∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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.
The Dickey Lie bracket on conserved currents is due to
and is reviewed in
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
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
Created on January 2, 2018 at 17:11:26. See the history of this page for a list of all contributions to it.