From double brackets to integrable systems

Maxime Fairon (University of Leeds)

Tuesday 5th March, 2019 16:00-17:00 Maths 311B

Abstract

Double brackets were introduced by M. Van den Bergh in his successful attempt to understand the Poisson geometry of (multiplicative) quiver varieties directly at the level of the path algebra of quivers. I will begin with a review of the basics of this theory and its relation to usual geometric structures. As a first application, I will explain how the double Poisson bracket on the path algebra of an extended Jordan (or one-loop) quiver can be used to easily derive integrable systems of Calogero-Moser type. As a second application, I will explain the corresponding relation between double quasi-Poisson brackets and  Ruijsenaars-Schneider systems based on recent works with O. Chalykh (Leeds).

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