$W$, and therefore they generate $\sS(W)$. Similarly, the $y_{i,j}$ span $W^*$, and therefore they generate $\sP(W)$. Moreover, $$x_{i,j}=(-1)^{|i|\cdot |j|}x_{j,i},\ y_{i,j}=(-1)^{|i|\cdot |j|}y_{j,i},\ 
\text{and } \lag y_{i,j},x_{p,q}\rag=\delta_{i,p}\delta_{j,q}
+(-1)^{|i|\cdot |j|}\delta_{i,q}\delta_{j,p}.$$

The action of $\g g$ on $\sS(W)$ can be realized by the polarization operators $$\label{polrhoEi-j}
\rho(E_{i,j}):=\sum_{r\in\mathcal{I}_{m,n}}x_{i,r}\sfD_{{j,r}}
\ \text{ for }i,j\in\mathcal I_{m,n},$$ where $\sfD_{{i,j}}:\sS(W)\to \sS(W)$ is the superderivation of parity $%
|i|+|j|$ uniquely defined by $$\label{pari-j}
\sfD_{i,j}(x_{k,l}):=\delta_{i,k}\delta_{j,l}+(-1)^{|i|\cdot|j|}\delta_{i,l}%
\delta_{j,k} \text{ for }i,j,k,l\in\mathcal{I}_{m,n}.$$ Similarly, the action of $\g g$ on $\sP(W)$ is realized by the polarization operators $$\label{ppooll}
\check\rho(E_{i,j})=-(-1)^{|i|\cdot|j|} \sum_{r\in\mathcal{I}_{m,n}}
(-1)^{|r|}y_{r,j}\partial_{{r,i}}\ \text{ for }
i,j\in\mathcal I_{m,n},$$ where $\partial_{{i,j}}:=\partial_{x_{i,j}}$ is the superderivation of $%
\sP(W)$ corresponding to $x_{i,j}\in W$, as in . Let $U=U_\eev\oplus U_\ood$ be a $\Ztwo$-graded vector space. For every $\cA%
\subseteq\mathrm{End}_\C(U)$, we set $$\cA^{\prime }:= \{B\in\mathrm{End}_\C(U)\ :\ [A,B]=0 \,\text{ for every }A\in%
\cA
\}.$$

\[rmk-dblcomm\] Fix finite dimensional $\Ztwo$-graded vector spaces $U$ and $U'$, and set $$\cA:=\mathrm{End}_\C(U)\otimes 1_{U'}\subset
\mathrm{End}_\C(U\otimes U').$$ Then $\cA'=1_U\otimes \mathrm{End}_\C(U')$.

\[UAcomm\] Let $U=U_\eev\oplus U_\ood$ be a finite dimensional $\Ztwo$-graded vector space, and let $\cA\subseteq\mathrm{End}_\C(U)_\eev$ be a semisimple associative algebra. Then $(\cA')'=\cA$.

Since $\cA$ is semisimple and purely even, both $U_\eev$ and $U_\ood$ can be expressed as direct sums of irreducible $\cA$-modules. It follows that $U\cong\bigoplus_{\tau} 
U_\tau\otimes V_\tau,
$ where the $U_\tau$ 