Quantum R-matrix identities and Interacting Integrable Tops

Abstract: Integrability of classical integrable systems, for example, multi-particle Calogero–Moser system, is based on some functional identities on rational, trigonometric, or elliptic functions, which ensure the existence of Lax pair and the Poisson commutativity of integrals of motion. It appears that some quantum R-matrices satisfy the matrix analogues of the relations, known as associative Yang–Baxter equation and its degenerations. This fact allows us to use such quantum R-matrices in Lax pairs instead of scalar functions and construct new classical integrable systems.

I will describe the example of the application of quantum R-matrices relations in classical integrability, introducing the system of interacting integrable tops, generalizing both Calogero–Moser systems of particles and Euler tops. I will also show how the resulting integrable structures simultaneously contain the properties of particle and top systems. If time permits, I briefly discuss the quantization of these structures, in the elliptic case it leads to quadratic quantum algebras which generalize both Sklyanin algebra and Felder elliptic quantum group.

mathematical physicsalgebraic geometrydifferential geometrydynamical systemssymplectic geometry

Audience: researchers in the topic

Junior Global Poisson Workshop II