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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2605.05398 |
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| _version_ | 1866917480518123520 |
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| author | Nguyên, Hai Châu |
| author_facet | Nguyên, Hai Châu |
| contents | We study equivariant vector bundles over configuration spaces with diagonals included, viewed as orbifold quotients $M^n/\mathfrak{S}_n$ by permutation groups. Working in the equivalent language of equivariant vector bundles, we construct an induced-equivariance functor and prove its adjunction with restriction. We then define Hadamard and Cauchy tensor products and show that they form a symmetric $2$-monoidal structure. We construct the corresponding tensor and symmetric algebra bundles and prove that, for a local vector bundle $V \rightarrow M$, the bundle $\mathbf{S}^{\boxtimes} \big( \mathbf{S}^{\otimes}(V) \big)$ is the free commutative $2$-algebra generated by $V$. Finally, we show that any skew-symmetric bundle map $k : V \boxtimes V \rightarrow \mathbf{I}_{\otimes}$ induces a compatible Poisson bracket on this $2$-algebra bundle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05398 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivariant Poisson 2-Algebra Bundles over Configuration Spaces Nguyên, Hai Châu Mathematical Physics Category Theory Differential Geometry Quantum Algebra Symplectic Geometry 17B63 (Primary), 55R91, 55R80, 18M05, 18M60, 70S99, 16W25 (Secondary) We study equivariant vector bundles over configuration spaces with diagonals included, viewed as orbifold quotients $M^n/\mathfrak{S}_n$ by permutation groups. Working in the equivalent language of equivariant vector bundles, we construct an induced-equivariance functor and prove its adjunction with restriction. We then define Hadamard and Cauchy tensor products and show that they form a symmetric $2$-monoidal structure. We construct the corresponding tensor and symmetric algebra bundles and prove that, for a local vector bundle $V \rightarrow M$, the bundle $\mathbf{S}^{\boxtimes} \big( \mathbf{S}^{\otimes}(V) \big)$ is the free commutative $2$-algebra generated by $V$. Finally, we show that any skew-symmetric bundle map $k : V \boxtimes V \rightarrow \mathbf{I}_{\otimes}$ induces a compatible Poisson bracket on this $2$-algebra bundle. |
| title | Equivariant Poisson 2-Algebra Bundles over Configuration Spaces |
| topic | Mathematical Physics Category Theory Differential Geometry Quantum Algebra Symplectic Geometry 17B63 (Primary), 55R91, 55R80, 18M05, 18M60, 70S99, 16W25 (Secondary) |
| url | https://arxiv.org/abs/2605.05398 |