#### This page is no longer maintained! Please follow the new double brackets webpage for an updated list.

Double brackets have been introduced by Michel Van den Bergh in 2004 in the preprint *Double Poisson algebras*. This page records all* their appearances in different areas of mathematics. I chose to sort references according to 4 topics:

I began the minor project of gathering all uses of double brackets during my PhD, and I wrote a few pages about this in my thesis. Hence, this idea is continued thanks to this webpage, and I hope that it will be helpful to other researchers.

A *double bracket* is a bilinear map $\{\!\{-,- \}\!\}:A \times A \to A \otimes A$ such that

(i) $\{\!\{a,b \}\!\}=-\{\!\{b,a\}\!\}^\circ\,,$ (ii) $\{\!\{a,bc \}\!\}=b \{\!\{a,c\}\!\}+\{\!\{a,b\}\!\}c\,.$

M. Van den Bergh, *Double Poisson algebras*. [arXiv] [doi] [zbMATH] [MathSciNet]

#### Study of double brackets

Reference | Keywords |
---|---|

A. Alekseev, N. Kawazumi, Y. Kuno, and F. Naef, Goldman-Turaev formality implies Kashiwara-Vergne, Quantum Topol. 11 (2020), no. 4, 657--689 |
Double Poisson cohomology for linear double Poisson bracket |

S. Arthamonov, Modified double Poisson brackets, J. Algebra 492 (2017), 212--233 |
Double brackets (modification of antisymmetry), Link to representation spaces |

R. Bielawski, Quivers and Poisson structures, Manuscripta Math. 141 (2013), no. 1-2, 29--49 |
Double Poisson bracket, Quivers |

M. Fairon, Double quasi-Poisson brackets : fusion and new examples, Alg. Represent. Theor. 24, 911--958 (2021) |
Double quasi-Poisson bracket, Fusion, Classification |

M. Fairon, Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems, Annales Henri Lebesgue 5, 179--262 (2022) |
Morphisms of double brackets, Fusion |

M. Fairon and C. McCulloch, Around Van den Bergh's double brackets for different bimodule structures, Comm. Algebra 51, 1673--1706 (2023) |
Double brackets with different Leibniz rules, Equivalences of double brackets, Gradient double Poisson algebras |

M. Fairon and D. Valeri, Double Multiplicative Poisson Vertex Algebras, Int. Math. Res. Not. IMRN 17, 14991--15072 (2023) |
Local lattice double Poisson algebra, Classification |

A.V. Odesskii, V.N. Rubtsov, and V.V. Sokolov, Double Poisson brackets on free associative algebras, Noncommutative birational geometry, representations and combinatorics, Contemp. Math., vol. 592, Amer. Math. Soc., Providence, RI, 2013, pp. 225--239. |
Linear double Poisson bracket, Quadratic double Poisson bracket, Classification, Compatible double Poisson bracket |

A.V. Odesskii, V.N. Rubtsov, and V.V. Sokolov, Parameter-dependent associative Yang- Baxter equations and Poisson brackets, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 9, 1460036, 18. |
Parameter-dependent double bracket, Parameter-dependent double Poisson bracket, Classification |

G. Olshanski and N. Safonkin, Double Poisson brackets and involutive representation spaces, Preprint, arXiv:2310.01086. |
Involution-adapted double brackets, double brackets in type B-C-D, Link to involutive representation spaces |

A. Pichereau and G. Van de Weyer, Double Poisson cohomology of path algebras of quivers, J. Algebra 319 (2008), no. 5, 2166--2208 | Linear double Poisson bracket, Double Poisson cohomology, Classification |

G. Powell, On double Poisson structures on commutative algebras, J. Geom. Phys. 110 (2016), 1--8. |
Double Poisson bracket, Polynomial ring |

V.V. Sokolov, Classification of constant solutions of the associative Yang-Baxter equation on Mat3, Theoret. and Math. Phys. 176 (2013), no. 3, 1156--1162 |
Quadratic double Poisson bracket, Classification |

V. Turaev, Poisson-Gerstenhaber brackets in representation algebras, J. Algebra 402 (2014), 435--478 |
Double bracket over a commutative ring |

M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360 (2008), no. 11, 5711--5769 |
Double Poisson bracket, Double quasi-Poisson bracket, Hamitlonian algebra, quasi-Hamitlonian algebra, Double Schouten-Nijenhuis bracket, Quivers, Link to representation spaces |

