1330.16020
MathSciNet: MR3340544
Digital Object Identifier: 10.2140/ant.2015.9.547

Subjects:
Primary: 16T05
Secondary: 16T30

Rights: Copyright © 2015 Mathematical Sciences Publishers

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Vol.9 • No. 3 • 2015
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