different from zero. The expression for $\alpha_k$ reduces then to (43) since $k+1$ must be even in (49).

Although $\alpha_k = 0$ for $k$ odd, $k > 1$, one easily sees that $\alpha_k < 0$ for $k$ even. This is true for $k =
2$ as $\alpha_2 = -1/12$. Let $p$ be even, and suppose $\alpha_m < 0$ for $1 < m < p$, $m$ even. Then, all terms in the sum in the recurrence relation (\[eq:AlpRec\]) are strictly positive, so that $\alpha_p < 0$. Since $\alpha_p \neq 0$, the quadratic algebra (\[eq:maxquad\]) provides examples of systems with arbitrarily high covariant rank.

Note also that the minimal rank is equal to one: indeed, the non covariant BRST charge given by $$\tilde{\Omega}={{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} + {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}} +  ({{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_1 - {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_2)/2 =
\eta^a G_a - G_a \eta^a \eta^1 {{\cal P}}_1$$ is nilpotent and $$\delta({{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_1 - {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_2) = 0$$ so $\tilde{\Omega}$ is indeed a valid BRST charge. This shows that the minimal and covariant ranks are in general different.

