A classification of coadjoint orbits carrying Gibbs ensembles
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915716679073792 |
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| author | Neeb, Karl-Hermann |
| author_facet | Neeb, Karl-Hermann |
| contents | A coadjoint orbit $O_λ\subseteq {\mathfrak g}^*$ of a Lie group $G$ is said to carry a Gibbs ensemble if the set of all $x \in {\mathfrak g}$, for which the function
$α\mapsto e^{-α(x)}$ on the orbit is integrable with respect to the
Liouville measure, has non-empty interior $Ω_λ$.
We describe a classification of all coadjoint orbits of finite-dimensional
Lie algebras with this property. In the context of Souriau's
Lie group thermodynamics, the subset $Ω_λ$
is the geometric temperature, a parameter space for a family
of Gibbs measures on the coadjoint orbit. The
corresponding Fenchel--Legendre transform maps
$Ω_λ/{\mathfrak z}({\mathfrak g})$ diffeomorphically
onto the interior of the convex hull of the coadjoint orbit
$O_λ$. This provides an interesting perspective on the
underlying information geometry.
We also show that already
the integrability of $e^{-α(x)}$ for one $x \in {\mathfrak g}$ implies
that $Ω_λ\not=\emptyset$ and that, for general Hamiltonian
actions, the existence of Gibbs measures implies that the range
of the momentum maps consists of coadjoint orbits $O_λ$ as above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04934 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A classification of coadjoint orbits carrying Gibbs ensembles Neeb, Karl-Hermann Symplectic Geometry Mathematical Physics Primary 37J37, Secondary 22F30, 53D20, 58F05, 70H33, 82B05 A coadjoint orbit $O_λ\subseteq {\mathfrak g}^*$ of a Lie group $G$ is said to carry a Gibbs ensemble if the set of all $x \in {\mathfrak g}$, for which the function $α\mapsto e^{-α(x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $Ω_λ$. We describe a classification of all coadjoint orbits of finite-dimensional Lie algebras with this property. In the context of Souriau's Lie group thermodynamics, the subset $Ω_λ$ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps $Ω_λ/{\mathfrak z}({\mathfrak g})$ diffeomorphically onto the interior of the convex hull of the coadjoint orbit $O_λ$. This provides an interesting perspective on the underlying information geometry. We also show that already the integrability of $e^{-α(x)}$ for one $x \in {\mathfrak g}$ implies that $Ω_λ\not=\emptyset$ and that, for general Hamiltonian actions, the existence of Gibbs measures implies that the range of the momentum maps consists of coadjoint orbits $O_λ$ as above. |
| title | A classification of coadjoint orbits carrying Gibbs ensembles |
| topic | Symplectic Geometry Mathematical Physics Primary 37J37, Secondary 22F30, 53D20, 58F05, 70H33, 82B05 |
| url | https://arxiv.org/abs/2601.04934 |