$\varphi:~{}U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)\overset{\sim}{\longrightarrow}(Q_{\chi}^{\chi})^{\text{ad}{\mathfrak{m}}_{\mathds{k}}}$

in Proposition 2.21 is even, it follows from

$\varphi(\theta)=\varphi(\theta_{\bar{0}})+\varphi(\theta_{\bar{1}})=\theta_{\bar{0}}(\bar{1}_{\chi})+\theta_{\bar{1}}(\bar{1}_{\chi})\in(Q_{\chi}^{\chi})^{\text{ad}{\mathfrak{m}}_{\mathds{k}}}$

that
$\varphi(\theta_{\bar{0}})\in(Q_{\chi}^{\chi})^{\text{ad}{\mathfrak{m}}_{\mathds{k}}}_{\bar{0}}$
and
$\varphi(\theta_{\bar{1}})\in(Q_{\chi}^{\chi})^{\text{ad}{\mathfrak{m}}_{\mathds{k}}}_{\bar{1}}$,
i.e. $\theta_{\bar{0}}\in U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{0}}$
and $\theta_{\bar{1}}\in U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{1}}$.

Thus, for any $\theta\in U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$, if the
leading term of $\theta(\bar{1}_{\chi})$ is $\bar{x}_{i}$ with $1\leqslant
i\leqslant l$ (thereby in $(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$), one can
choose $\theta_{\bar{0}}$ as the desired element in
$U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{0}}$. Similarly, if the leading
term of $\theta(\bar{1}_{\chi})$ is $\bar{y}_{i}$ with $1\leqslant i\leqslant
q$ (thereby in $({\mathfrak{g}}_{\mathds{k}})_{\bar{1}}$), one can choose
$\theta_{\bar{1}}$ as the desired element in
$U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{1}}$. ∎

Recall that $\\{\bar{x}_{1},\cdots,\bar{x}_{l}\\}$ and
$\\{\bar{y}_{1},\cdots,\bar{y}_{q}\\}$ are ${\mathds{k}}$-basis of
$({\mathfrak{g}}^{e}_{\mathds{k}})_{\bar{0}}$ and
$({\mathfrak{g}}^{e}_{\mathds{k}})_{\bar{1}}$, respectively; and there is
another element
$\bar{v}_{\frac{\textsf{r}+1}{2}}\in({\mathfrak{g}}_{\mathds{k}}(-1)_{\bar{1}}^{\prime})^{\bot}\subseteq{\mathfrak{g}}_{\mathds{k}}(-1)_{\bar{1}}$
when r is odd. Set

$\bar{Y}_{i}:=\left\\{\begin{array}[]{ll}\bar{x}_{i}&\text{if}~{}1\leqslant
i\leqslant l;\\\ \bar{y}_{i-l}&\text{if}~{}l+1\leqslant i\leqslant l+q;\\\
\bar{v}_{\frac{\textsf{r}+1}{2}}&\text{if}~{}i=l+q+1\mbox{ whenever
}\textsf{r}\mbox{ is odd}.\end{array}\right.$

With the new notations, the subalgebra $\mathfrak{g}_{\mathds{k}}^{e}$ admits
a basis

$\bar{Y}_{1},\cdots,\bar{Y}_{l+q}.$

Moreover, the element $\bar{Y}_{l+q+1}\notin\mathfrak{g}_{\mathds{k}}^{e}$
occurs only in the case when r is odd. We can assume that the homogenous
element $\bar{Y}_{i}$ is in $\mathfrak{g}_{\mathds{k}}(m_{i})$ for $1\leqslant
i\leqslant l+q+1$. Let $\bar{\theta}_{i}$ denote the image of $\theta_{i}\in
U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$ in
$\text{gr}\,(U_{\chi}({\mathfrak{g}}_{\mathds{k}},e))$.

Now we are in the position to introduce the main result of this section, which
provides the generators and relations for the reduced $W$-superalgebra
$U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$, along with the PBW basis. We will
exploit the arguments of [33, Theorem 3.4] in our super case. We will further
see variation arising from the change of the parity of
$\textsf{r}=\dim\mathfrak{g}_{\mathds{k}}(-1)_{\bar{1}}$. In the remaining
part of the paper, we always set $q^{\prime}=q$ if r is even, and
$q^{\prime}=q+1$ if r is odd.

######  Theorem 3.7.

All the elements given in Corollary 3.6 constitute a set of generators of
$U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$. Their relations are presented as
below.

  * (1)

For $1\leqslant i,j\leqslant l+q^{\prime}$ , we have

$[\theta_{i},\theta_{j}]=\theta_{i}\cdot\theta_{j}-(-1)^{|\theta_{i}||\theta_{j}|}\theta_{j}\cdot\theta_{i}=(-1)^{|\theta_{i}|\cdot|\theta_{j}|}\theta_{j}\circ\theta_{i}-\theta_{i}\circ\theta_{j}\in
H^{m_{i}+m_{j}+2}.$

Here and thereinafter $\theta_{i}\cdot\theta_{j}$ is just the algebra
multiplication of two elements in $U_{\chi}(\mathfrak{g}_{\mathds{k}},e)$
while $\theta_{i}\circ\theta_{j}$ means the composition of two transformations
on $Q_{\chi}^{\chi}$.

  * (2)

The Lie bracket relations in the Lie superalgebra
$\mathfrak{g}_{\mathds{k}}^{e}$ with

$[\bar{Y}_{i},\bar{Y}_{j}]=\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\bar{Y}_{k}\quad\text{for}~{}1\leqslant
i,j\leqslant l+q$

imply

$\displaystyle[\theta_{i},\theta_{j}]$
$\displaystyle\equiv\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\theta_{k}+q_{ij}(\theta_{1},\cdots,\theta_{l+q^{\prime}})~{}(\mbox{mod
}H^{m_{i}+m_{j}+1}),$

where $q_{ij}$ with $1\leqslant i,j\leqslant l+q$ is a truncated super-
polynomial in $l+q^{\prime}$ variables whose constant term and linear part are
both zero. When r is odd, we have additionally

$[\theta_{l+q+1},\theta_{l+q+1}]=\text{id}.$ (3.13)

  * (3)

The monomials
$\bar{\theta}_{1}^{a_{1}}\cdots\bar{\theta}_{l}^{a_{l}}\bar{\theta}_{l+1}^{b_{1}}\cdots\bar{\theta}_{l+q^{\prime}}^{b_{q^{\prime}}}$
and
$\theta_{1}^{a_{1}}\cdots\theta_{l}^{a_{l}}\theta_{l+1}^{b_{1}}\cdots\theta_{l+q^{\prime}}^{b_{q^{\prime}}}$
form bases of $\text{gr}\,(U_{\chi}({\mathfrak{g}}_{\mathds{k}},e))$ and
$U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$ respectively, where $0\leqslant
a_{i}\leqslant p-1$ for $1\leqslant i\leqslant l$, and $b_{i}\in\\{0,1\\}$ for
$1\leqslant i\leqslant q^{\prime}$.

