Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
Annales Henri Lebesgue, Volume 5 (2022), pp. 179-262.

KeywordsDouble bracket, Hamiltonian algebra, Quasi-Hamiltonian algebra, Non-commutative geometry, Integrable system

### Abstract

Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the ${H}_{0}$-Poisson structures of Crawley–Boevey. We prove in particular that the double (quasi-) Poisson algebra structure defined by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as an undirected graph, up to isomorphism. We derive from our results a representation theoretic description of action-angle duality for several classical integrable systems.

### References

[AKSM02] Alekseev, Anton Yu.; Kosmann-Schwarzbach, Yvette; Meinrenken, Eckard Quasi-Poisson manifolds, Can. J. Math., Volume 54 (2002) no. 1, pp. 3-29 | DOI | MR | Zbl

[Art18] Arthamonov, Semen Generalized quasi Poisson structures and noncommutative integrable systems, Ph. D. Thesis, The State University of New Jersey, Rutgers, New Brunswick, USA (2018)

[BEF20] Braverman, Alexander; Etingof, Pavel; Finkelberg, Michael Cyclotomic double affine Hecke algebras, Ann. Sci. Éc. Norm. Supér., Volume 53 (2020) no. 5, pp. 1249-1312 (with an Appendix by H. Nakajima and D. Yamakawa) | DOI | MR | Zbl

[Bie13] Bielawski, Roger Quivers and Poisson structures, Manuscr. Math., Volume 141 (2013) no. 1-2, pp. 29-49 | DOI | MR | Zbl

[BLB02] Bocklandt, Raf; Le Bruyn, Lieven Necklace Lie algebras and noncommutative symplectic geometry, Math. Z., Volume 240 (2002) no. 1, pp. 141-167 | DOI | MR | Zbl

[BP11] Bielawski, Roger; Pidstrygach, Victor On the symplectic structure of instanton moduli spaces, Adv. Math., Volume 226 (2011) no. 3, pp. 2796-2824 | DOI | MR | Zbl

[Brü01] Brüstle, Thomas Kit algebras, J. Algebra, Volume 240 (2001) no. 1, pp. 1-24 | DOI | MR | Zbl

[BW00] Berest, Yuri; Wilson, George Automorphisms and ideals of the Weyl algebra, Math. Ann., Volume 318 (2000) no. 1, pp. 127-147 | DOI | MR | Zbl

[Cal71] Calogero, Francesco A. Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., Volume 12 (1971), pp. 419-436 | DOI | MR

[CB11] Crawley-Boevey, William Poisson structures on moduli spaces of representations, J. Algebra, Volume 325 (2011) no. 1, pp. 205-215 | DOI | MR | Zbl

[CBH98] Crawley-Boevey, William; Holland, Martin P. Noncommutative deformations of Kleinian singularities, Duke Math. J., Volume 92 (1998) no. 3, pp. 605-635 | MR | Zbl

[CBS06] Crawley-Boevey, William; Shaw, Peter Multiplicative preprojective algebras, middle convolution and the Deligne–Simpson problem, Adv. Math., Volume 201 (2006) no. 1, pp. 180-208 | DOI | MR | Zbl

[CF17] Chalykh, Oleg; Fairon, Maxime Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models, J. Geom. Phys., Volume 121 (2017), pp. 413-437 | DOI | MR | Zbl

[CF20] Chalykh, Oleg; Fairon, Maxime On the Hamiltonian formulation of the trigonometric spin Ruijsenaars–Schneider system, Lett. Math. Phys., Volume 110 (2020) no. 11, pp. 2893-2940 | DOI | MR | Zbl

[CS17] Chalykh, Oleg; Silantyev, Alexey KP hierarchy for the cyclic quiver, J. Math. Phys., Volume 58 (2017) no. 7, 071702, 31 pages | MR | Zbl

[Dix68] Dixmier, Jacques Sur les algèbres de Weyl, Bull. Soc. Math. Fr., Volume 96 (1968), pp. 209-242 | DOI | Numdam | MR | Zbl

[Eti07] Etingof, Pavel Calogero–Moser systems and representation theory, Zürich Lectures in Advanced Mathematics, European Mathematical Society, 2007 | DOI | Zbl

[Fai19a] Fairon, Maxime Multiplicative quiver varieties and integrable particle systems, Ph. D. Thesis, University of Leeds, Leeds, United Kingdom (2019)

[Fai19b] Fairon, Maxime Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers, J. Integrable Sys., Volume 4 (2019), xyz008, 55 pages | MR | Zbl

[Fai21] Fairon, Maxime Double quasi-Poisson brackets: fusion and new examples, Algebr. Represent. Theory, Volume 24 (2021) no. 4, pp. 911-958 | DOI | MR | Zbl

[FGNR00] Fock, Vladimir V.; Gorsky, Alexander; Nekrasov, Nikita; Rubtsov, Vladimir N. Duality in integrable systems and gauge theories, J. High Energy Phys., Volume 4 (2000) no. 7, 07(2000)028 | MR | Zbl