M. Van den Bergh, Non-commutative quasi-Hamiltonian spaces, Poisson geometry in mathematics and physics, Contemp. Math., vol. 450, Amer. Math. Soc., Providence, RI, 2008, 273--299 |
Non-degenerate double quasi-Poisson bracket, Quivers |

G. Van de Weyer, Double Poisson structures on finite dimensional semi-simple algebras, Algebr. Represent. Theory 11 (2008), no. 5, 437--460. | Double Poisson bracket, Semi-simple algebra, Necklace Lie algebra, Double Poisson cohomology |

#### Algebra

Reference | Keywords |
---|---|

J. Alev and G. Van de Weyer, On the structure of the necklace Lie algebra, preprint arXiv:0801.1621 |
Necklace Lie algebra |

L. Alvarez-Consul and D. Fernandez, Non-commutative Courant algebroids and quiver algebras, preprint arXiv:1705.04285 |
Twisted double Lie–Rinehart algebra |

L. Alvarez-Consul, D. Fernandez and R. Heluani, Noncommutative Poisson vertex algebras and Courant-Dorfman algebras, Adv. Math. 433, Paper No. 109269 (2023) |
Double Poisson vertex algebra, Double Courant-Dorfman algebra, Calculus over double derivations |

S. Arthamonov, Generalized quasi Poisson structures and noncommutative integrable systems, The State University of New Jersey, Rutgers, 2018. Thesis (Ph.D.). Available at rutgers-lib/57470/. |
Double quasi-Poisson bracket on categories |

J. Avan, E. Ragoucy, and V. Rubtsov, Quantization and dynamisation of trace-Poisson brackets, Comm. Math. Phys. 341 (2016), no. 1, 263--287 |
Trace-Poisson bracket, Quantization |

Y. Berest, X. Chen, F. Eshmatov, and A. Ramadoss, Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras, Mathematical aspects of quantization, Contemp. Math., vol. 583, Amer. Math. Soc., Providence, RI, 2012, pp. 219- 246. |
Double Poisson brackets on DGA, Derived NC Poisson algebra, Calabi-Yau algebra |

Y. Berest, A.C. Ramadoss, and Y. Zhang, Dual Hodge decompositions and derived Poisson brackets, Selecta Math. (N.S.) 23 (2017), no. 3, 2029--2070 |
Double Poisson brackets on DGA, Derived NC Poisson algebra |

Y. Berest, A.C. Ramadoss, and Y. Zhang, Hodge decomposition of string topology, Hodge decomposition of string topology. Forum Math. Sigma 9 (2021), Paper No. e33, 31 pp |
Derived NC Poisson algebra, String topology bracket |

M. Boucrot, Algèbres de Frobenius et structure de double algèbre sur l'homologie de Hochschild, Mémoire de Master, Université Grenoble Alpes (2021) |
Double algebra |

T. Bozec, D. Calaque and S. Scherotzke, Calabi-Yau structures on (quasi-)bisymplectic algebras, Forum Math. Sigma 11, Paper No. e87 (2023) |
Double (quasi-)Poisson algebras, (quasi-)bisymplectic algebras, Calabi-Yau structures |

T. Bozec, M. Fairon and A. Moreau, Functorial constructions related to double Poisson vertex algebras, Preprint arXiv:2307.06071 |
Double Poisson (vertex) algebras, $H_0$-Poisson (vertex) structures, NC Hamiltonian reduction |

M. Casati and J.P. Wang, Hamiltonian structures for integrable nonabelian difference equations, Commun. Math. Phys. 392 (2022), No. 1, 219--278 |
Double quasi-Poisson algebra, Multiplicative double Poisson vertex algebra |

S. Chemla, Differential calculus over double Lie algebroids, J. Noncommut. Geom. 14 (2020), no. 1, 191--222 |
Double Lie-Rinehart algebra (called double Lie algebroid), Complex from a double Lie-Rinehart algebra |

X. Chen, A. Eshmatov, F. Eshmatov, and S. Yang, The derived non-commutative Poisson bracket on Koszul Calabi-Yau algebras, J. Noncommut. Geom. 11 (2017), no. 1, 111--160. |
Double Poisson brackets for bimodules (over DGA), Derived NC Poisson algebra, Calabi-Yau algebra |

X. Chen and F. Eshmatov, Calabi-Yau algebras and the shifted noncommutative symplectic structure, Adv. Math. 367 (2020), 107126, 40 pp |
Derived NC Poisson algebra, Shifted NC symplectic algebra, Calabi-Yau algebra |

