Compatible Poisson structures on multiplicative quiver varieties
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915987750649856 |
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| author | Fairon, Maxime |
| author_facet | Fairon, Maxime |
| contents | Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa through quasi-Hamiltonian reduction. In this note, we include the Poisson structure as part of a pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of Hamiltonian quasi-Poisson structures which has dimension $\ell(\ell-1)/2$, where $\ell$ is the number of arrows in the underlying quiver. For each element of the pencil, we exhibit the corresponding compatible symplectic or quasi-Hamiltonian structure. We comment on analogous constructions for character varieties and quiver varieties. This formalism is applied to the spin Ruijsenaars-Schneider phase space in order to explain the compatibility of two Poisson structures that have recently appeared in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18751 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Compatible Poisson structures on multiplicative quiver varieties Fairon, Maxime Symplectic Geometry Mathematical Physics Exactly Solvable and Integrable Systems 53D20, 16G20, 70G55, 37J35 Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa through quasi-Hamiltonian reduction. In this note, we include the Poisson structure as part of a pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of Hamiltonian quasi-Poisson structures which has dimension $\ell(\ell-1)/2$, where $\ell$ is the number of arrows in the underlying quiver. For each element of the pencil, we exhibit the corresponding compatible symplectic or quasi-Hamiltonian structure. We comment on analogous constructions for character varieties and quiver varieties. This formalism is applied to the spin Ruijsenaars-Schneider phase space in order to explain the compatibility of two Poisson structures that have recently appeared in the literature. |
| title | Compatible Poisson structures on multiplicative quiver varieties |
| topic | Symplectic Geometry Mathematical Physics Exactly Solvable and Integrable Systems 53D20, 16G20, 70G55, 37J35 |
| url | https://arxiv.org/abs/2310.18751 |