Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912155578662912 |
|---|---|
| author | Schipper, Floor Brown, Mollie S Jagoe Kiselev, Arthemy V |
| author_facet | Schipper, Floor Brown, Mollie S Jagoe Kiselev, Arthemy V |
| contents | Kontsevich constructed a map between `good' graph cocycles $γ$ and infinitesimal deformations of Poisson bivectors on affine manifolds, that is, Poisson cocycles in the second Lichnerowicz--Poisson cohomology. For the tetrahedral graph cocycle $γ_3$ and for the class of Nambu-determinant Poisson bivectors $P$ over $\mathbb{R}^2$, $\mathbb{R}^3$ and $\mathbb{R}^4$, we know the fact of trivialization, $\dot{P}=[[ P, \vec{X}^{γ_3}_{\text{dim}}]]$, by using dimension-dependent vector fields $\vec{X}^{γ_3}_{\text{dim}}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields $\vec{X}^{γ_3}_{\text{dim}}$ are unique modulo Hamiltonian vector fields $\vec{X}_{H}=d_P(H)= [[ P, H]]$, where $d_P$ is the Lichnerowicz--Poisson differential and where the Hamiltonians $H$ are also represented by Kontsevich (micro-)graphs. However, we find that the choice of Kontsevich (micro-)graphs to represent the aforementioned multivectors is not unique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects Schipper, Floor Brown, Mollie S Jagoe Kiselev, Arthemy V Quantum Algebra 53D55, 17B63, 05C22, 68R10, 53D17 Kontsevich constructed a map between `good' graph cocycles $γ$ and infinitesimal deformations of Poisson bivectors on affine manifolds, that is, Poisson cocycles in the second Lichnerowicz--Poisson cohomology. For the tetrahedral graph cocycle $γ_3$ and for the class of Nambu-determinant Poisson bivectors $P$ over $\mathbb{R}^2$, $\mathbb{R}^3$ and $\mathbb{R}^4$, we know the fact of trivialization, $\dot{P}=[[ P, \vec{X}^{γ_3}_{\text{dim}}]]$, by using dimension-dependent vector fields $\vec{X}^{γ_3}_{\text{dim}}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields $\vec{X}^{γ_3}_{\text{dim}}$ are unique modulo Hamiltonian vector fields $\vec{X}_{H}=d_P(H)= [[ P, H]]$, where $d_P$ is the Lichnerowicz--Poisson differential and where the Hamiltonians $H$ are also represented by Kontsevich (micro-)graphs. However, we find that the choice of Kontsevich (micro-)graphs to represent the aforementioned multivectors is not unique. |
| title | Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects |
| topic | Quantum Algebra 53D55, 17B63, 05C22, 68R10, 53D17 |
| url | https://arxiv.org/abs/2409.15932 |