###### Proof.

The arguments for both cases of r being even and odd are almost the same, with
the latter case being more complicated. We will only consider the latter case,
i.e. the case when r is odd. Firstly, it can be easily verified that

$[\theta_{i},\theta_{j}]=\theta_{i}\cdot\theta_{j}-(-1)^{|\theta_{i}||\theta_{j}|}\theta_{j}\cdot\theta_{i}=(-1)^{|\theta_{i}|\cdot|\theta_{j}|}\theta_{j}\circ\theta_{i}-\theta_{i}\circ\theta_{j}$
(3.14)

by the definition of the $\mathds{k}$-algebra
$U_{\chi}(\mathfrak{g}_{\mathds{k}},e)$.

Now let us next prove (3). Firstly recall the elements
$\theta_{1},\cdots,\theta_{l}\in
U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{0}}$ and
$\theta_{l+1},\cdots,\theta_{l+q+1}\in
U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{1}}$ in Corollary 3.6. Combing
with (3.14), the arguments by induction on $|\mathbf{a}|+|\mathbf{b}|+c$ show
that

$\begin{array}[]{cl}&\theta_{1}^{a_{1}}\cdots\theta_{l}^{a_{l}}\theta_{l+1}^{b_{1}}\cdots\theta_{l+q}^{b_{q}}\theta_{l+q+1}^{c}(\bar{1}_{\chi})\\\
=&(\bar{Y}_{1}^{a_{1}}\cdots\bar{Y}_{l}^{a_{l}}\bar{Y}_{l+1}^{b_{1}}\cdots\bar{Y}_{l+q}^{b_{q}}\bar{Y}_{l+q+1}^{c}+\sum\limits_{\mbox{\tiny$\begin{array}[]{c}|(\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g})|_{e}=|(\mathbf{a},\mathbf{b},\mathbf{0},c\mathbf{e}_{\frac{\textsf{r}+1}{2}})|_{e},\\\
|\mathbf{i}|+|\mathbf{j}|+|\mathbf{f}|+|\mathbf{g}|>|\mathbf{a}|+|\mathbf{b}|+c\end{array}$}}\lambda^{\mathbf{a},\mathbf{b},\mathbf{0},c\mathbf{e}_{\frac{\textsf{r}+1}{2}}}_{\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g}}\bar{x}^{\mathbf{i}}\bar{y}^{\mathbf{j}}\bar{u}^{\mathbf{f}}\bar{v}^{\mathbf{g}}\\\
&+\text{terms of lower}~{}e\text{-degree})\otimes\bar{1}_{\chi}\end{array}$

for any
$(\mathbf{a},\mathbf{b},c)=(a_{1},\cdots,a_{l},b_{1},\cdots,b_{q},c)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}$
(the induction step is based on Corollary 3.6 and Lemma 3.2). Due to
Proposition 3.5 we have that

$\theta_{1}^{a_{1}}\cdots\theta_{l+q+1}^{c}=\mu_{\mathbf{a},\mathbf{b},c}\bar{h}_{\mathbf{a},\mathbf{b},c}+\sum\limits_{(\mathbf{i},\mathbf{j},k)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}}\mu_{\mathbf{i},\mathbf{j},k}\bar{h}_{\mathbf{i},\mathbf{j},k},\qquad\mu_{\mathbf{a},\mathbf{b},c}\neq
0,$ (3.15)

where $\mu_{\mathbf{i},\mathbf{j},k}=0$ unless
$(n(\bar{h}_{\mathbf{i},\mathbf{j},k}),N(\bar{h}_{\mathbf{i},\mathbf{j},k}))\prec(n(\bar{h}_{\mathbf{a},\mathbf{b},c}),N(\bar{h}_{\mathbf{a},\mathbf{b},c}))$.

Since this establishes for any
$(\mathbf{a},\mathbf{b},c)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}$,
the monomials $\theta_{1}^{a_{1}}\cdots\theta_{l+q+1}^{c}$ with
$(\mathbf{a},\mathbf{b},c)=(a_{1},\cdots,a_{l},b_{1},\cdots,b_{q},c)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}$
form a basis of $U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$. It follows from the
proof of Proposition 3.5 that for any positive integer $M$, the cosets
$\bar{h}_{\mathbf{i},\mathbf{j},k}+H^{M-1}$ with
$(\mathbf{i},\mathbf{j},k)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}$
and $\sum\limits_{1\leqslant f\leqslant
l}i_{f}(m_{f}+2)+\sum\limits_{1\leqslant g\leqslant q}j_{g}(m_{g+l}+2)+k=M$
(recall that $m_{f}^{\prime}s,~{}m_{g+l}^{\prime}s$ are the weights of
$\bar{Y}_{f}$’s ($1\leqslant f\leqslant l$) and $\bar{Y}_{g+l}$’s ($1\leqslant
g\leqslant q$), respectively) form a basis of
$\text{gr}(U_{\chi}({\mathfrak{g}}_{\mathds{k}},e))$. Due to (3.15) and Lemma
3.2 the cosets
$\theta_{1}^{i_{1}}\cdots\theta_{l}^{i_{l}}\theta_{l+1}^{j_{1}}\cdots\theta_{l+q}^{j_{q}}\theta_{l+q+1}^{k}+H^{M-1}$
with
$(\mathbf{i},\mathbf{j},k)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}$
and $\sum\limits_{1\leqslant f\leqslant
l}i_{f}(m_{f}+2)+\sum\limits_{1\leqslant g\leqslant q}j_{g}(m_{g+l}+2)+k=M$
have the same property.

Now for the completion of the proof (3), we note that

$\bar{\theta}_{1}^{i_{1}}\cdots\bar{\theta}_{l}^{i_{l}}\bar{\theta}_{l+1}^{j_{1}}\cdots\bar{\theta}_{l+q}^{j_{q}}\bar{\theta}_{l+q+1}^{k}=\theta_{1}^{i_{1}}\cdots\theta_{l}^{i_{l}}\theta_{l+1}^{j_{1}}\cdots\theta_{l+q}^{j_{q}}\theta_{l+q+1}^{k}+H^{M-1}$

for any
$(\mathbf{i},\mathbf{j},k)\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}$
with $\sum\limits_{1\leqslant f\leqslant
l}i_{f}(m_{f}+2)+\sum\limits_{1\leqslant g\leqslant q}j_{g}(m_{g+l}+2)+k=M$.
Hence the monomials listed in the first part of statement (3) constitute a
basis of $\text{gr}(U_{\chi}({\mathfrak{g}}_{\mathds{k}},e))$. We complete the
proof for (3).

