is abelian. It is equal to one otherwise. For non linear Poisson algebras, the rank is more useful. All values of the rank compatible with the trivial inequality $$rank \leq n-1$$ may occur. Thus, the rank of the BRST generator provides a non trivial characterization of Poisson algebras. Conversely, non linear Poisson algebras yield an interesting illustration of the full BRST machinery where higher order terms besides ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}$ are required in $\Omega$ to achieve nilpotency.

Upper bound on the rank
=======================

One can understand the fact that the rank of a Lie algebra is at most equal to one by introducing a degree in $\Bbb{C}\,({{\cal P}}_a) \otimes
\Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$ different from the ghost degree as follows.

Assume that one can assign a “degree” $n_a \geq 1$ to the generators $G_a$ in such a way that the bracket decreases the degree by at least one, $$\label{deg-g=n}
\deg G_a = n_a, \; \deg([G_a, G_b]) \leq n_a + n_b - 1.$$ Then, one can bound the covariant and minimal ranks of the algebra by $\sum_a (n_a - 1) + 1$, $$r \leq \sum_a (n_a - 1) + 1$$

In the case of a Lie algebra, one 