nLab Poisson vertex algebra

Idea

A Poisson vertex algebra is a unital commutative associative algebra with a derivation and a $\lambda$-bracket; it has to satisfy the axioms of a Lie conformal algebra; and the $\lambda$-bracket and the multiplication form a Poisson algebra (i.e. satisfy the Leibniz rule). It is a Poisson analogue of a conformal Lie algebra.

• Haisheng Li, Vertex algebras and vertex Poisson algebras, Commun. Contemp. Math. http://arxiv.org/abs/math/0209310
• A. De Sole, V. G. Kac, M.; Wakimoto, On classification of Poisson vertex algebras, Transform. Groups 15 (2010), no. 4, 883–907 MR2012a:17051 doi
• B. Bakalov, A. De Sole, Non-linear Lie conformal algebras with three generators, Selecta Math. (N.S.) 14 (2009), no. 2, 163–198 dpo MR2009m:17024
• Alberto De Sole, Victor G. Kac, Daniele Valeri, Classical W-algebras and Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, arxiv/1207.6286
• Gaywalee Yamskulna, Vertex Poisson algebras associated with Courant algebroids and their deformations; I, math.QA/0509122
• A. De Sole, Poisson vertex algebras in the theory of Hamiltonian equations, Talk at Algebraic Lie Theory, Newton Inst. U. Cambridge 2009, video

On non-local Poisson vertex algebras with applications to the study of integrability of Hamiltonian PDEs in

• Alberto De Sole, Victor G. Kac, Non-local Hamiltonian structures and applications to the theory of integrable systems I, arxiv/1210.1688; II, arxiv/1211.2391; merged version, arxiv/1302.0148

• Aliaa Barakat, Alberto De Sole, Victor G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, arxiv/0907.1275

• Alberto De Sole, Victor G. Kac, Daniele Valeri, Poisson vertex algebras and Hamiltonian PDE, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2306.09709]