Now we prove the statement (2). As in the proof of Lemma 3.2, the induction on
$|\mathbf{d}|$, then on $|\mathbf{c}|$, yields

$\begin{array}[]{ll}&(\rho_{\chi}(\bar{u}^{\mathbf{c}}\bar{v}^{\mathbf{d}}))(\bar{x}^{\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}^{\prime}}\bar{u}^{\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}^{\prime}}\otimes\bar{1}_{\chi})\\\
=&(K^{\prime}\bar{x}^{\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}^{\prime}}\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i,j}\kappa_{i,j}\bar{x}^{\mathbf{a}^{\prime}-\mathbf{e}_{i}}\bar{y}^{\mathbf{b}^{\prime}}\rho_{\chi}([\bar{u}_{j},\bar{x}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}-\mathbf{e}_{j}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}\\\
&+\sum\limits_{i,j}\lambda_{i,j}\bar{x}^{\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}^{\prime}-\mathbf{e}_{i}}\rho_{\chi}([\bar{u}_{j},\bar{y}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}-\mathbf{e}_{j}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i,j}\omega_{i,j}\bar{x}^{\mathbf{a}^{\prime}-\mathbf{e}_{i}}\bar{y}^{\mathbf{b}^{\prime}}\rho_{\chi}([\bar{v}_{j},\bar{x}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\cdot\\\
&\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}-\mathbf{e}_{j}}+\sum\limits_{i,j}\nu_{i,j}\bar{x}^{\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}^{\prime}-\mathbf{e}_{i}}\rho_{\chi}([\bar{v}_{j},\bar{y}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}-\mathbf{e}_{j}}+\text{terms
of}~{}e\text{-degree}\leqslant|(\mathbf{a}^{\prime},\mathbf{b}^{\prime},\\\
&\mathbf{c}+\mathbf{c}^{\prime},\mathbf{d}+\mathbf{d}^{\prime})|_{e}-3)\otimes\bar{1}_{\chi};\end{array}$

it follows that

$\begin{array}[]{ll}&(\rho_{\chi}(\bar{x}^{\mathbf{a}}\bar{y}^{\mathbf{b}}\bar{u}^{\mathbf{c}}\bar{v}^{\mathbf{d}}))(\bar{x}^{\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}^{\prime}}\bar{u}^{\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}^{\prime}}\otimes\bar{1}_{\chi})\\\
=&(K\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}}\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i<j}\alpha_{i,j}\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}-\mathbf{e}_{i}-\mathbf{e}_{j}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}}\rho_{\chi}([\bar{x}_{i},\bar{x}_{j}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i,j}\beta_{i,j}\cdot\\\
&\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}-\mathbf{e}_{i}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}-\mathbf{e}_{j}}\rho_{\chi}([\bar{x}_{i},\bar{y}_{j}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i<j}\gamma_{i,j}\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}-\mathbf{e}_{i}-\mathbf{e}_{j}}\rho_{\chi}([\bar{y}_{i},\bar{y}_{j}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\cdot\\\
&\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i,j}\kappa^{\prime}_{i,j}\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}-\mathbf{e}_{i}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}}\rho_{\chi}([\bar{u}_{j},\bar{x}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}-\mathbf{e}_{j}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i,j}\lambda^{\prime}_{i,j}\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}-\mathbf{e}_{i}}\cdot\\\
&\rho_{\chi}([\bar{u}_{j},\bar{y}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}-\mathbf{e}_{j}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}}+\sum\limits_{i,j}\omega^{\prime}_{i,j}\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}-\mathbf{e}_{i}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}}\rho_{\chi}([\bar{v}_{j},\bar{x}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}-\mathbf{e}_{j}}+\sum\limits_{i,j}\nu^{\prime}_{i,j}\cdot\\\
&\bar{x}^{\mathbf{a}+\mathbf{a}^{\prime}}\bar{y}^{\mathbf{b}+\mathbf{b}^{\prime}-\mathbf{e}_{i}}\rho_{\chi}([\bar{v}_{j},\bar{y}_{i}])\bar{u}^{\mathbf{c}+\mathbf{c}^{\prime}}\bar{v}^{\mathbf{d}+\mathbf{d}^{\prime}-\mathbf{e}_{j}}+\text{terms
of}~{}e\text{-degree}\leqslant|(\mathbf{a}+\mathbf{a}^{\prime},\mathbf{b}+\mathbf{b}^{\prime},\mathbf{c}\\\
&+\mathbf{c}^{\prime},\mathbf{d}+\mathbf{d}^{\prime})|_{e}-3)\otimes\bar{1}_{\chi};\end{array}$

where all the coefficients above are in $\mathds{k}$, and
$K^{\prime},K\in{\mathds{k}}$ are in the same sense as in Lemma 3.2. Together
with Corollary 3.6, for $1\leqslant i,j\leqslant l+q$ we have

$\begin{array}[]{cl}&[\theta_{i},\theta_{j}](\bar{1}_{\chi})\\\
=&(\theta_{i}\cdot\theta_{j})(\bar{1}_{\chi})-(-1)^{|\theta_{i}||\theta_{j}|}(\theta_{j}\cdot\theta_{i})(\bar{1}_{\chi})\\\
=&([\bar{Y}_{i},\bar{Y}_{j}]+\sum\limits_{\mbox{\tiny$\begin{array}[]{c}|\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g}|_{e}=m_{i}+m_{j}+2,\\\
|\mathbf{i}|+|\mathbf{j}|+|\mathbf{f}|+|\mathbf{g}|\geqslant
2\end{array}$}}\mu_{\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g}}\bar{x}^{\mathbf{i}}\bar{y}^{\mathbf{j}}\bar{u}^{\mathbf{f}}\bar{v}^{\mathbf{g}}+\sum\limits_{|(\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g})|_{e}<m_{i}+m_{j}+2}\mu_{\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g}}\bar{x}^{\mathbf{i}}\bar{y}^{\mathbf{j}}\bar{u}^{\mathbf{f}}\bar{v}^{\mathbf{g}})\\\
&\otimes\bar{1}_{\chi},\end{array}$ (3.16)

where $\mu_{\mathbf{i},\mathbf{j},\mathbf{f},\mathbf{g}}\in{\mathds{k}}$.
Since
$[\bar{Y}_{i},\bar{Y}_{j}]=\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\bar{Y}_{k}$
for $1\leqslant i,j\leqslant l+q$ by the assumption in the theorem, taking
$\pi_{m_{i}+m_{j}+2,m_{i}+m_{j}}$ on both sides of (3.16), we have

