Finally, it is easy to modify slightly the basic brackets so as to induce non zero covariant ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ with $k$ odd. One simply replaces (\[eq:maxquad\]) by $$\begin{aligned}
[G_{n-1}, G_n] &=& G_{n-1} G_n = - [G_n, G_{n-1}] \\{}
[G_1, G_n] &=& - G_1 G_n, \\{}
[G_1, G_{\alpha}] &=& G_1 G_{\alpha} \quad (\alpha \neq n).\end{aligned}$$

Purely Quadratic Algebras
-------------------------

A generalization of the above is the pure quadratic algebra. The basic Poisson brackets are then given by $$[G_{d_1}, G_{d_2}] = D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4}$$ where $D_{d_1 d_2}^{d_3 d_4}$ is antisymmetric in $d_1, d_2$ and symmetric in $d_3, d_4$. The Jacobi identity implies that $$D_{d_4 d_1}^{d_5 d_6} D_{d_2 d_3}^{d_4 d_7} + \mbox{symm}(d_5, d_6,
d_7) + \mbox{cyclic}(d_1,d_2,d_3) = 0$$

The first orders of the covariant BRST charge are given by $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}=
\[G_{d_{1}}\,\eta^{d_{1}}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}=
\[\frac{D_{d_{1}d_{2}}^{
           d_{4}d_{3}}\,G_{d_{4}}\,\eta^{d_{1}}\,
        \eta^{d_{2}}\,{{\cal P}}_{d_{3}}}{
        2}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (2)} \\ \Omega \end{array}}}=
\[\frac{D_{d_{2}d_{3}}^{
           d_{6}d_{5}}\,D_{d_{6}d_{1}}^{
                           d_{7}d_{4}}\,G_{d_{7}}\,
        \eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
        {{\cal P}}_{d_{4}}\,{{\cal P}}_{d_{5}}}{
        6}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (3)} \\ \Omega \end{array}}}=
\[\frac{-
        \(D_{d_{2}d_{3}}^{
             d_{8}d_{7}}\,D_{d_{4}d_{1}}^{
                             d_{9}d_{5}}\,D_{d_{8}d_{9}}^{
          d_{10}d_{6}}\,G_{d_{10}}\,\eta^{d_{1}}\,
          \eta^{d_{2}}\,\eta^{d_{3}}\,\eta^{d_{4}}\,
          {{\cal P}}_{d_{5}}\,{{\cal P}}_{d_{6}}\,{{\cal P}}_{d_{7}}
        \)
        }{
        24}
\]
\cr}$$ 