Isomorphisms between the Batalin–Vilkovisky antibracket and the Poisson bracket

One may introduce at least three different Lie algebras in any Lagrangian field theory: (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin–Vilkovisky antibracket, which has attracted considerable interest recently; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number −1 and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number −1, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from −1, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin, and Vilkovisky.