Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models

Maxime Fairon, University of Leeds. Part of the integrable systems seminars series.

It has been observed by Van den Bergh that given an associative algebra, it is possible to define on it a structure called a double Poisson bracket, such that it induces a Poisson bracket on the corresponding representation spaces of the algebra. In particular, this can be defined for the preprojective algebra of an arbitrary quiver, and it yields the standard symplectic form on the corresponding quiver varieties. In his work, he also introduced the concept of a double quasi-Poisson bracket, and showed that the multiplicative preprojective algebra of an arbitrary quiver can be endowed with such a structure, defining the so-called multiplicative quiver varieties from its representations. My aim is to explain what are the different notions that appear both at the algebraic and at the geometric levels, and see their relation to the theory of integrable systems.

As an application, I will describe how the RS model can be recovered from the one-loop quiver with a simple framing, and how to get new generalisations by looking at cyclic quivers. I will also explain how this construction can be naturally adapted to spin versions. In particular, I will outline how this formalism gives the complete set of Poisson brackets for the (complex) trigonometric spin RS model, answering a problem posed by Arutyunov and Frolov about 20 years ago.

This is joint work with Oleg Chalykh.

Maxime Fairon, University of Leeds