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

We introduce a family of quivers Z(r) (labeled by a natural number r >= 1) and study the non-commutative symplectic geometry of the corresponding doubles Q(r). We show that the group of non-commutative symplectomorphisms of the path algebra CQ(r) contains two copies of the group GL(r) over a ring of polynomials in one indeterminate, and that a particular subgroup P-r (which contains both of these copies) acts on the completion e(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 e(n,r). This generalizes some known results for the cases r = 1 and r = 2. (C) 2015 Elsevier B.V. All rights reserved. (AU)