$\pi_{m_{i}+m_{j}+2,m_{i}+m_{j}}(([\theta_{i},\theta_{j}]-\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\theta_{k})(\bar{1}_{\chi}))=0.$

On the other hand, the arguments in the proof for (3) show that there exists a
unique truncated super-polynomial $\tilde{q}_{ij}$ in
$\bar{Y}_{1},\cdots,\bar{Y}_{l+q+1}$ such that

$[\theta_{i},\theta_{j}]-\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\theta_{k}=\tilde{q}_{ij}(\theta_{1},\cdots,\theta_{l+q+1}).$

Taking the last three equalities into account, we have that the linear part of
$\tilde{q}_{ij}$ involves only those $\bar{Y}_{1},\cdots,\bar{Y}_{l+q+1}$
whose weights $<m_{i}+m_{j}$. Hence there exists a truncated super-polynomial
$q_{ij}$ in $l+q+1$ variables with initial form of degree at least $2$ such
that

$[\theta_{i},\theta_{j}]-\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\theta_{k}-q_{ij}(\theta_{1},\cdots,\theta_{l+q+1})\in
H^{m_{i}+m_{j}+1}$

for $1\leqslant i,j\leqslant l+q$.

It is immediate from the assumption in §2.2 that

$[\bar{v}_{\frac{\textsf{r}+1}{2}},\bar{v}_{\frac{\textsf{r}+1}{2}}]\otimes\bar{1}_{\chi}=1\otimes\langle\bar{v}_{\frac{\textsf{r}+1}{2}},\bar{v}_{\frac{\textsf{r}+1}{2}}\rangle\bar{1}_{\chi}=1\otimes\bar{1}_{\chi}.$

We have further

$[\theta_{l+q+1},\theta_{l+q+1}](\bar{1}_{\chi})=2\bar{v}_{\frac{\textsf{r}+1}{2}}^{2}\otimes\bar{1}_{\chi}=[\bar{v}_{\frac{\textsf{r}+1}{2}},\bar{v}_{\frac{\textsf{r}+1}{2}}]\otimes\bar{1}_{\chi}=1\otimes\bar{1}_{\chi},$

i.e. $[\theta_{l+q+1},\theta_{l+q+1}]=\text{id}$. The statement (2) is proved.

Finally, by the equations in (3.14) and (3.16), the statement (1) in the
theorem follows.

Summing up, we complete the proof. ∎

######  Remark 3.8.

(1) It is notable that Theorem 3.7 shows that the parity of r plays the key
role for the construction of reduced $W$-superalgebras, which makes the
structure and representation theory of reduced $W$-superalgebras distinguished
from that of reduced $W$-algebras (see [33, §3] for ordinary reduced
$W$-algebras).

(2) Recall that all the indices $i,j$ in Theorem 3.7(2) are assumed to be in
the set $\\{1,\cdots,l+q\\}$. In this case, there exist truncated super-
polynomials $\bar{F}_{ij}$ of $l+q^{\prime}$ indeterminants over $\mathds{k}$
($i,j=1,\cdots,l+q$) with the first $l$ indeterminants being even, and the
others being odd, such that

$\displaystyle[\theta_{i},\theta_{j}]=\bar{F}_{i,j}(\theta_{1},\cdots,\theta_{l+q^{\prime}}),\quad
i,j=1,\cdots,l+q,$ (3.17)

while
$\bar{F}_{i,j}(\theta_{1},\cdots,\theta_{l+q^{\prime}})\equiv\sum_{k=1}^{l+q}\alpha_{ij}^{k}\theta_{k}+q_{ij}(\theta_{1},\cdots,\theta_{l+q^{\prime}})(\mbox{mod
}H^{m_{i}+m_{j}+1})$ for $i,j=1,\cdots,l+q$ (Note that hereinafter the super
polynomials in (3.17) are “formal” ones, which can be endowed with real
meaning when considered as the image under “gradation” associated with Kazhdan
filtration. This point is similar to the one happening in the ordinary Lie
algebra case (cf. [37, §2.2])).

(3) For the case when r is odd, if one of the indices $i,j$ happens to be
$l+q+1$, then it is hard to derive an explicit formulas for
$[\bar{Y}_{i},\bar{Y}_{l+q+1}]$ and $[\bar{Y}_{l+q+1},\bar{Y}_{j}]$ since
$\mathfrak{g}_{\mathds{k}}^{e}\oplus\mathds{k}\bar{v}_{\frac{\textsf{r}+1}{2}}$
is not necessary a Lie superalgebra. However, by Lemma 3.2 and Theorem 3.7(3)
one can still find truncated super-polynomials $\bar{F}_{i,l+q+1}$’s and
$\bar{F}_{l+q+1,j}$’s in $l+q+1$ indeterminants over $\mathds{k}$ ($1\leqslant
i,j\leqslant l+q$) such that

$\displaystyle[\theta_{i},\theta_{l+q+1}]=\bar{F}_{i,l+q+1}(\theta_{1},\cdots,\theta_{l+q+1}),\quad[\theta_{l+q+1},\theta_{j}]=\bar{F}_{l+q+1,j}(\theta_{1},\cdots,\theta_{l+q+1}),$
(3.18)

where the $e$-degree for all the monomials in the $\mathds{k}$-span of
$\bar{F}_{i,l+q+1}$’s and $\bar{F}_{l+q+1,j}$’s is less than $m_{i}+1$ and
$m_{j}+1$, respectively.

(4) It is notable that
$[\theta_{i},\theta_{j}]=-(-1)^{|\theta_{i}||\theta_{j}|}[\theta_{j},\theta_{i}]$
for any $1\leqslant i,j\leqslant l+q^{\prime}$. In particular, one can deduce
that $[\theta_{i},\theta_{i}]=0$ for $1\leqslant i\leqslant l$ as $\theta_{i}$
is always an even element in $U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$.
Therefore, after deleting all the redundant commutating relations in (3.13),
(3.17) and (3.18), the remaining ones are with indices $i,j$ satisfying
$1\leqslant i<j\leqslant l+q^{\prime}$ and $l+1\leqslant i=j\leqslant
l+q^{\prime}$.

######  Remark 3.9.

More generally, one can consider a reduced $W$-superalgebra
$U_{\eta}(\mathfrak{g}_{\mathds{k}},e)$ associated with a $p$-character
$\eta\in\chi+(\mathfrak{m}_{\mathds{k}}^{\perp})_{\bar{0}}$ (see Definition
2.20). Proposition 2.21 and Theorem 2.24(1) enable us to extend the arguments
in this section for the case $U_{\chi}(\mathfrak{g}_{\mathds{k}},e)$ to the
general case $U_{\eta}(\mathfrak{g}_{\mathds{k}},e)$. Especially, we can
rewrite Theorem 3.7(3) as follows.

