Poisson–Nijenhuis structures on quiver path algebras

Abstract

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].

Modified double Poisson brackets

“…One of the major advantages of the double Poisson bracket as opposed to an H 0 -Poisson structure is that for a finitely generated associative algebra it is defined completely by its' action on generators. This allows one to provide numerous explicit examples of double Poisson brackets [PVdW08,BT16] and even carry out certain partial classification problems [ORS13].…”