nLab Dickey Lie bracket

Idea

In Lagrangian field theory the Dickey bracket is a canonical Lie bracket on conserved currents. Under the relation between conserved currents and infinitesimal symmetries of the Lagrangian the Dickey bracket lifts the canonical Lie bracket on infinitesimal symmetries to their associated conserved currents (Noether's theorem), exhibiting a Lie algebra extension.

References

The Dickey Lie bracket on conserved currents is due to

• Leonid Dickey, Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, Vol. 12 (World Scientific 1991).

and is reviewed in

• Glenn Barnich, Marc Henneaux, section 3 of Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket, Journal of Mathematical Physics 37, 5273-5296 (1996) (arXiv:hep-th/9601124, DOI 10.1063/1.531726)

The statement that the Dickey bracket Lie algebra of currents is a central Lie algebra extension of the algebra of symmetries by de Rham cohomology of the jet bundle appears as theorem 11.2 in (Part II of)

• Alexandre Vinogradov, The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. the linear theory, Journal of Mathematical Analysis and Applications 100, 1-40 (1984) (doi90071-4))

• Alexandre Vinogradov, The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, Journal of Mathematical Analysis and Applications 100, Issue 1, 30 April 1984, Pages 41-129 (publisher)

and is stated as exercise 2.28 on p. 203 of

• Alexandre Vinogradov, Joseph Krasil'shchik (eds.), Symmetries and conservation laws for differential equations of mathematical physics, vol. 182 of Translations of Mathematical Monographs, AMS (1999)

A lift of the Dickey Lie bracket on cohomologically trivial spaces to an equivalent L-infinity equivalent L-infinity bracket is constructed, under some assumptions, in

• Glenn Barnich, Ronald Fulp, Tom Lada, Jim Stasheff, The sh Lie structure of Poisson brackets in field theory, Communications in Mathematical Physics 191, 585-601 (1998) (arXiv:hep-th/9702176)

• Martin Markl, Steve Shnider, Differential Operator Endomorphisms of an Euler-Lagrange Complex, Contemporary Mathematics, Volume 231, 1999 (arXiv:9808105)

The cohomologically non-trivial lift is discussed in

• Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges, J. High Energ. Phys. (2017) 2017: 87 (arXiv:1507.08692)