Algebra & Number Theory
- Algebra Number Theory
- Volume 9, Number 3 (2015), 547-583.
The characteristic polynomial of the Adams operators on graded connected Hopf algebras
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.
Algebra Number Theory, Volume 9, Number 3 (2015), 547-583.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
Secondary: 16T30: Connections with combinatorics
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