quite cumbersome. However, because the procedure is purely algorithmic, it can be performed by means of an algebraic program like REDUCE.

The cohomology of the Poisson algebra may be defined to be the cohomology of the BRST differential $s$ in the algebra $\Bbb{C}\,({{\cal P}}_a)
\otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$. Because $s$ contains $\delta$ as its piece of lowest antighost number (with ${\mbox{antigh}}({{\cal P}}_a) = 1, {\mbox{antigh}}(\mbox{anything else}) = 0$), and because $\delta$ provides a resolution of the zero-dimensional point $G_a =
0$, standard arguments show that the cohomology of $s$ is isomorphic to the cohomology of the differential $s'$ in $\Bbb{C}\,(\eta^a)$, $$\label{eq:sPrime}
s' \eta^a = 1/2 C^a{}_{bc} \eta^b \eta^c$$ where $C^a{}_{bc}$ is defined by $$C^a{}_{bc} = \left.\frac{\partial C_{bc}}{\partial G_a}\right|_{G = 0}$$

The $C^a{}_{bc}$ fulfill the Jacobi identity so that $s'^2 = 0$. Hence, they are the structure constants of a Lie algebra, which is called the Lie algebra underlying the given Poisson algebra.

Because of (\[eq:sPrime\]), the BRST cohomology of a Poisson algebra is isomorphic to the cohomology of the underlying Lie algebra. For a different and more thorough treatment of Poisson cohomology, see [@Hue:].

Rank
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Again in analogy with the terminology used in the theory of constrained systems, we 