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

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 $T^∗U(n)$ and its holomorphic analogue $T^∗GL(n,\mathbb C)$, as well as $T^∗GL(n,\mathbb C)_{\mathbb R}$, 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 $T^∗U(n)$ and $T^∗GL(n,\mathbb C)$ 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 $T^∗GL(n,\mathbb C)_{\mathbb R}$ by the group $U(n) \times U(n)$ gives a generalized Sutherland model coupled to two $u(n)^∗$-valued spins. We also show that a bi-Hamiltonian structure on the associative algebra $gl(n,\mathbb R)$ that appeared in the context of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^∗GL(n,\mathbb R)$. Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles, without realizing the bi-Hamiltonian aspect.

References

[1] L. Feher, Reduction of a bi-Hamiltonian hierarchy on $T^∗U(n)$ 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 $u(n)^∗$-valued spins from Poisson reduction, Nonlinearity 35, 2971-3003 (2022).

[4] L. Feher and B. Juhasz, A note on quadratic Poisson brackets on $gl(n,\mathbb R)$ related to Toda lattices, Lett. Math. Phys. 112:45 (2022).

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

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BIMSA Integrable Systems Seminar

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