On a family of quivers related to the Gibbons-Hermsen system

We introduce a family of quivers Zr (labeled by a natural number r≥1) and study the non-commutative symplectic geometry of the corresponding doubles Qr. We show that the group of non-commutative symplectomorphisms of the path algebra CQr contains two copies of the group GLr over a ring of polynomials in one indeterminate, and that a particular subgroup Pr (which contains both of these copies) acts on the completion Cn,r of the phase space of the n-particles, rank r Gibbons-Hermsen integrable system and connects each pair of points belonging to a certain dense open subset of Cn,r. This generalizes some known results for the cases r=1 and r=2.