Letters in Mathematical Physics

, Volume 107, Issue 7, pp 1265–1291 | Cite as

Poisson–Nijenhuis structures on quiver path algebras

  • Claudio Bartocci
  • 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 Bartocci et al. (Int Math Res Not 2010:279–296, 2010. arXiv:0902.0953).


Quiver representations Bihamiltonian structures Integrable systems Calogero-Moser system 

Mathematics Subject Classification

16G20 53D17 70H06 



This work was partially supported by the PRIN “Geometria delle varietà algebriche” and by the University of Genoa’s research Grant “Aspetti matematici della teoria dei campi interagenti e quantizzazione per deformazione.” The authors are grateful to the referee, whose suggestions have been incorporated in Sect. 4. A.T. would like to thank the Department of Mathematics at the University of Genoa for the kind hospitality during the period in which this paper was written.


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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di GenovaGenoaItaly

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