---
author:
- Alain Dresse
- |
    Marc Henneaux[Also at Centro de Estudios Científicos de Santiago, Casilla 16443, Santiago 9, Chile]{}\
    Faculté des Sciences, Université Libre de Bruxelles,\
    Campus Plaine C.P. 231, B-1050 Bruxelles (Belgium)
title: BRST Structure of Polynomial Poisson Algebras
---

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\def$[by-1)]{} $${\relax} \def$$ \#1 \#1[0=\#1sp0 by 30]{}

Introduction
============

Polynomial algebras with a Lie bracket fulfilling the derivation property $$[f g, h] = f [g,h] + [f,h]g$$ are called polynomial Poisson algebras and play an increasingly important role in various areas of theoretical physics [@Nak:; @Pri:; @Skl:; @Zam:; @FatZam:; @Oh:; @TarTakFad:; @BakMat:; @BhaRam:; @GraZhe:]. In terms of a set of independent generators $G_a$, $a = 1, \ldots, n$, the brackets are given by $$\label{basic_bracket}
[G_a, G_b] = C_{ab}(G)$$ where $C_{ab} = - C_{ba}$ are polynomials in the $G$’s[^1]. If the polynomials $C_{ab}(G)$ vanish when the $G$’s are set equal to zero, i.e. if they have no constant part, the polynomial algebra is said to be first class, in analogy with the terminology for constrained Hamiltonian systems (see e.g. [@HenTei:QuaGauSys]). An important class of first class Poisson algebras are symmetric algebras over a finite dimensional Lie algebra. In that case, the bracket (\[basic\_bracket\]) belongs 