_There exist even elements $\theta_{1},\cdots,\theta_{l}\in
U_{\eta}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{0}}$ and odd elements
$\theta_{l+1},\cdots,\theta_{l+q^{\prime}}\in
U_{\eta}({\mathfrak{g}}_{\mathds{k}},e)_{\bar{1}}$ in the same sense as in
Corollary 3.6, such that the monomials_

$\theta_{1}^{a_{1}}\cdots\theta_{l}^{a_{l}}\theta_{l+1}^{b_{1}}\cdots\theta_{l+q^{\prime}}^{b_{q^{\prime}}}$

_with $0\leqslant a_{k}\leqslant p-1$ for $1\leqslant k\leqslant l$ and
$b_{k}\in\\{0,1\\}$ for $1\leqslant k\leqslant q^{\prime}$ form a
${\mathds{k}}$-basis of $U_{\eta}({\mathfrak{g}}_{\mathds{k}},e)$._

##  4\. The structure of finite $W$-superalgebras over $\mathbb{C}$

Maintain the notations as the previous sections. Especially, let
$\mathfrak{g}$ be a basic Lie superalgebra over $\mathbb{C}$ with a given
nilpotent element $e\in\mathfrak{g}_{\bar{0}}$, $A$ be an admissible ring
associated with the pair $(\mathfrak{g},e)$.

In this concluding section, we will establish the structure theory of finite
$W$-superalgebra $U(\mathfrak{g},e)$ over $\mathbb{C}$, parallel to that over
$\mathds{k}$. The approach is the procedure of “admissible” via the admissible
ring $A$ associated to the basic Lie superalgebra $\mathfrak{g}$, which was
exploited by Premet in the study of finite $W$-algebras (cf. [33]). The PBW
theorem of finite $W$-superalgebra $U({\mathfrak{g}},e)$ over $\mathbb{C}$
will be present in §4.2.

We need recall some primary conventions parallel to §3.1. For example, for
$k\in\mathbb{Z}_{+}$, define

$\begin{array}[]{llllll}\mathbb{Z}_{+}^{k}&:=&\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in\mathbb{Z}_{+}\\},&\Lambda^{\prime}_{k}&:=&\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in\\{0,1\\}\\}\end{array}$

with $1\leqslant j\leqslant k$. As in §2.2 and §2.5, there is a set of
“monomials basis” in $Q_{\chi}$ over $\mathbb{C}$ as follows

$x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}\otimes
1_{\chi}:=x_{1}^{a_{1}}\cdots x_{m}^{a_{m}}y_{1}^{b_{1}}\cdots
y_{n}^{b_{n}}u_{1}^{c_{1}}\cdots u_{s}^{c_{s}}v_{1}^{d_{1}}\cdots
v_{t}^{d_{t}}\otimes 1_{\chi},$

where $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$ runs over
$\mathbb{Z}_{+}^{m}\times\Lambda^{\prime}_{n}\times\mathbb{Z}_{+}^{s}\times\Lambda^{\prime}_{t}$.
(recall that
$t=\lfloor\frac{\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}}{2}\rfloor$)

Assume that $\text{wt}(x_{i})=k_{i}$, $\text{wt}(y_{j})=k^{\prime}_{j}$, i.e.
$x_{i}\in{\mathfrak{g}}(k_{i})_{\bar{0}}$ and
$y_{j}\in{\mathfrak{g}}(k^{\prime}_{j})_{\bar{1}}$ where $1\leqslant
i\leqslant m$ and $1\leqslant j\leqslant n$. Given
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\mathbb{Z}_{+}^{m}\times\Lambda^{\prime}_{n}\times\mathbb{Z}_{+}^{s}\times\Lambda^{\prime}_{t}$,
set

$|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}:=\sum_{i=1}^{m}a_{i}(k_{i}+2)+\sum_{i=1}^{n}b_{i}(k^{\prime}_{i}+2)+\sum_{i=1}^{s}c_{i}+\sum_{i=1}^{t}d_{i},$

and say $x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}$ to have
$e$-degree $|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}$, which is
written as
$\text{deg}_{e}(x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}})=|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}$.

###  4.1. Some Lemmas

Parallel to Lemma 3.1, one can easily obtain that

######  Lemma 4.1.

For any homogeneous element $w\in U({\mathfrak{g}})_{i}$
($i\in\mathbb{Z}_{2}$), we have

$w\cdot
x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}=\sum_{\mathbf{i}\in\mathbb{Z}_{+}^{m}}\sum_{\mathbf{j}\in\Lambda_{n}^{\prime}}\left(\begin{array}[]{@{\hspace{0pt}}c@{\hspace{0pt}}}\mathbf{a}\\\
\mathbf{i}\end{array}\right)x^{\mathbf{a}-\mathbf{i}}y^{\mathbf{b}-\mathbf{j}}\cdot[wx^{\mathbf{i}}y^{\mathbf{j}}]\cdot
u^{\mathbf{c}}v^{\mathbf{d}},$

where
$\mathbf{a}\choose\mathbf{i}$$=\prod\limits_{l^{\prime}=1}^{m}$$a_{l^{\prime}}\choose
i_{l^{\prime}}$, and

$[wx^{\mathbf{i}}y^{\mathbf{j}}]=k_{1,b_{1},j_{1}}\cdots
k_{n,b_{n},j_{n}}(-1)^{|\mathbf{i}|}(\text{ad}y_{n})^{j_{n}}\cdots(\text{ad}y_{1})^{j_{1}}(\text{ad}x_{m})^{i_{m}}\cdots(\text{ad}x_{1})^{i_{1}}(w),$

in which the coefficients
$k_{1,b_{1},j_{1}},\cdots,k_{n,b_{n},j_{n}}\in\mathbb{C}$ (note that
$b_{1},\cdots,b_{n},j_{1},\cdots,j_{n}\in\\{0,1\\}$) are defined by

$k_{t^{\prime},0,0}=1,k_{t^{\prime},0,1}=0,k_{t^{\prime},1,0}=(-1)^{|w|+j_{1}+\cdots+j_{{t^{\prime}}-1}},k_{t^{\prime},1,1}=(-1)^{|w|+1+j_{1}+\cdots+j_{{t^{\prime}}-1}},$

where $1\leqslant{t^{\prime}}\leqslant n$ ($j_{0}$ is interpreted as $0$).

