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The Adams operators on connected graded Hopf algebras
Rings and Algebras · Submitted 21 Feb 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main result is that for any of such $H$, there exist a PBW type homogeneous basis and a natural total order on it such that the restrictions $\Psi_n|_{H_m}$ of the Adams operators are simultaneously upper triangularizable with respect to this ordered basis.
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The Adams operators on a Hopf algebra $H$ are the convolution powers of the identity map of $H$. They are also called Hopf powers or Sweedler powers. It is a natural family of operators on $H$ that contains the antipode. We study the linear properties of the Adams operators when $H=\bigoplus_{m\in \mathbb{N}} H_m$ is connected graded. The main result is that for any of such $H$, there exist a PBW type homogeneous basis and a natural total order on it such that the restrictions $\Psi_n|_{H_m}$ of the Adams operators are simultaneously upper triangularizable with respect to this ordered basis. Moreover, the diagonal coefficients are determined in terms of $n$ and a combinatorial number assigned to the basis elements. As an immediate consequence, we obtain a complete description of the characteristic polynomial of $\Psi_n|_{H_m}$, both on eigenvalues and their multiplicities, when $H$ is locally finite and the base field is of characteristic zero. It recovers the main result of the paper [2] by Aguiar and Lauve, where the approach is different from ours.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Adams operators on a Hopf algebra $H$ are the convolution powers of the identity map of $H$. | background 1.00 | 0.07 | 0.02 | asserted 1.00 |
| s1 | They are also called Hopf powers or Sweedler powers. | background 1.00 | 0.05 | 0.01 | asserted 1.00 |
| s2 | It is a natural family of operators on $H$ that contains the antipode. | background 1.00 | 0.09 | 0.02 | asserted 1.00 |
| s3 | We study the linear properties of the Adams operators when $H=\bigoplus_{m\in \mathbb{N}} H_m$ is connected graded. | contribution 0.94 | 0.70 | 0.03 | asserted 0.92 |
| s4 | The main result is that for any of such $H$, there exist a PBW type homogeneous basis and a natural total order on it such that the restrictions $\Psi_n|_{H_m}$ of the Adams operators are simultaneously upper triangularizable with respect to this ordered basis. | result 0.99 | 0.94 | 0.98 | asserted 1.00 |
| s5 | Moreover, the diagonal coefficients are determined in terms of $n$ and a combinatorial number assigned to the basis elements. | result 1.00 | 0.93 | 0.06 | asserted 1.00 |
| s6 | As an immediate consequence, we obtain a complete description of the characteristic polynomial of $\Psi_n|_{H_m}$, both on eigenvalues and their multiplicities, when $H$ is locally finite and the base field is of characteristic zero. | result 0.88 | 0.92 | 0.07 | asserted 1.00 |
| s7 | It recovers the main result of the paper [2] by Aguiar and Lauve, where the approach is different from ours. | interpretation 0.67 | 0.27 | 0.03 | asserted 1.00 |
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Original abstract
The Adams operators on a Hopf algebra $H$ are the convolution powers of the identity map of $H$. They are also called Hopf powers or Sweedler powers. It is a natural family of operators on $H$ that contains the antipode. We study the linear properties of the Adams operators when $H=\bigoplus_{m\in \mathbb{N}} H_m$ is connected graded. The main result is that for any of such $H$, there exist a PBW type homogeneous basis and a natural total order on it such that the restrictions $\Psi_n|_{H_m}$ of the Adams operators are simultaneously upper triangularizable with respect to this ordered basis. Moreover, the diagonal coefficients are determined in terms of $n$ and a combinatorial number assigned to the basis elements. As an immediate consequence, we obtain a complete description of the characteristic polynomial of $\Psi_n|_{H_m}$, both on eigenvalues and their multiplicities, when $H$ is locally finite and the base field is of characteristic zero. It recovers the main result of the paper [2] by Aguiar and Lauve, where the approach is different from ours.
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Hopf Algebras in Combinatorics · Applications to the Hopf algebra of permutations · 1.00
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