Braid group symmetries on Poisson algebras arising from quantum symmetric pairs
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909686977003520 |
|---|---|
| author | Song, Jinfeng Zhang, Weinan |
| author_facet | Song, Jinfeng Zhang, Weinan |
| contents | Let $(\mathrm{U},\mathrm{U}^\imath)$ be the quantum symmetric pair of arbitrary finite type and $G^*$ be the associated dual Poisson-Lie group. Generalizing the work of De Concini and Procesi, the first author introduced an integral form for the $\imath$quantum group $\mathrm{U}^\imath$ and its semi-classical limit was shown to be the coordinate algebra for a Poisson homogeneous space of $G^*$. In this paper, we establish (relative) braid group symmetries and PBW bases on this integral form of $\mathrm{U}^\imath$. By taking the semi-classical limit, we obtain braid group symmetries and polynomial generators on the associated Poisson algebra. These symmetries further allow us to describe the Poisson brackets explicitly. Examples of such Poisson structures include Dubrovin-Ugaglia Poisson brackets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Braid group symmetries on Poisson algebras arising from quantum symmetric pairs Song, Jinfeng Zhang, Weinan Quantum Algebra Representation Theory Symplectic Geometry Primary 17B37, 17B63 Let $(\mathrm{U},\mathrm{U}^\imath)$ be the quantum symmetric pair of arbitrary finite type and $G^*$ be the associated dual Poisson-Lie group. Generalizing the work of De Concini and Procesi, the first author introduced an integral form for the $\imath$quantum group $\mathrm{U}^\imath$ and its semi-classical limit was shown to be the coordinate algebra for a Poisson homogeneous space of $G^*$. In this paper, we establish (relative) braid group symmetries and PBW bases on this integral form of $\mathrm{U}^\imath$. By taking the semi-classical limit, we obtain braid group symmetries and polynomial generators on the associated Poisson algebra. These symmetries further allow us to describe the Poisson brackets explicitly. Examples of such Poisson structures include Dubrovin-Ugaglia Poisson brackets. |
| title | Braid group symmetries on Poisson algebras arising from quantum symmetric pairs |
| topic | Quantum Algebra Representation Theory Symplectic Geometry Primary 17B37, 17B63 |
| url | https://arxiv.org/abs/2507.09456 |