The sh Lie structure of Poisson brackets in field theory

TL;DR

This paper constructs an sh Lie algebra structure from a homological resolution of a Lie algebra, applying it to Poisson brackets in field theory, including BRST and BV brackets, revealing higher order algebraic structures.

Contribution

It introduces a general method to derive sh Lie algebra structures from homological resolutions and applies it to various brackets in field theory.

Findings

Constructs an sh Lie algebra on the graded differential algebra of horizontal forms.

Applies the construction to Poisson brackets in local functionals.

Extends the approach to graded brackets like BRST and BV brackets.

Abstract

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 suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.