arXiv subjects
Publications and source records attributed to Alberto Tacchella.
We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero-Moser and Gibbons-Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in [3].
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory associative geometry. We argue that this formalism sheds a new light on some classic solution methods in the theory of finite-dimensional integrable dynamical systems.
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 $r=2$.
We show that there exists a morphism between a group $Γ^{\mathrm{alg}}$ 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 $Γ^{\mathrm{alg}}$ 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.
We derive some rational solutions for the multicomponent and matrix KP hierarchies generalising an approach by Wilson. Connections with the multicomponent version of the KP/CM correspondence are discussed.