U^* \to
\Hom_\C(U,U^{\prime }) \ ,\ 
u^{\prime}\otimes u^*\mapsto T_{u^{\prime}\otimes u^*},$$ where $T_{u^{\prime}\otimes u^*}(u):=\lag u^*,u\rag u^{\prime }$ for all $u\in U$, is an isomorphism in the category $\SVec$.

The natural representation (in the symmetric monoidal category $\SVec$) of the symmetric group $S_d$ on $U^{\otimes d}$ is explicitly given by $$\label{dfTsigm}
\sigma\mapsto T_{U,d}^\sigma\ \,,\,\ T^{\sigma}_{U,d}( v_1\otimes \cdots
\otimes v_{d}) := (-1)^{\seps\left(\sigma^{-1};v_1,\ldots,v_{d}\right)}
v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(d)}\,,$$ where $$\label{sepssign}
\seps(\sigma;v_1,\ldots,v_d):= \sum_{\substack{ 1\leq r<s\leq d \\ %
\sigma(r)>\sigma(s) }} |v_{\sigma(r)}|\cdot|v_{\sigma(s)}|.$$ The supersymmetrization map $\ssym^d_U:U^{\otimes d}\to U^{\otimes d}$ is defined by $$\label{symdUeq}
\ssym^d_U :=\frac{1}{d!} \sum_{\sigma\in S_d} T^\sigma_{U,d}.$$ We set $$\sS^d(U):=\ssym^d_U\left(U^{\otimes d}\right)
,\  
\sS(U):=\bigoplus_{d\ge 0}\sS
^d(U),\
\sP(U):=\sS(U^*), 
\text{ and }
\sP^d(U):=\sS^d( U^*).$$ Every $\eta\in U^*_\eev$ extends canonically to a homomorphism of superalgebras $$\label{dfheta}
\sfh_\eta:\sS(U)\to \C\cong \C^{1|0}$$ defined by $\sfh_\eta \big(
\ssym^d_U(u_1\otimes \cdots \otimes u_d) \big)
:=\eta(u_1)\cdots \eta(u_d) $ for $d\geq 0$ and $u_1,\ldots,u_d\in U$.

If $\g g$ is a Lie superalgebra, then $\bfU(\g g)$ denotes the universal enveloping algebra of $\g g$, and $$\C=\bfU^0(\g g)\subset
\bfU^1(\g g)\subset\cdots
\subset
\bfU^d(\g g)\subset\cdots$$ denotes the standard filtration of $\bfU(\g g)$. If $(\pi,U)$ is a $\g g$-module, we define $$U^\g g:=\{u\in U\ :\ \pi(x)u=0\text{ for every }x\in\g g\}.$$ If $(\pi^{\prime },U^{\prime })$ is another $\g g$-module, then $\Hom_\C%
(U,U^{\prime })$ is also a 