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

Abstract

We introduce a family of quivers $Z_r$ (labeled by a natural number $r\ge 1$) 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 $r=2$.

29 pages. v3: keeps some introductory material left out of the journal version