Let $\tilde{\rho}_{\chi}$ denote the representation of $U({\mathfrak{g}})$ in
$\text{End}Q_{\chi}$, then we have

######  Lemma 4.2.

Let
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}),~{}(\mathbf{a}^{\prime},\mathbf{b}^{\prime},\mathbf{c}^{\prime},\mathbf{d}^{\prime})\in\mathbb{Z}_{+}^{m}\times\Lambda^{\prime}_{n}\times\mathbb{Z}_{+}^{s}\times\Lambda^{\prime}_{t}$
be such that
$|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}=A,~{}|(\mathbf{a}^{\prime},\mathbf{b}^{\prime},\mathbf{c}^{\prime},\mathbf{d}^{\prime})|_{e}=B$.
Then

$\begin{array}[]{ccl}(\tilde{\rho}_{\chi}(x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}))(x^{\mathbf{a}^{\prime}}y^{\mathbf{b}^{\prime}}u^{\mathbf{c}^{\prime}}v^{\mathbf{d}^{\prime}}\otimes
1_{\chi})&=&(Cx^{\mathbf{a}+\mathbf{a}^{\prime}}y^{\mathbf{b}+\mathbf{b}^{\prime}}u^{\mathbf{c}+\mathbf{c}^{\prime}}v^{\mathbf{d}+\mathbf{d}^{\prime}}+\text{terms
of}~{}e\text{-degree}\\\ &&\leqslant A+B-2)\otimes 1_{\chi},\end{array}$

where the coefficient $C\in\\{-1,0,1\\}\subseteq\mathbb{C}$ is subject to the
following items:

  * (1)

$C=0$ if and only if
$(\mathbf{b}+\mathbf{b}^{\prime},\mathbf{d}+\mathbf{d}^{\prime})\notin\Lambda^{\prime}_{n}\times\Lambda^{\prime}_{t}$;

  * (2)

When
$(\mathbf{b}+\mathbf{b}^{\prime},\mathbf{d}+\mathbf{d}^{\prime})\in\Lambda^{\prime}_{n}\times\Lambda^{\prime}_{t}$,
then
$C=(-1)^{\tau(\mathbf{b},\mathbf{d},\mathbf{b}^{\prime},\mathbf{d}^{\prime})}$.
Here $\tau(\mathbf{b},\mathbf{d},\mathbf{b}^{\prime},\mathbf{d}^{\prime})$ has
the same meaning as in Lemma 3.2.

For the proof of the above lemma, we can repeat that of Lemma 3.2 applying
Lemma 4.1 in place of Lemma 3.1. We omit the details.

Now we can get the $\mathbb{C}$-analogue of the leading-terms lemma. Recall
any $0\neq h\in U({\mathfrak{g}},e)$ is uniquely determined by its value
$h(1_{\chi})\in Q_{\chi}$. For $h\neq 0$ we let $n(h),~{}N(h)$ and
$\Lambda_{h}^{\text{max}}$ be the same meaning as in §3.1.6.

######  Lemma 4.3.

Let $h\in U({\mathfrak{g}},e)\backslash\\{0\\}$ and
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda^{\text{max}}_{h}$.
Then $\mathbf{c}=\mathbf{0}$, and
$\mathbf{a}\in\mathbb{Z}_{+}^{l}\times\\{\mathbf{0}\\},~{}\mathbf{b}\in\Lambda^{\prime}_{q}\times\\{\mathbf{0}\\}$.
Moreover, the sequence $\mathbf{d}$ satisfies

(1) $\mathbf{d}=\mathbf{0}$ when dim ${\mathfrak{g}}(-1)_{\bar{1}}$ is even;

(2)
$\mathbf{d}=\\{\mathbf{0}\\}_{\frac{\textsf{r}-1}{2}}\times\Lambda^{\prime}_{1}$
when dim ${\mathfrak{g}}(-1)_{\bar{1}}$ is odd.

###### Proof.

Repeat verbatim the proof of Lemma 3.3 but apply Lemma 4.2 in place of Lemma
3.2. ∎

######  Remark 4.4.

In [30, Theorem 2.5], Poletaeva-Serganova also formulate the similar statement
to Lemma 4.3 bypassing Lemma 4.1 and Lemma 4.2. However, Lemma 4.1 and Lemma
4.2 will play the key role for the construction theory of finite
$W$-superalgebra $U({\mathfrak{g}},e)$ in Theorem 4.5.

###  4.2. The PBW structure theory of finite $W$-superalgebras over
$\mathbb{C}$

For $k\in\mathbb{Z}_{+}$ we denote by $\tilde{H}^{k}$ the linear span of all
$0\neq h\in U({\mathfrak{g}},e)$ with $n(h)\leqslant k$. Put
$\tilde{H}^{-1}=0$. It follows from Lemma 4.2 that the subspaces
$\\{\tilde{H}^{i}\,|\,i\in\mathbb{Z}_{+}\\}$ form a filtration of the algebra
$U({\mathfrak{g}},e)$. Moreover, Lemma 4.2 implies that the graded algebra
$\text{gr}(U({\mathfrak{g}},e))=\bigoplus\limits_{i\geqslant
0}\tilde{H}^{i}/\tilde{H}^{i-1}$ is supercommutative.

Recall that $\\{x_{1},\cdots,x_{l}\\}$ and $\\{y_{1},\cdots,y_{q}\\}$ are
$\mathbb{C}$-basis of ${\mathfrak{g}}^{e}_{\bar{0}}$ and
${\mathfrak{g}}^{e}_{\bar{1}}$, respectively. When dim
${\mathfrak{g}}(-1)_{\bar{1}}$ is odd, there is
$v_{\frac{\textsf{r}+1}{2}}\in{\mathfrak{g}}(-1)_{\bar{1}}\cap({\mathfrak{g}}(-1)^{\prime}_{\bar{1}})^{\bot}$.
Set

$Y_{i}:=\left\\{\begin{array}[]{ll}x_{i}&\text{if}~{}1\leqslant i\leqslant
l;\\\ y_{i-l}&\text{if}~{}l+1\leqslant i\leqslant l+q;\\\
v_{\frac{\textsf{r}+1}{2}}&\text{if}~{}i=l+q+1\mbox{ whenever
}\textsf{r}\mbox{ is odd}.\end{array}\right.$

By assumption it is immediate that $Y_{i}\in{\mathfrak{g}}^{e}$ for
$1\leqslant i\leqslant l+q$. The term $Y_{l+q+1}\notin{\mathfrak{g}}^{e}$
occurs only when $\textsf{r}=\dim\mathfrak{g}(-1)_{\bar{1}}$ is odd. Assume
that $Y_{i}$ falls in ${\mathfrak{g}}(m_{i})$ for $1\leqslant i\leqslant
l+q+1$. In the sequent arguments, we further set

$q^{\prime}:=\begin{cases}q&\mbox{ if }\textsf{r}\mbox{ is even; }\cr
q+1&\mbox{ if }\textsf{r}\mbox{ is odd}.\end{cases}$

Let $\tilde{\Theta}_{i}$ denote the image of $\Theta_{i}\in
U({\mathfrak{g}},e)$ in $\text{gr}\,(U({\mathfrak{g}},e))$. We are in the
position to present the PBW structure of finite $W$-superalgebras over
$\mathbb{C}$. We will further understand, as seeing in the last section, that
the variation of the parity of r gives rise to the change of the structure of
finite $W$-superalgebras.

