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| 開催情報 | 土曜日 13:30～16:00 数理科学研究科棟(駒場) 117号室 |
|---|---|
| 担当者 | 神保道夫、国場敦夫、山田裕二、武部尚志、高木太一郎、白石潤一 |
| セミナーURL | https://www.ms.u-tokyo.ac.jp/~takebe/iat/index-j.html |
2023年12月15日(金)
13:00-14:30 数理科学研究科棟(駒場) 056号室
Laszlo Feher 氏 (University of Szeged, Hungary)
Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)
Laszlo Feher 氏 (University of Szeged, Hungary)
Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)
[ 講演概要 ]
We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models
built on collective spin variables.
Our basic observation was that the cotangent bundle $T^*\mathrm{U}(n)$ and its holomorphic analogue $T^* \mathrm{GL}(n,{\mathbb C})$,
as well as $T^*\mathrm{GL}(n,{\mathbb C})_{\mathbb R}$, carry a natural quadratic Poisson bracket,
which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation
from an associated Heisenberg double.
Then, the reductions of $T^*{\mathrm{U}}(n)$ and $T^*{\mathrm{GL}}(n,{\mathbb C})$ by the conjugation actions of the
corresponding groups lead to the real and holomorphic spin Sutherland models, respectively, equipped
with a bi-Hamiltonian structure. The reduction of $T^*{\mathrm{GL}}(n,{\mathbb C})_{\mathbb R}$ by the group $\mathrm{U}(n) \times \mathrm{U}(n)$ gives
a generalized Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins.
We also show that
a bi-Hamiltonian structure on the associative algebra ${\mathfrak{gl}}(n,{\mathbb R})$ that appeared in the context
of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*{\mathrm{GL}}(n,{\mathbb R})$.
Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles,
without realizing the bi-Hamiltonian aspect.
Finally, if time permits, the degenerate integrability of some of the reduced systems
will be explained as well.
[1] L. Feher, Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$
to spin Ruijsenaars--Sutherland models, Lett. Math. Phys. 110, 1057-1079 (2020).
[2] L. Feher, Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case, Ann. Henri Poincar\'e 22, 4063-4085 (2021).
[3] L. Feher, Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n)^*$-valued spins from Poisson reduction,
Nonlinearity 35, 2971-3003 (2022).
[4] L. Feher and B. Juhasz,
A note on quadratic Poisson brackets on $\mathfrak{gl}(n,\mathbb{R})$ related to Toda lattices,
Lett. Math. Phys. 112:45 (2022).
[5] L. Feher,
Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of
compact Lie groups, arXiv:2309.16245
We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models
built on collective spin variables.
Our basic observation was that the cotangent bundle $T^*\mathrm{U}(n)$ and its holomorphic analogue $T^* \mathrm{GL}(n,{\mathbb C})$,
as well as $T^*\mathrm{GL}(n,{\mathbb C})_{\mathbb R}$, carry a natural quadratic Poisson bracket,
which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation
from an associated Heisenberg double.
Then, the reductions of $T^*{\mathrm{U}}(n)$ and $T^*{\mathrm{GL}}(n,{\mathbb C})$ by the conjugation actions of the
corresponding groups lead to the real and holomorphic spin Sutherland models, respectively, equipped
with a bi-Hamiltonian structure. The reduction of $T^*{\mathrm{GL}}(n,{\mathbb C})_{\mathbb R}$ by the group $\mathrm{U}(n) \times \mathrm{U}(n)$ gives
a generalized Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins.
We also show that
a bi-Hamiltonian structure on the associative algebra ${\mathfrak{gl}}(n,{\mathbb R})$ that appeared in the context
of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*{\mathrm{GL}}(n,{\mathbb R})$.
Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles,
without realizing the bi-Hamiltonian aspect.
Finally, if time permits, the degenerate integrability of some of the reduced systems
will be explained as well.
[1] L. Feher, Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$
to spin Ruijsenaars--Sutherland models, Lett. Math. Phys. 110, 1057-1079 (2020).
[2] L. Feher, Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case, Ann. Henri Poincar\'e 22, 4063-4085 (2021).
[3] L. Feher, Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n)^*$-valued spins from Poisson reduction,
Nonlinearity 35, 2971-3003 (2022).
[4] L. Feher and B. Juhasz,
A note on quadratic Poisson brackets on $\mathfrak{gl}(n,\mathbb{R})$ related to Toda lattices,
Lett. Math. Phys. 112:45 (2022).
[5] L. Feher,
Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of
compact Lie groups, arXiv:2309.16245
