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BIRS Workshop Lecture Videos

Discrete projective polygons and Hamiltonian structures Mari-Beffa, Gloria


Decades ago Semenov-Tian-Shansky defined the so-called twisted Poisson structure on $G^N$, where $G$ is a Poisson Lie group. We will review the definition and show that in the case $G = PSL(n+1)$, it can be reduced to the moduli space of projective polygons (under the projective action), as defined by the discrete projective curvatures. We will also show that any reduced Hamiltonian evolution is induced on the curvatures by a simple polygon evolution that can be defined directly from the variation of the Hamiltonian function. We will also define a second structure reducing a right invariant tensor, which is proven to be compatible with the previous reduction for dimensions 2 and 3, and conjectured to be for any dimension. The pair are Hamiltonian structures for integrable discretizations of W_n algebras in any dimension. Thus, one can write a very simple realization of this integrable system as an evolution of projective polygons. This is joint work with Jing Ping Wang.

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