######  Theorem 4.5.

Let $U({\mathfrak{g}},e)$ be a finite $W$-superalgebra over $\mathbb{C}$. The
following PBW structural statements for $U(\mathfrak{g},e)$ hold,
corresponding to the cases when $\textsf{r}=\dim\mathfrak{g}(-1)_{{\bar{1}}}$
is even and when r is odd, respectively.

  * (1)

There exist homogeneous elements $\Theta_{1},\cdots,\Theta_{l}\in
U({\mathfrak{g}},e)_{\bar{0}}$ and
$\Theta_{l+1},\cdots,\Theta_{l+q^{\prime}}\in U({\mathfrak{g}},e)_{\bar{1}}$
such that

$\begin{array}[]{ll}&\Theta_{k}(1_{\chi})\\\
=&(Y_{k}+\sum\limits_{\mbox{\tiny$\begin{array}[]{c}|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_{e}=m_{k}+2,\\\
|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\geqslant
2\end{array}$}}\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}+\sum\limits_{|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_{e}<m_{k}+2}\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}})\otimes
1_{\chi}\end{array}$

for $1\leqslant k\leqslant l+q$, where
$\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\in\mathbb{Q}$, and
$\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}=0$ if
$a_{l+1}=\cdots=a_{m}=b_{q+1}=\cdots=b_{n}=c_{1}=\cdots=c_{s}=d_{1}=\cdots=d_{\lceil\frac{\textsf{r}}{2}\rceil}=0$.

Additionally set $\Theta_{l+q+1}(1_{\chi})=v_{\frac{\textsf{r}+1}{2}}\otimes
1_{\chi}$ when r is odd.

  * (2)

The monomials
$\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q^{\prime}}^{b_{q^{\prime}}}$
with $a_{i}\in\mathbb{Z}_{+},~{}b_{j}\in\Lambda^{\prime}_{1}$ for $1\leqslant
i\leqslant l$ and $1\leqslant j\leqslant q^{\prime}$ form a basis of
$U({\mathfrak{g}},e)$ over $\mathbb{C}$.

  * (3)

For $1\leqslant i\leqslant l+q^{\prime}$, the elements
$\tilde{\Theta}_{i}=\Theta_{i}+\tilde{H}^{m_{i}+1}\in\text{gr}\,(U({\mathfrak{g}},e))$
are algebraically independent and generate $\text{gr}\,(U({\mathfrak{g}},e))$.
In particular, $\text{gr}\,(U({\mathfrak{g}},e))$ is a graded polynomial
superalgebra with homogeneous generators of degrees
$m_{1}+2,\cdots,m_{l+q^{\prime}}+2$.

  * (4)

For $1\leqslant i,j\leqslant l+q^{\prime}$, we have

$[\Theta_{i},\Theta_{j}]\in\tilde{H}^{m_{i}+m_{j}+2}.$

Moreover, if the elements $Y_{i},~{}Y_{j}\in{\mathfrak{g}}^{e}$ for
$1\leqslant i,j\leqslant l+q$ satisfy
$[Y_{i},Y_{j}]=\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}Y_{k}$ in
${\mathfrak{g}}^{e}$, then

$\begin{array}[]{llllll}&[\Theta_{i},\Theta_{j}]&\equiv&\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}\Theta_{k}+q_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}})&(\text{mod}~{}\tilde{H}^{m_{i}+m_{j}+1}),\end{array}$
(4.1)

where $q_{ij}$ is a super-polynomial in $l+q^{\prime}$ variables in
$\mathbb{Q}$ whose constant term and linear part are zero.

  * (5)

When r is odd, if one of $i$ and $j$ happens to be $l+q+1$, we can find super-
polynomials $F_{i,l+q+1}$ and $F_{l+q+1,j}$ in $l+q+1$ invariants over
$\mathbb{Q}$ ($1\leqslant i,j\leqslant l+q$) such that

$\begin{array}[]{lll}[\Theta_{i},\Theta_{l+q+1}]&=&F_{i,l+q+1}(\Theta_{1},\cdots,\Theta_{l+q+1}),\\\
\,[\Theta_{l+q+1},\Theta_{j}]&=&F_{l+q+1,j}(\Theta_{1},\cdots,\Theta_{l+q+1}),\end{array}$
(4.2)

where the $e$-degree for all the monomials in the $\mathbb{C}$-span of
$F_{i,l+q+1}$’s and $F_{l+q+1,j}$’s is less than $m_{i}+1$ and $m_{j}+1$,
respectively.

Moveover, if $i=j=l+q+1$, then

$[\Theta_{l+q+1},\Theta_{l+q+1}]=\text{id}.$ (4.3)

In [33, §4], Premet obtained the PBW theorem for the finite $W$-algebras over
$\mathbb{C}$ through the procedure of “admissible” with aid of reduced
$W$-algebras over ${\mathds{k}}$. Since the “modulo $p$” version of Theorem
4.5 has been formulated in Theorem 3.7, and the choice of admissible ring $A$
has nothing to do with the super property of ${\mathfrak{g}}$, we can prove
the theorem by the same spirit of Premet’s argument for the ordinary finite
$W$-algebra case based on the results of reduced $W$-superalgebras over
${\mathds{k}}$ in §3. Explicitly speaking, in virtue of the lemmas introduced
in §4, one can translate the formulas in Statement (1) of the theorem to a
system of linear equations, then (1) follows by the discussion on the
existence of solution for these equations over $\mathbb{Q}$ by the knowledge
of field theory. The remaining consequence can be easily obtained by the same
consideration as Theorem 3.7 and Remark 3.8. Now we only need to give a
sketchy proof.

