nLab double Poisson structure

Related notions include double derivation, shifted symplectic structure, Poisson algebra, noncommutative geometry, Kontsevich-Rosenberg principle

• Michel Van den Bergh, Double Poisson algebras, Trans. of Amer. Math. Soc. (2008) 5711–5769
• J. P. Pridham, Shifted bisymplectic and double Poisson structures on noncommutative derived prestacks, arXiv:2008.11698
• A. Odesskii, V. Rubtsov, V. Sokolov, Double Poisson brackets on free associative algebras, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. 592, Amer. Math. Soc. (2013) 225–239 doi
• A. Odesskii, V. Rubtsov, V. Sokolov, Parameter-dependent associative Yang–Baxter equations and Poisson brackets, Int. J. Geom. Meth. Mod. Phys. 11:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 doi
• A. Pichereau, G. Van de Weyer, Double Poisson cohomology of path algebras of quivers, J. Algebra 319:5 (2008) 2166–2208.

Double Poisson vertex algebra generalize double Poisson algebras:

• Alberto De Sole, Victor G. Kac, Daniele Valeri, Double Poisson vertex algebras and non-commutative Hamiltonian equations, Adv. Math. 281 (2015) 1025–1099 doi
• Luis Álvarez-Cónsul, David Fernández, Reimundo Heluani, Noncommutative Poisson vertex algebras and Courant-Dorfman algebras, arXiv:2106.00270

There is also a viewpoint of double algebras

• M.E.Goncharov, P.S.Kolesnikov, Simple finite-dimensional double algebras, J. Algebra 500 (2018) 425–438 doi

There are some restrictions and no-go theorems for commutative algebras

• Geoffrey Powell, On double Poisson structures on commutative algebras, J. Geom. Phys. 110 (2016) 1–8 doi

category: algebra