Canonical Transformations of Local Functionals and sh-Lie Structures

TL;DR

This paper explores how canonical automorphisms of vector bundles in Lagrangian field theories preserve the Poisson bracket and induce sh-Lie structures, including their behavior under group actions and reductions.

Contribution

It characterizes the relationship between automorphisms of vector bundles and the induced sh-Lie structures in local functionals, extending understanding beyond Poisson algebra frameworks.

Findings

Canonical automorphisms preserve the Poisson bracket structure.

Group actions induce transformations on the sh-Lie algebra.

Conditions for sh-Lie structure reduction under group actions are established.

Abstract

In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms. We determine how such automorphisms relate to the corresponding sh-Lie structure. If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra. We determine conditions under which the sh-Lie structure induces an sh-Lie structure on a corresponding reduced space where the reduction is determined by the action of the group. These results are not directly a consequence of the corresponding theorems on Poisson manifolds as none of the algebraic structures are Poisson algebras.