We introduce a family of quivers $Z_{ r}$ (labeled by a natural number $r\geq 1$) and study the non-commutative symplectic geometry of the corresponding doubles $\mathbf{ Q}_{ r}$. We show that the group of non-commutative symplectomorphisms of the path algebra $\mathbb{ C}\mathbf{ Q}_{ r}$ contains two copies of the group $\mathrm{ GL}_{ r}$ over a ring of polynomials in one indeterminate, and that a particular subgroup $\mathcal{ P}_{ r}$ (which contains both of these copies) acts on the completion $\mathcal{ C}_{ n,r}$ 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 $\mathcal{ C}_{ n,r}$. This generalizes some known results for the cases $r=1$ and $r=2$.
Categories: math.SG math-ph math.MP