[FH21] Fernández, David; Herscovich, Estanislao Double quasi-Poisson algebras are pre-Calabi–Yau (2021) (https://arxiv.org/abs/2002.10495)

[FK12] Fehér, László; Klimčík, Ctirad Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction, Nuclear Phys., B, Volume 860 (2012) no. 3, pp. 464-515 | DOI | MR | Zbl

[FK13] Fehér, László; Klimčík, Ctirad The Ruijsenaars self-duality map as a mapping class symplectomorphism, Lie theory and its applications in physics. IX international workshop. Based on the 9th workshop on Lie theory and its applications in physics, Varna, Bulgaria, June 20–26, 2011 (Springer Proceedings in Mathematics & Statistics), Volume 36 (2013), pp. 423-437 | MR | Zbl

[FM19] Fehér, László; Marshall, Ian Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${BC}_{n}$ Sutherland system, Ann. Henri Poincaré, Volume 20 (2019) no. 4, pp. 1217-1262 | DOI | MR | Zbl

[FR99] Fock, Vladimir V.; Roslyĭ, A. Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix, Moscow Seminar in Mathematical Physics (American Mathematical Society Translations, Series 2), Volume 191, American Mathematical Society, 1999, pp. 67-86 | MR | Zbl

[GH84] Gibbons, John; Hermsen, Theo A generalisation of the Calogero–Moser system, Physica D, Volume 11 (1984) no. 3, pp. 337-348 | DOI | MR | Zbl

[Gin01] Ginzburg, Victor Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett., Volume 8 (2001) no. 3, pp. 377-400 | DOI | MR | Zbl

[Gin12] Ginzburg, Victor Lectures on Nakajima’s quiver varieties, Geometric methods in representation theory. I Selected papers based on the presentations at the summer school, Grenoble, France, June 16 – July 4, 2008 (Séminaires et Congrès), Volume 24-pt. 1, Société Mathématique de France, 2012, pp. 145-219 | Zbl

[Gol03] Goldman, William M. The modular group action on real $SL\left(2\right)$-characters of a one-holed torus, Geom. Topol., Volume 7 (2003), pp. 443-486 | DOI | MR | Zbl

[GR01] Gorskiĭ, Aleksandr S.; Rubtsov, Vladimir N. Dualities in integrable systems: geometrical aspects, Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000). Proceedings of the NATO Advanced Research workshop on dynamical symmetries of integrable quantum field theory and lattice models, Kiev, Ukraine, September 25-30, 2000 (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 35 (2001), pp. 173-198 | MR | Zbl

[Ili00] Iliev, Plamen $q$-KP hierarchy, bispectrality and Calogero–Moser systems, J. Geom. Phys., Volume 35 (2000) no. 2-3, pp. 157-182 | DOI | MR | Zbl

[Kir16] Kirillov, Alexander Jr. Quiver representations and quiver varieties, Graduate Studies in Mathematics, 174, American Mathematical Society, 2016 | DOI | Zbl

[KKS78] Kazhdan, David; Kostant, Bertram; Sternberg, Shlomo Hamiltonian group actions and dynamical systems of Calogero type, Commun. Pure Appl. Math., Volume 31 (1978) no. 4, pp. 481-507 | DOI | MR | Zbl

[Kon93] Kontsevich, Maxim Formal (non)-commutative symplectic geometry, The Gelfand Mathematical Seminars, 1990–1992 (Gelfand, I. M.; Corwin, L.; Lepowsky, J., eds.), Birkhäuser, 1993, pp. 173-187 | DOI | Zbl

[KPS94] Kirkman, Ellen E.; Procesi, Claudio; Small, Lance W. A $q$-analog for the Virasoro algebra, Commun. Algebra, Volume 22 (1994) no. 10, pp. 3755-3774 | DOI | MR | Zbl

[KR00] Kontsevich, Maxim; Rosenberg, Alexander L. Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996-1999, Birkhäuser, 2000, pp. 85-108 (dedicated to the memory of Chih-Han Sah. Boston) | DOI | Zbl

[KZ95] Krichever, Igor M.; Zabrodin, Anton V. Spin generalization of the Ruijsenaars–Schneider model, the nonabelian two-dimensionalized Toda lattice, and representations of the Sklyanin algebra, Usp. Mat. Nauk, Volume 50 (1995) no. 6 (306), pp. 3-56 | MR | Zbl

[Mos75] Moser, Jürgern Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., Volume 16 (1975), pp. 197-220 | DOI | MR | Zbl

[MT13] Mencattini, Igor; Tacchella, Alberto A note on the automorphism group of the Bielawski–Pidstrygach quiver, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 9 (2013), 037, 13 pages | MR | Zbl

[MT14] Massuyeau, Gwénaël; Turaev, Vladimir Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not., Volume 2014 (2014) no. 1, pp. 1-64 | DOI | MR | Zbl