X. Chen, H.-L. Her, S. Sun, and X. Yang, A double Poisson algebra structure on Fukaya categories, J. Geom. Phys. 98 (2015), 57--76 |
Double Poisson brackets on DGA, Fukaya category |

S. D'Alesio, Noncommutative derived Poisson reduction, preprint arXiv:2012.04451 |
Derived NC Poisson algebra, Derived NC Hamiltonian reduction, BRST complex |

S. D'Alesio, Derived Representation Schemes, Nakajima Quiver Varieties and Noncommutative Derived Poisson Reduction, ETH Zurich, 2022. Thesis (Ph.D.). Available at doi.org/10.3929/ethz-b-000580686 |
Derived NC Poisson algebra, Derived NC Hamiltonian reduction, BRST complex |

A. De Sole, V.G. Kac, and D. Valeri, Double Poisson vertex algebras and non-commutative Hamiltonian equations, Adv. Math. 281 (2015), 1025--1099 |
Double (Lie) algebra, Double Poisson vertex algebra |

M. Fairon and D. Fernandez, On the noncommutative Poisson geometry of certain wild character varieties, preprint arXiv:2103.10117 |
Boalch algebra, Fission algebra, Multiplicative preprojective algebra |

M. Fairon and D. Fernandez, Euler continuants in noncommutative quasi-Poisson geometry, Forum Math. Sigma 10 (2022), Paper No. e88, 54 pp |
Boalch algebra, Fission algebra, Euler continuants |

M. Fairon and D. Valeri, Double Multiplicative Poisson Vertex Algebras, Int. Math. Res. Not. IMRN 17, 14991--15072 (2023) |
Local lattice double Poisson algebra, Double multiplicative Poisson vertex algebra |

D. Fernandez and E. Herscovich, Cyclic A-infinity-algebras and double Poisson algebras, J. Noncommut. Geom. 15 (2021), no. 1, 241--278 |
Double Poisson-infinity algebra, Pre-Calabi-Yau algebra |

D. Fernandez and E. Herscovich, Double quasi-Poisson algebras are pre-Calabi-Yau, Int. Math. Res. Not. IMRN, rnab115 (2021) |
Double quasi-Poisson algebra, Pre-Calabi-Yau algebra |

M. Goncharov and V. Gubarev, Double Lie algebras of a nonzero weight, Adv. Math. 409 (2022), Paper No. 108680 |
Double (Lie) algebra, Rota-Baxter operator, Modified double Poisson algebra |

M.E. Goncharov and P.S. Kolesnikov, Simple finite-dimensional double algebras, J. Algebra 500 (2018), 425--438 |
Double (Lie) algebra |

V. Gubarev, An example of a simple double Lie algebra, Siberian Electronic Mathematical Reports 18, iss. 2, 834-844 (2021) |
Double (Lie) algebra, Rota-Baxter operator |

N. Iyudu, M. Kontsevich and Y. Vlassopoulos, Pre-Calabi-Yau algebras as noncommutative Poisson structures, J. Algebra 567 (2021), 63--90 |
Double Poisson bracket, Pre-Calabi-Yau algebra |

N. Iyudu and M. Kontsevich, Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology, preprint arXiv:2011.11888 |
Double Poisson bracket, Pre-Calabi-Yau algebra |

N. Iyudu, M. Kontsevich and Y. Vlassopoulos, Pre-Calabi-Yau algebras and double Poisson brackets, preprint arXiv:1906.07134 |
Double Poisson bracket, Pre-Calabi-Yau algebra |

J. Leray, Approche fonctorielle et combinatoire de la properade des algebres double Poisson, Universite d'Angers, 2017. Thesis (Ph.D.). Available at archives-ouvertes:tel-01719403. | Double Poisson algebra, Protoperad, Double Lie-Rinehart algebra, Shifted double Lie-Rinehart algebra |

J. Leray, Shifted double Lie-Rinehart algebras, Theory Appl. Categ. 35 (2020), Paper No. 17, 594--621 |
Double Lie-Rinehart algebra, Shifted double Lie-Rinehart algebra |

J. Leray, Protoperads II: Koszul duality, J. Éc. polytech. Math. 7 (2020), 897--941 |
Double Poisson algebra, Protoperad |

J. Leray and B. Vallette, Pre-Calabi--Yau algebras and homotopy double Poisson gebras, preprint arXiv:2203.05062 |
Double Poisson algebra, Curved homotopy double Poisson (al)gebra, Pre-Calabi-Yau algebra |

