Because the degree of a ghost number one object is necessarily greater than or equal to $1/2$, the condition $\deg {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k+1$ (if ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \neq 0$) implies ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} = 0$ for $k > 0$.

Self-reproducing algebras
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While Lie algebras are characterized by the existence of a degreee that is decreased by the bracket, one may easily construct examples of Poisson algebras for which such a degree does not exist. The simplest ones are quadratic algebras for which $[G_a, G_b]$ is proportional to $G_a, G_b$ $$[G_a, G_b] = M_{ab} G_a G_b \quad\quad\mbox{no summation on $a,b$}$$ with $M_{ab} = -M_{ba}$. The Jacobi identity is fulfilled for arbitrary $M$’s. Since $\deg(G_a G_b) = n_a + n_b$, the inequality (\[deg-g=n\]) is violated for any choice of $n_a$. Because $[G_a, G_b]$ is proportional to $G_a G_b$, we shall call these algebras “self-reproducing algebras”.

The most general self-reproducing algebra with three generators is given by $$\begin{aligned}
[G_1, G_2] &=& \alpha\, G_1 G_2 \\{}
[G_2, G_3] &=& \beta \, G_2 G_3 \\{}
[G_3, G_1] &=& \gamma \, G_1 G_3.\end{aligned}$$ This Poisson algebra can be realized on a 