[Nak94] Nakajima, Hiraku Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J., Volume 76 (1994) no. 2, pp. 365-416 | MR | Zbl

[Obl04] Oblomkov, Alexei Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory, Volume 8 (2004), pp. 243-266 | DOI | MR | Zbl

[ORS13] Odesskii, Alexander V.; Rubtsov, Vladimir N.; Sokolov, Vladimir V. Double Poisson brackets on free associative algebras, Noncommutative birational geometry, representations and combinatorics. Proceedings of the AMS special session on noncommutative birational geometry, representations and cluster algebras, Boston, MA, USA, January 6–7, 2012 (Contemporary Mathematics), Volume 592 (2013), pp. 225-239 | MR | Zbl

[ORS14] Odesskii, Alexander V.; Rubtsov, Vladimir N.; Sokolov, Vladimir V. Parameter-dependent associative Yang–Baxter equations and Poisson brackets, Int. J. Geom. Methods Mod. Phys., Volume 11 (2014) no. 9, 1460036, 18 pages | MR | Zbl

[Pow16] Powell, Geoffrey On double Poisson structures on commutative algebras, J. Geom. Phys., Volume 110 (2016), pp. 1-8 | DOI | MR | Zbl

[Pus12] Pusztai, Béla Gábor The hyperbolic ${BC}_{n}$ Sutherland and the rational ${BC}_{n}$ Ruijsenaars–Schneider–van Diejen models: Lax matrices and duality, Nuclear Phys., Volume 856 (2012) no. 2, pp. 528-551 | DOI | Zbl

[PVdW08] Pichereau, Anne; Van de Weyer, Geert Double Poisson cohomology of path algebras of quivers, J. Algebra, Volume 319 (2008) no. 5, pp. 2166-2208 | DOI | MR | Zbl

[Res03] Reshetikhin, Nikolai Yu. Degenerate integrability of the spin Calogero–Moser systems and the duality with the spin Ruijsenaars systems, Lett. Math. Phys., Volume 63 (2003) no. 1, pp. 55-71 | DOI | MR | Zbl

[Res16] Reshetikhin, Nikolai Yu. Degenerately integrable systems, J. Math. Sci., New York, Volume 213 (2016) no. 15, pp. 769-785 | DOI | MR | Zbl

[RS86] Ruijsenaars, Simon N. M.; Schneider, Harald A new class of integrable systems and its relation to solitons, Ann. Phys., Volume 170 (1986) no. 2, pp. 370-405 | DOI | MR | Zbl

[Rui88] Ruijsenaars, Simon N. M. Action-angle maps and scattering theory for some finite-dimensional integrable systems I. The pure soliton case, Commun. Math. Phys., Volume 115 (1988), pp. 127-165 | DOI | MR | Zbl

[Sch08] Schiffmann, Olivier Variétés carquois de Nakajima (d’après Nakajima, Lusztig, Varagnolo, Vasserot, Crawley-Boevey, et al.), Séminaire Bourbaki. Vol. 2006/2007. Exposés 967–981 (Astérisque), Volume 317, Société Mathématique de France, 2008, pp. 295-344 (Exp. No. 976) | Zbl

[Sil18] Silantyev, A. V. Reflection functor in the representation theory of preprojective algebras for quivers and integrable systems, Phys. Part. Nuclei, Volume 49 (2018), pp. 397-430 | DOI

[Sut71] Sutherland, Bill Exact results for a quantum many-body problem in one dimension, Phys. Rev. A, Volume 4 (1971) no. 5, pp. 2019-2021 | DOI

[Tac15] Tacchella, A. On a family of quivers related to the Gibbons–Hermsen system, J. Geom. Phys., Volume 93 (2015), pp. 11-32 | DOI | MR | Zbl

[VdB08a] Van den Bergh, Michel Double Poisson algebras, Trans. Am. Math. Soc., Volume 360 (2008) no. 11, pp. 5711-5769 | DOI | MR | Zbl

[VdB08b] Van den Bergh, Michel Non-commutative quasi-Hamiltonian spaces, Poisson geometry in mathematics and physics. Proceedings of the international conference, Tokyo, Japan, June 5–9, 2006 (Contemporary Mathematics), Volume 450 (2008), pp. 273-299 | MR | Zbl

[VdW08] Van de Weyer, Geert Double Poisson structures on finite dimensional semi-simple algebras, Algebr. Represent. Theory, Volume 11 (2008) no. 5, pp. 437-460 | MR | Zbl

[Wil98] Wilson, George Collisions of Calogero–Moser particles and an adelic Grassmannian (With an appendix by I. G. Macdonald), Invent. Math., Volume 133 (1998) no. 1, pp. 1-41 | DOI | Zbl

[Yam08] Yamakawa, Daisuke Geometry of multiplicative preprojective algebra, IMRP, Int. Math. Res. Pap., Volume 2008 (2008), rpn008, 77 pages | MR | Zbl