A generalized Poisson bracket and an associated natural one-form

Journal of Mathematical Physics, 2015

How to give a natural geometric definition of a covariant Poisson bracket in classical field theory has for a long time been an open problem-as testified by the extensive literature on "multisymplectic Poisson brackets", together with the fact that all these proposals suffer from serious defects. On the other hand, the functional approach does provide a good candidate which has come to be known as the Peierls-De Witt bracket and whose construction in a geometrical setting is now well understood. Here, we show how the basic "multisymplectic Poisson bracket" already proposed in the 1970s can be derived from the Peierls-De Witt bracket, applied to a special class of functionals. This relation allows to trace back most (if not all) of the problems encountered in the past to ambiguities (the relation between differential forms on multiphase space and the functionals they define is not one-to-one) and also to the fact that this class of functionals does not form a Poisson subalgebra.

Journal of Physics A: Mathematical and Theoretical, 2012

We introduce a bracket on 1-forms defined on J ∞ (S 1 , R n ), the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of F -manifolds with compatible flat connection (M, ∇, •) are Hamiltonian in a generalized sense. Moreover, we show that if a metric g compatible with ∇ is also invariant with respect to •, then this generalized Hamiltonian set-up reduces to the standard one.

Journal of Geometry and Physics, 1996

We prove that the Poisson algebra of certain non compact symplectic man ifolds is isomorphic to a Lie algebra of vector fields on a smooth manifold. We also prove the integrability of the Poisson algebra of functions with compact supports. Finally we discuss and extend the notion of integrability of infinite dimensional Lie algebras.

Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established. Mathematics Subject Classifications (2000): Primary: 58A50, 58F05; Secondary: 58A10, 53C15.

2016

We introduce a bracket on 1-forms defined on J ∞ (S 1 , R n), the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of F-manifolds with compatible flat connection (M, ∇, •) are Hamiltonian in a generalized sense. Moreover, we show that if a metric g compatible with ∇ is also invariant with respect to •, then this generalized Hamiltonian setup reduces to the standard one.

Annals of Physics, 1976

The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments. The aim of the present paper is to suggest a general theory of Poisson brackets * This work has been sponsored by the Consiglio nazionale delle Ricerche, Gruppo per la Fisica-Matematica.

Journal of Symplectic Geometry, 2004

arXiv (Cornell University), 1998

We consider the algebra of spatial diffeomorphisms and gauge transformations in the canonical formalism of General Relativity in the Ashtekar and ADM variables. Modifying the Poisson bracket by including surface terms in accordance with our previous proposal allows us to consider all local functionals as differentiable. We show that closure of the algebra under consideration can be achieved by choosing surface terms in the expressions for the generators prior to imposing any boundary conditions. An essential point is that the Poisson structure in the Ashtekar formalism differs from the canonical one by boundary terms.

2008

In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi-Yau manifold.

Duke Mathematical Journal, 2002

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection. Let M be a paracompact smooth d-dimensional manifold. The Lie bracket of vector fields extends to a bracket, the Schouten-Nijenhuis bracket, on the graded commutative algebra Γ(M, ∧ • T M ) of multivector fields so that: This bracket defines a graded super Lie algebra structure on Γ(M, ∧ • T M ), with the shifted grading deg A Poisson structure on M is a bivector field α ∈ Γ(M, ∧ 2 T M ) obeying [α, α] = 0. This identity for α, which we can regard as a bilinear form on the cotangent bundle, implies that {f, g} = α(df, dg) is a Poisson bracket on the algebra C ∞ (M ) of smooth real-valued function. If such a bivector field is given, we say that M is a Poisson manifold. Following [2], we introduce the notion of (deformation) quantization of the algebra of functions on a Poisson manifold. A quantization of the algebra of smooth functions C ∞ (M ) on the Poisson manifold M is a topological algebra A over the ring of formal power series R [[ǫ]] in a formal variable ǫ with product ⋆, together with an R-algebra ǫ j P j (f, g) for some bidifferential operators P j : C ∞ (M ) 2 → C ∞ (M ) with P j (f, 1) = P j (1, g) = 0 and P 1 (f, g) -P 1 (g, f ) = 2α(df, dg). If we fix a section as in (ii), we obtain a star product on C ∞ (M ), i.e. a formal series P ǫ = ǫP 1 + ǫ 2 P 2 + • • • whose coefficients P j are bidifferential operators C ∞ (M ) 2 → C ∞ (M ) so that f ⋆ M g := f g + P ǫ (f, g) extends to an associative R[[ǫ]]-bilinear product on C ∞ (M ) [[ǫ]] with unit 1 ∈ C ∞ (M ) and such that f ⋆ M g -g ⋆ M f = 2ǫα(df, dg) mod ǫ 2 .