Poisson reduction of the space of polygons

A family of Poisson structures, parametrised by an arbitrary odd periodic function $ϕ$, is defined on the space $\cW$ of twisted polygons in $\RR^ν$. Poisson reductions with respect to two Poisson group actions on $\cW$ are described. The $ν=2$ and $ν=3$ cases are discussed in detail and the general $ν$ case in less detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice structure and some extended Toda lattice structures. A general result is proved showing that, for any $ν$, to certain concrete choices of $ϕ$ there correspond compatible Poisson structures which generate all the extended bigraded Toda hierarchies of a suitable size.

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