Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912678357762048 |
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| author | Buhovsky, Lev Stokić, Maksim |
| author_facet | Buhovsky, Lev Stokić, Maksim |
| contents | On an open, connected symplectic manifold $(M,ω)$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fréchet Lie group with Lie algebra $C^{\infty}_c(M)$ and adjoint action given by pullbacks. We prove that this action is flexible: for any non-constant $u \in C^{\infty}(M)$, every $f \in C^{\infty}_c(M)$ can be expressed as a weighted finite sum of elements from the adjoint orbit of $u$, with total weight bounded by constant multiple of $\|f\|_{\infty} + \|f\|_{L^1}$. Consequently, all $\mathrm{Ham}(M,ω)$-invariant norms on $C^{\infty}_c(M)$ are dominated by a sum of $L^{\infty}$ and $L^1$ norms. As an application, we classify up to equivalence all bi-invariant pseudo-metrics on the group of Hamiltonian diffeomorphisms of an exact symplectic manifold, answering a question of Eliashberg and Polterovich. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics Buhovsky, Lev Stokić, Maksim Symplectic Geometry 53D05, 22E65, 58D19 On an open, connected symplectic manifold $(M,ω)$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fréchet Lie group with Lie algebra $C^{\infty}_c(M)$ and adjoint action given by pullbacks. We prove that this action is flexible: for any non-constant $u \in C^{\infty}(M)$, every $f \in C^{\infty}_c(M)$ can be expressed as a weighted finite sum of elements from the adjoint orbit of $u$, with total weight bounded by constant multiple of $\|f\|_{\infty} + \|f\|_{L^1}$. Consequently, all $\mathrm{Ham}(M,ω)$-invariant norms on $C^{\infty}_c(M)$ are dominated by a sum of $L^{\infty}$ and $L^1$ norms. As an application, we classify up to equivalence all bi-invariant pseudo-metrics on the group of Hamiltonian diffeomorphisms of an exact symplectic manifold, answering a question of Eliashberg and Polterovich. |
| title | Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics |
| topic | Symplectic Geometry 53D05, 22E65, 58D19 |
| url | https://arxiv.org/abs/2510.26590 |