Double brackets in mathematics


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 began the project of gathering all uses of double brackets during my PhD, and I wrote a few pages about this in my thesis. Hence, this page continues the project succinctly.


The subjects used below are :

  1. Study of double brackets
  2. Algebra
  3. Geometry and Topology
  4. Integrable Systems and Mathematical Physics

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, Accepted in: Ann. Henri Lebesgue; arXiv:2008.01409 Morphisms of double brackets, Fusion
M. Fairon and D. Valeri, Double Multiplicative Poisson Vertex Algebras, Preprint arXiv:2110.03418 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
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, Russian version appears in Teoret. Mat. Fiz. 176 (2013), no. 3, 385--392 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
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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
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, preprint arXiv:2002.06596 Derived NC Poisson algebra, String topology bracket
M. Casati and J.P. Wang, Hamiltonian structures for integrable nonabelian difference equations, Preprint arXiv:2101.06191 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)
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. in Math. (2020) 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
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, preprint arXiv:2105.04858 Boalch algebra, Fission algebra, Euler continuants
M. Fairon and D. Valeri, Double Multiplicative Poisson Vertex Algebras, Preprint arXiv:2110.03418 Local lattice double Poisson algebra, Double multiplicative Poisson vertex algebra
D. Fernandez and R. Heluani, Noncommutative Poisson vertex algebras and Courant-Dorfman algebras, preprint arXiv:2106.00270 Double Poisson vertex algebra, Double Courant–Dorfman 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, preprint arXiv:2002.10495 Double quasi-Poisson algebra, Pre-Calabi-Yau algebra
M. Goncharov and V. Gubarev, Double Lie algebras of a nonzero weight, preprint arXiv:2104.13678 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, Example of simple double Lie algebra, preprint arXiv:2104.03605 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
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. (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
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
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, Weak Calabi-Yau structures and moduli of representations, preprint arXiv:1802.05398 Graded double Poisson algebra, Pre-Calabi-Yau algebra
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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, preprint arXiv:2105.04858 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, 138 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
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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, Preprint arXiv:2101.06191 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
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, Accepted in: Ann. Henri Lebesgue; arXiv:2008.01409 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, Preprint arXiv:2110.03418 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
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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 an important open question regarding them :


Last update : 5 October 2021


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