G. Massuyeau and V. Turaev, Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not. IMRN (2014), no. 1, 1--64 |
Fox pairing, Quasi-Poisson algebra, Double quasi-Poisson algebra |

G. Massuyeau and V. Turaev, Brackets in representation algebras of Hopf algebras, J. Noncommut. Geom. 12 (2018), no. 2, 577--636 |
Hopf algebras, Fox pairing, Graded double brackets, Link to representation spaces relative to a bialgebra |

F. Naef, Poisson brackets in Kontsevich's 'Lie World', J. Geom. Phys. 155, 103741, 13 pp (2020) |
Double Poisson brackets on Lie algebras |

A.V. Odesskii, V.N. Rubtsov, and V.V. Sokolov, Double Poisson brackets on free associative algebras, Noncommutative birational geometry, representations and combinatorics, Contemp. Math., vol. 592, Amer. Math. Soc., Providence, RI, 2013, pp. 225--239. |
Associative Yang-Baxter Equation, Trace bracket, Trace-Poisson bracket |

A.V. Odesskii, V.N. Rubtsov, and V.V. Sokolov, Parameter-dependent associative Yang- Baxter equations and Poisson brackets, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 9, 1460036, 18. |
Double (Lie) algebra, Parameter-dependent Associative Yang-Baxter Equation |

G. Olshanski, The centralizer construction and Yangian-type algebras, preprint arXiv:2208.04809 |
Yangian-type quantization of the KKS double Poisson bracket |

G. Olshanski and N. Safonkin, Remarks on Yangian-type algebras and double Poisson brackets, preprint arXiv:2308.13325 |
Yangian-type quantization of linear double Poisson bracket |

J.P. Pridham, Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks, Preprint, arXiv:2008.11698. |
shifted double Poisson structures (over DGA), Shifted Poisson structures |

V. Roubtsov and R. Suchánek, Lectures on Poisson algebras, Groups, Invariants, Integrals, and Mathematical Physics, 2023, pp 41–116 |
(modified) double Poisson algebras |

T. Schedler, Poisson algebras and Yang-Baxter equations, Advances in quantum computation, Contemp. Math., vol. 482, Amer. Math. Soc., Providence, RI, 2009, pp. 91--106. |
Double (Lie) algebra, Associative Yang-Baxter Equation, Double Poisson-infinity algebra |

V. Turaev, Poisson-Gerstenhaber brackets in representation algebras, J. Algebra 402 (2014), 435--478 |
Link to representation spaces relative to a coalgebra |

M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360 (2008), no. 11, 5711--5769 |
Deformed preprojective algebra, Multiplicative preprojective algebra, Double Lie-Rinehart algebra (called double Lie algebroid) |

W.-K. Yeung, Pre-Calabi-Yau structures and moduli of representations, preprint arXiv:1802.05398
(Original title: |
Graded double Poisson algebra, Pre-Calabi-Yau algebra. |

#### Geometry and Topology

Reference | Keywords |
---|---|

A. Alekseev, N. Kawazumi, Y. Kuno, and F. Naef, The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem, Adv. Math. 326 (2018), 1--53. |
Fundamental group of (genus zero) surface, Goldman-Turaev Lie bialgebra, Kashiwara-Vergne problem |

A. Alekseev, N. Kawazumi, Y. Kuno, and F. Naef, The Goldman-Turaev Lie bialgebras and the Kashiwara-Vergne problem in higher genera, preprint arXiv:1804.09566 |
Fundamental group of surface, Goldman-Turaev Lie bialgebra, Kashiwara-Vergne problem |

A. Alekseev, N. Kawazumi, Y. Kuno, and F. Naef, Goldman-Turaev formality implies Kashiwara-Vergne, Quantum Topol. 11 (2020), no. 4, 657--689 |
Fundamental group of surface, Goldman-Turaev Lie bialgebra, Kashiwara-Vergne problem |

A. Alekseev and F. Naef, Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection, C. R. Math. Acad. Sci. Paris 355 (2017), no. 11, 1138--1147 |
Fundamental group of (genus zero) surface, Goldman-Turaev Lie bialgebra, Kashiwara-Vergne problem, Knizhnik-Zamolodchikov connection |

L. Alvarez-Consul and D. Fernandez, Noncommutative bi-symplectic NQ-algebras of weight 1, Discrete Contin. Dyn. Syst. (2015), no. Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 19--28 |
NC N-graded manifold, Double Poisson algebra, NC bi-symplectic NQ-algebra |

