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The
characteristic polynomial of the Adams operators on graded
connected Hopf algebras
Marcelo Aguiar and Aaron Lauve |
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Vol. 9 (2015), No. 3, 547–583
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Abstract |
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The Adams operators on a Hopf algebra are the convolution powers of the identity of . They are also called Hopf powers or Sweedler powers. We study the Adams operators when 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 on each homogeneous component of . The eigenvalues are powers of . The multiplicities are independent of , and in fact only depend on the dimension sequence of . These results apply in particular to the antipode of , as the case . 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 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 -Hopf algebras. |
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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
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Mathematical Subject Classification 2010
Primary: 16T05
Secondary: 16T30
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Milestones
Received: 27 March 2014
Revised: 23 October 2014
Accepted: 1 December 2014
Published: 17 April 2015
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Authors |
| Marcelo Aguiar | |
| Department of Mathematics Cornell University Ithaca, NY 14853 United States |
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| Aaron Lauve | |
| Department of Mathematics and
Statistics Loyola University Chicago Chicago, IL 60660 United States |
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