A Note on the Automorphism Group of the Bielawski-Pidstrygach Quiver

TL;DR

This paper establishes a connection between a group introduced by Wilson and a quotient of the tame symplectic automorphism group of a quiver's path algebra, revealing its transitive action on the Gibbons-Hermsen phase space.

Contribution

It demonstrates a morphism linking Wilson's group to a quotient of the tame automorphism group, and shows the generated subgroup acts transitively on the phase space of the integrable system.

Findings

Existence of a morphism between Wilson's group and a quotient of the tame automorphism group.

The subgroup generated acts transitively on the phase space of the Gibbons-Hermsen system.

The automorphism subgroup can send any regular, semisimple point to any other.

Abstract

We show that there exists a morphism between a group $\Gamma^{\mathrm{alg}}$ introduced by G. Wilson and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by Bielawski and Pidstrygach. The latter is known to act transitively on the phase space $\mathcal{C}_{n,2}$ of the Gibbons-Hermsen integrable system of rank 2, and we prove that the subgroup generated by the image of $\Gamma^{\mathrm{alg}}$ together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of $\mathcal{C}_{n,2}$, the subgroup contains an element sending the first point to the second.