## Infinite Analysis Seminar Tokyo

Seminar information archive ～12/08｜Next seminar｜Future seminars 12/09～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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### 2023/12/15

13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)

**Laszlo Feher**(University of Szeged, Hungary)Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)

[ Abstract ]

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