Poisson algebras can be of arbitrarily high rank. We also point out that BRST concepts provide intrinsic characterizations of Poisson algebras.

In the next section, we briefly review the BRST construction. We then discuss how it applies to Poisson algebras, even when the generators $G_a$ are not realized as phase space functions of some dynamical system. We analyze the BRST cohomology and introduce the concepts of covariant and minimal ranks, for which an elementary theorem is proven. Quadratic algebras are then shown to provide models with arbitrarily high rank. These contain “self-reproducing” algebras for which the bracket of $G_a$ with $G_b$ is proportional to the product $G_a G_b$. The first few terms in the BRST generator are also computed for more general algebras by means of a program written in REDUCE. The paper ends with some concluding remarks on the quantum case.

A Brief Survey of the BRST Formalism
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We follow the presentation of [@HenTei:QuaGauSys], to which we refer for details and proofs. Given a set of independent functions $G_a(q,p)$ defined in some phase space $P$ with local coordinates $(q^i, p_i)$ and fulfilling the first class property $[G_a, G_b] \approx 0$, where $\approx$ denotes equality on the surface $G_a(q,p) 