Related papers: Brackets in the jet-bundle approach to field theor…
A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines…
The Schouten bracket (or antibracket) plays a central role in the Poisson formalism and the Batalin-Vilkovisky quantization of gauge systems. There are several (in)equivalent ways to realize this concept on jet spaces. In this paper, we…
We construct a Leibniz bracket on the space $\Omega^\bullet (J^k (\pi))$ of all differential forms over the finite-dimensional jet bundle $J^k (\pi)$. As an example, we write Maxwell equations with sources in the covariant…
Natural analogs of Lie brackets on affine bundles are studied, based on natural examples from differential geometry and analytical mechanics. In particular, a close relation to Lie algebroids and, by a sort of duality, to affine analogs of…
Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well…
The geometric interpretation of the Batalin-Vilkovisky antibracket as the Schouten bracket of functional multivectors is examined in detail. The identification is achieved by the process of repeated contraction of even functional…
In this review paper we discuss the different interpretations of the concept of connection in a fiber bundle and in a jet bundle, and relate it with first and second-order systems of partial differential equations (PDE's) and multivector…
Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well…
We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a…
We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.
We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic…
We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovic-Witten and Zuckerman, from the…
A general approach is proposed to constructing covariant Poisson brackets in the space of histories of a classical field-theoretical model. The approach is based on the concept of Lagrange anchor, which was originally developed as a tool…
The jet bundle $J^kG$ of $k$-jets of curves in a Lie group $G$ has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie…
In this paper, we consider Hamiltonian structures of hydrodynamic type and some of their generalizations. In particular, we discuss the questions concerning the structure and special forms of the corresponding Poisson brackets and the…
The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are…
Dvali and Shifman have proposed a field-theoretic mechanism for localizing gauge fields to "branes" in higher dimensional spaces using confinement in a bulk gauge theory. The resulting objects have a number of qualitative features in common…
We analyse the problem of defining a Poisson bracket structure on the space of solutions of the equations of motions of first order Hamiltonian field theories. The cases of Hamiltonian mechanical point systems (as a (0 + 1)-dimensional…
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with…