what is kept fixed. For example, if there is one constraint $p_1 = 0$ on the four-dimensional phase space $(q^1, p_1),
(q^2, p_2)$, then $\partial p_2 / \partial p_1 = 0$ if one keeps $q^1,
q^2$ and $p_2$ fixed, but $\partial p_2 / \partial p_1 = 1$ if one keeps $q^1, q^2$ and $p_2 - p_1$ fixed. Note that the subsequent developments require only that the $C^a{}_{b c}$ be functions of the $G_a$, but not that these functions be polynomials. We consider here the polynomial case for the sole sake of simplicity.

As mentioned earlier, the solution (\[om-p\]) of the equation (\[delta-om=d\]) is not unique. We call it the “covariant solution” because the homotopy $\sigma$ defined by (\[sigma\]) is invariant under linear redefinitions of the generators.

[**Example:**]{} for a Lie algebra $$[G_a, G_b] = C^c{}_{ab} G_c$$ the covariant BRST generator is given by $$\label{L-A-omega}
\Omega = G_a \eta^a - 1/2 {{\cal P}}_a C^a{}_{bc} \eta^c \eta^b.$$ Its nilpotency expresses the Jacobi identity for the structure constants $C^a{}_{bc}$. One has ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ \Omega \end{array}}} = 0$ for $p \geq
2$.

In general, the BRST generator $\Omega$ for a generic Poisson algebra contains higher order terms whose calculation may be 