L. Alvarez-Consul and D. Fernandez, Non-commutative Courant algebroids and quiver algebras, preprint arXiv:1705.04285 |
NC Courant algebroid |

S. Arthamonov, Generalized quasi Poisson structures and noncommutative integrable systems, The State University of New Jersey, Rutgers, 2018. Thesis (Ph.D.). Available at rutgers-lib/57470/. |
Fundamental groupoid of ribbon graph |

S. Arthamonov, N. Ovenhouse, and M. Shapiro, Noncommutative Networks on a Cylinder, preprint arXiv:2008.02889 |
Networks on disk/cylinder, noncommutative Goldman's bracket |

M. Fairon, Double quasi-Poisson brackets : fusion and new examples, Alg. Represent. Theor. 24, 911--958 (2021) |
Fundamental group of surface, Representation variety, Multiplicative quiver variety |

M. Fairon and D. Fernandez, On the noncommutative Poisson geometry of certain wild character varieties, preprint arXiv:2103.10117 |
Multiplicative quiver variety, Wild character variety |

M. Fairon and D. Fernandez, Euler continuants in noncommutative quasi-Poisson geometry, Forum Math. Sigma 10 (2022), Paper No. e88, 54 pp |
Wild character variety, Sibuya variety |

D.A. Fernandez, Non-commutative symplectic NQ-geometry and Courant algebroids, Universidad Autonoma de Madrid, 2015. Thesis (Ph.D.). Available at hdl.handle.net/10486/671677 |
NC N-graded manifold, NC Courant algebroid |

G. Massuyeau and V. Turaev, Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not. IMRN (2014), no. 1, 1--64 |
Fundamental group of surface, Goldman's bracket, Representation variety |

G. Massuyeau and V. Turaev, Brackets in the Pontryagin algebras of manifolds, Mém. Soc. Math. Fr. (N.S.) (2017), no. 154, 138pp |
Pontryagin algebra of manifold, Chas-Sullivan string bracket |

N. Ovenhouse, Non-commutative integrability of Grassmann pentagram map, Adv. Math. 373 (2020), 107309, 56p |
Fundamental groupoid of ribbon graph, Goldman's bracket |

V. Turaev, Poisson-Gerstenhaber brackets in representation algebras, J. Algebra 402 (2014), 435--478 |
Equivariant Hamiltonian reduction |

M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360 (2008), no. 11, 5711--5769 |
Hamiltonian reduction, Quasi-Hamiltonian reduction, Quiver variety, Multiplicative quiver variety |

M. Van den Bergh, Non-commutative quasi-Hamiltonian spaces, Poisson geometry in mathematics and physics, Contemp. Math., vol. 450, Amer. Math. Soc., Providence, RI, 2008, 273--299 |
Quasi-Hamiltonian spaces |

#### Integrable Systems and Mathematical Physics

Reference | Keywords |
---|---|

S. Arthamonov, Noncommutative inverse scattering method for the Kontsevich system, Lett. Math. Phys. 105 (2015), no. 9, 1223--1251 |
Non-commutative Hamiltonian ODEs, Kontsevich system |

S. Arthamonov, Modified double Poisson brackets, J. Algebra 492 (2017), 212--233 |
Non-commutative Hamiltonian ODEs, Kontsevich system |

S. Arthamonov, Generalized quasi Poisson structures and noncommutative integrable systems, The State University of New Jersey, Rutgers, 2018. Thesis (Ph.D.). Available at rutgers-lib/57470/. |
Non-commutative Hamiltonian ODEs, Kontsevich system |

S. Arthamonov, N. Ovenhouse, and M. Shapiro, Noncommutative Networks on a Cylinder, preprint arXiv:2008.02889 |
Noncommutative r-matrix |

J. Avan, E. Ragoucy, and V. Rubtsov, Quantization and dynamisation of trace-Poisson brackets, Comm. Math. Phys. 341 (2016), no. 1, 263--287 |
Trace-Poisson bracket, Quantization |

M. Casati and J.P. Wang, Hamiltonian structures for integrable nonabelian difference equations, Commun. Math. Phys. 392 (2022), No. 1, 219--278 |
Non-commutative Hamiltonian difference equations |

O. Chalykh and M. Fairon, Multiplicative quiver varieties and generalised Ruijsenaars- Schneider models, J. Geom. Phys. 121 (2017), 413--437 |
Ruijsenaars-Schneider system, Cyclic quiver |

