to the linear span of the $G_a$’s, i.e. the $C_{ab}(G)$ are homogeneous of degree one in the $G$’s, $[G_a, G_b] = C_{ab}{}^c G_c$. We shall call this situation the “Lie algebra case”, and refer to the non Lie algebra case as the “open algebra case” using again terminology from the theory of first class constrained systems [@HenTei:QuaGauSys][^2].

The purpose of this paper is to investigate the BRST structure of first class Poisson algebras. The BRST formalism has turned out recently to be the arena of a fruitful interplay between physics and mathematics (see e.g. [@HenTei:QuaGauSys] and references therein). A crucial ingredient of BRST theory is the recursive pattern of homological perturbation theory [@Sta:] which allows one to construct the BRST generator step by step. In most applications, however, this recursive construction collapses almost immediately, and, to our knowledge, no example has been given so far for which the full BRST machinery is required (apart from the field-theoretical membrane models [@Hen:PhyLet; @FujKub:]). We show in this paper that Poisson algebras—actually, already quadratic Poisson algebras—offer splendid examples illustrating the complexity of the BRST construction. While Lie algebras yield a BRST generator of rank 1 (see e.g. [@HenTei:QuaGauSys]), the BRST charge for quadratic 