Observables of Relative Structures and Lie 2-algebras associated with Quasi-Hamiltonian $G$-spaces
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918138389463040 |
|---|---|
| author | Djounvouna, Dinamo |
| author_facet | Djounvouna, Dinamo |
| contents | A manifold is said to be $n$-plectic if it is equipped with a closed, nondegenerate $(n+1)$-form. This thesis develops the theory of \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed, nondegenerate \emph{relative} $(n+1)$-form defined with respect to a smooth map. Analogous to how $n$-plectic manifolds give rise to $L_\infty$-algebras of observables, we show that relative $n$-plectic structures naturally induce corresponding $L_\infty$-algebras. These structures provide a conceptual bridge between the frameworks of quasi-Hamiltonian $G$-spaces and $2$-plectic geometry.
As an application, we examine the relative $2$-plectic structure canonically associated to quasi-Hamiltonian $G$-spaces. We show that every quasi-Hamiltonian $G$-space defines a closed, nondegenerate relative $3$-form, and that the group action induces a Hamiltonian infinitesimal action compatible with this structure. We then construct explicit homotopy moment maps as $L_\infty$-morphisms from the Lie algebra $\mathfrak{g}$ into the Lie $2$-algebra of relative observables, extending the moment map formalism to the higher and relative geometric setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08153 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Observables of Relative Structures and Lie 2-algebras associated with Quasi-Hamiltonian $G$-spaces Djounvouna, Dinamo Symplectic Geometry A manifold is said to be $n$-plectic if it is equipped with a closed, nondegenerate $(n+1)$-form. This thesis develops the theory of \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed, nondegenerate \emph{relative} $(n+1)$-form defined with respect to a smooth map. Analogous to how $n$-plectic manifolds give rise to $L_\infty$-algebras of observables, we show that relative $n$-plectic structures naturally induce corresponding $L_\infty$-algebras. These structures provide a conceptual bridge between the frameworks of quasi-Hamiltonian $G$-spaces and $2$-plectic geometry. As an application, we examine the relative $2$-plectic structure canonically associated to quasi-Hamiltonian $G$-spaces. We show that every quasi-Hamiltonian $G$-space defines a closed, nondegenerate relative $3$-form, and that the group action induces a Hamiltonian infinitesimal action compatible with this structure. We then construct explicit homotopy moment maps as $L_\infty$-morphisms from the Lie algebra $\mathfrak{g}$ into the Lie $2$-algebra of relative observables, extending the moment map formalism to the higher and relative geometric setting. |
| title | Observables of Relative Structures and Lie 2-algebras associated with Quasi-Hamiltonian $G$-spaces |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2509.08153 |