shall call [*“covariant rank”*]{} of a first class polynomial Poisson algebra the degree in ${{\cal P}}_a$ of the covariant BRST generator. This concept is invariant under linear redefinitions of the generators because the covariant BRST generator is itself invariant if one transforms the ghosts and their momenta as $$\begin{aligned}
G_a &\rightarrow& \bar{G}_a = A_a{}^b G_b \\
{{\cal P}}_a &\rightarrow& \bar{{{\cal P}}}_a = A_a{}^b {{\cal P}}_b \\
\eta^a &\rightarrow& \bar{\eta}^a = (A^{-1})_b{}^a \eta^b\end{aligned}$$

We shall call [*“minimal rank”*]{} the degree in ${{\cal P}}_a$ of the solution of $[ \Omega, \Omega] = 0$ of lowest degree in ${{\cal P}}$ (i.e., one chooses at each stage ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$ in such a way that $\Omega$ has lowest possible degree in ${{\cal P}}$). It is easy to see that for a Lie algebra, the concepts of covariant and minimal ranks coincide. As we shall see on an explicit example below, they do not in the general case.

Now, for a Lie algebra, the rank is not particularly interesting in the sense that it does not tell much about the structure of the algebra : the rank of a Lie algebra is equal to zero if and only if the algebra 