O. Chalykh and M. Fairon, On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system, Lett. Math. Phys. 110 (2020), no. 11, 2893--2940 |
Spin Ruijsenaars-Schneider system, Arutyunov-Frolov Poisson bracket |

A. De Sole, V.G. Kac, and D. Valeri, Double Poisson vertex algebras and non-commutative Hamiltonian equations, Adv. Math. 281 (2015), 1025--1099 |
Non-commutative Magri scheme, Non-commutative Hamiltonian ODEs/PDEs |

A. De Sole, V.G. Kac, and D. Valeri, Poisson vertex algebras and Hamiltonian PDE, To appear in: Encyclopedia of Mathematical Physics 2nd ed. |
Non-commutative Hamiltonian ODEs/PDEs/DDEs |

M. Fairon, Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers, J. Integrable Systems, Volume 4, Issue 1, xyz008 (2019) |
Spin Ruijsenaars-Schneider system, Cyclic quiver |

M. Fairon, Multiplicative quiver varieties and integrable particle systems, PhD thesis, University of Leeds (2019). Available here |
Ruijsenaars-Schneider system, Spin Ruijsenaars-Schneider system, Arutyunov-Frolov Poisson bracket, Cyclic quiver |

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

M. Fairon and T. Görbe, Superintegrability of Calogero-Moser systems associated with the cyclic quiver, Nonlinearity 34, 7662--7682 (2021) |
Spin Calogero-Moser system, Cyclic quiver |

M. Fairon and D. Valeri, Double Multiplicative Poisson Vertex Algebras, Int. Math. Res. Not. IMRN 17, 14991--15072 (2023) |
Non-commutative Hamiltonian difference equations |

A.V. Odesskii, V.N. Rubtsov, and V.V. Sokolov, Double Poisson brackets on free associative algebras, Noncommutative birational geometry, representations and combinatorics, Contemp. Math., vol. 592, Amer. Math. Soc., Providence, RI, 2013, pp. 225--239. |
Non-commutative Magri scheme |

N. Ovenhouse, Non-commutative integrability of Grassmann pentagram map, Adv. Math. 373 (2020), 107309, 56p |
Grassmann pentagram map, Non-commutative discrete integrability |

B. Wang and S.-H. Li, On non-commutative leapfrog map, Preprint, arXiv:2310.01993 |
Non-commutative discrete integrability, Non-commutative leapfrog map |

#### Final remarks

Note that I have only considered works that *explicitly* use double brackets. For example, I have not considered works that are related to the quasi-Poisson bracket on multiplicative quiver varieties if they do not use the underlying double quasi-Poisson bracket at the level of the quivers.

I have also omitted works that use the bi-symplectic geometry of Crawley-Boevey, Etingof and Ginzburg, or that use the notion of H_0-Poisson structure of Crawley-Boevey. These two types of structures can be related to double Poisson brackets (see Van den Bergh's seminal 2008 paper) and are interesting in their own right. However, I did not take the opportunity to study them in greater details so I am not in the best position to create lists for them as the one above.

I hope that these references will convince mathematicians that double brackets are useful in many different subjects, and that they deserve to be studied independently. Finally, let me state some interesting open questions regarding them :

- What is the quantisation of a double Poisson bracket? (See [Odesskii, Rubtsov, Sokolov, 2013] based on a Mathoverflow question by D. Calaque.)
- Can one understand the Poisson structure of wild character varieties from a double quasi-Poisson bracket? (See my works with Fernandez for basic cases)
- Are there double brackets which are not (quasi-)Poisson but are useful in the theory of integrable systems? What about the (multiplicative) vertex case? (See [Casati, Wang, 2021] and my work with Valeri in that direction)
- Can one build a quasi-Hamiltonian algebra which is not differential, i.e. for which the double quasi-Poisson bracket does not come from a (noncommutative) bivector? (This would make Theorem 2.15 in [Fairon, 2021] strictly stronger than Theorem 5.3.2 in Van den Bergh's original work)
- What is the analogue of H0-Poisson structures in the (multiplicative) vertex case? (This is based on a question that I was asked by G. Powell. The Poisson vertex case was settled by Bozec, Moreau and myself; the multiplicative case is still open.)
- What is the obstruction to the existence of double (multiplicative) Poisson vertex algebras as deformations of double Poisson algebras? (This is based on a question that I was asked by V. Rubtsov)

(If you have some open questions, do not hesitate to send them and I will list them here.)

* If you think that something is missing from the list, please get in touch! Note however that I only record papers that make an explicit use of double brackets.

Last update : 18 October 2023

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