The Adams operators on connected graded Hopf algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917822655889408 |
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| author | Li, Y. -Y. Zhou, G. -S. |
| author_facet | Li, Y. -Y. Zhou, G. -S. |
| contents | 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 $Ψ_n|_{H_m}$ 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 $Ψ_n|_{H_m}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Adams operators on connected graded Hopf algebras Li, Y. -Y. Zhou, G. -S. Rings and Algebras 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 $Ψ_n|_{H_m}$ 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 $Ψ_n|_{H_m}$, 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. |
| title | The Adams operators on connected graded Hopf algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2402.13774 |