On compatible linear and quadratic Poisson brackets on $gl(N)$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915520200048640 |
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| author | Panasyuk, Andriy Skrypnyk, Taras |
| author_facet | Panasyuk, Andriy Skrypnyk, Taras |
| contents | In the present paper, using two constant tensors $c$ and $b$ on $sl(N)\otimes sl(N)$ satisfying certain linear-quadratic equation and a technique of Poisson bivectors and Schouten brackets, we explicitly construct quadratic Poisson bracket on the space $ (sl(N)+\mathbb{C})^*$ which is compatible with the standard Lie--Poisson bracket on $gl(N)^* \simeq (sl(N)\oplus \mathbb{C})^*$. The case of $N=3$ is considered in details. The relation of the proposed brackets with the generalized classical Sklyanin algebras is explained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On compatible linear and quadratic Poisson brackets on $gl(N)$ Panasyuk, Andriy Skrypnyk, Taras Mathematical Physics Differential Geometry 37J35, 14H70 In the present paper, using two constant tensors $c$ and $b$ on $sl(N)\otimes sl(N)$ satisfying certain linear-quadratic equation and a technique of Poisson bivectors and Schouten brackets, we explicitly construct quadratic Poisson bracket on the space $ (sl(N)+\mathbb{C})^*$ which is compatible with the standard Lie--Poisson bracket on $gl(N)^* \simeq (sl(N)\oplus \mathbb{C})^*$. The case of $N=3$ is considered in details. The relation of the proposed brackets with the generalized classical Sklyanin algebras is explained. |
| title | On compatible linear and quadratic Poisson brackets on $gl(N)$ |
| topic | Mathematical Physics Differential Geometry 37J35, 14H70 |
| url | https://arxiv.org/abs/2509.24054 |