###### Proof.

Repeat verbatim the proof of Theorem 4.6 in [33] but apply Theorem 3.7 (also
Remark 3.8), Lemma 4.2 and Lemma 4.3 in place of Theorem 3.4, Lemma 4.4 and
Lemma 4.5 in [33] respectively, then the consequence follows. For more details
one refers to [33, Theorem 4.6]. ∎

######  Remark 4.6.

Recall that we calculated the dimension of
$U_{\chi}({\mathfrak{g}}_{\mathds{k}},e)$ for the proof of Proposition 3.5.
However, since the dimension of finite $W$-superalgebra $U({\mathfrak{g}},e)$
over $\mathbb{C}$ is infinite, similar conclusion as Proposition 3.5 for the
$\mathbb{C}$-algebra $U(\mathfrak{g},e)$ can not be obtained directly.
Therefore, one can not establish Theorem 4.5 directly by the same means as
Theorem 3.7, bypassing the knowledge of the $\mathds{k}$-algebra
$U_{\chi}(\mathfrak{g}_{\mathds{k}},e)$ in §3. It is also remarkable that all
the coefficients for the generators of $U({\mathfrak{g}},e)$ we obtained in
Theorem 4.5(1) are over $\mathbb{Q}$.

By Theorem 4.5(4), there exist super-polynomials $F_{ij}$’s of $l+q^{\prime}$
indeterminants over $\mathbb{Q}$ for $i,j=1,\cdots,l+q$ with the first $l$
indeterminants being even, and the others being odd, such that

$[\Theta_{i},\Theta_{j}]=F_{i,j}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}}),\quad
i,j=1,\cdots,l+q,$

while
$F_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}})\equiv\sum_{k=1}^{l+q}\alpha_{ij}^{k}\Theta_{k}+q_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}})(\mbox{mod
}\tilde{H}^{m_{i}+m_{j}+1})$ for $i,j=1,\cdots,l+q$, and here the super-
polynomials are still under the same sense as in Remark 3.8(2). Note that by
Theorem 4.5(5), when r is odd there are still super-polynomials $F_{i,j}$’s of
$l+q+1$ indeterminants over $\mathbb{Q}$ with $i$ or $j$ equals $l+q+1$. By
the analogous arguments of [35, Lemma 4.1], one can prove the following
result.

######  Theorem 4.7.

The finite $W$-superalgebra $U(\mathfrak{g},e)$ over $\mathbb{C}$ is generated
by the $\mathbb{Z}_{2}$-homogeneous elements $\Theta_{1},\cdots,\Theta_{l}\in
U(\mathfrak{g},e)_{\bar{0}}$ and $\Theta_{l+1},\cdots,\Theta_{l+q^{\prime}}\in
U(\mathfrak{g},e)_{\bar{1}}$ subject to the relations

$[\Theta_{i},\Theta_{j}]=F_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}})$

with $1\leqslant i,j\leqslant l+q^{\prime}$.

###  4.3. Revisit to Theorem 0.1

Define the element $\Theta\in U({\mathfrak{g}},e)$ by letting
$\Theta(1_{\chi})=v_{\frac{\textsf{r}+1}{2}}\otimes 1_{\chi}$ for the case
when dim ${\mathfrak{g}}(-1)_{\bar{1}}$ is odd. From consequences of Lemma 4.3
and Theorem 4.5, Theorem 0.1 can be easily obtained. We omit the detailed
proof.

###  4.4. Some other definition of finite $W$-superalgebras

Recall in [46, Remark 70] Wang defined a reduced $W$-superalgebra over
${\mathds{k}}=\overline{\mathbb{F}}_{p}$ in another way, i.e.
$W^{\prime}_{\chi,{\mathds{k}}}:=(Q^{\chi}_{\chi})^{\text{ad}\,{\mathfrak{m}}^{\prime}_{{\mathds{k}}}}$,
which seems to make better sense (there this superalgebra is called the
modular $W$-superalgebra). In light of this (also cf. [14]), we can also
define the corresponding superalgebra over $\mathbb{C}$.

######  Definition 4.8.

Let $\mathfrak{g}$ be a basic Lie superalgebra over $\mathbb{C}$, and retain
the data appearing as before. Define the $\mathbb{C}$-algebra

$W^{\prime}_{\chi}:=(U({\mathfrak{g}})/I_{\chi})^{\text{ad}\,{\mathfrak{m}}^{\prime}}\cong
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}\equiv\\{\bar{y}\in
U({\mathfrak{g}})/I_{\chi}\mid[a,y]\in I_{\chi},\forall
a\in{\mathfrak{m}}^{\prime}\\},$

where $\bar{y}_{1}\cdot\bar{y}_{2}:=\overline{y_{1}y_{2}}$ for all
$\bar{y}_{1},\bar{y}_{2}\in W^{\prime}_{\chi}$.

When $\textsf{r}=\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}$ is even, since
${\mathfrak{m}}^{\prime}={\mathfrak{m}}$ by the definition, the isomorphism in
Theorem 2.12 shows that $U({\mathfrak{g}},e)\cong W^{\prime}_{\chi}$ as
$\mathbb{C}$-algebras. However, the situation changes in the case when
$\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}$ is odd. Since ${\mathfrak{m}}$ is a
proper subalgebra of ${\mathfrak{m}}^{\prime}$, it follows that
$W^{\prime}_{\chi}$ is a subalgebra of
$Q_{\chi}^{\text{ad}{\mathfrak{m}}}=U({\mathfrak{g}},e)$. In the following, we
give a precise connection between the $\mathbb{C}$-algebras
$W^{\prime}_{\chi}$ and $U(\mathfrak{g},e)$.

######  Proposition 4.9.

For the case when $\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}$ is odd, the
$\mathbb{C}$-algebra $W^{\prime}_{\chi}\cong
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$ is a proper subalgebra of
$U({\mathfrak{g}},e)\cong Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$. Moreover, we
have

$Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}=[v_{\frac{\textsf{r}+1}{2}},Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}].$

###### Proof.

At first, we claim that $Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$ is
properly contained in $Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$. Recall Theorem
4.5(1) shows that $\Theta_{l+q+1}(1_{\chi})=v_{\frac{\textsf{r}+1}{2}}\otimes
1_{\chi}\in Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$. By the definition we have
$v_{\frac{\textsf{r}+1}{2}}\in{\mathfrak{m}}^{\prime}$, and
$[v_{\frac{\textsf{r}+1}{2}},v_{\frac{\textsf{r}+1}{2}}\otimes
1_{\chi}]=[v_{\frac{\textsf{r}+1}{2}},v_{\frac{\textsf{r}+1}{2}}]\otimes
1_{\chi}=1\otimes\chi([v_{\frac{\textsf{r}+1}{2}},v_{\frac{\textsf{r}+1}{2}}])1_{\chi}=1\otimes
1_{\chi}$. Thus $v_{\frac{\textsf{r}+1}{2}}\otimes 1_{\chi}\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$, but not in
$Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$.

Next we consider the relation between the $\mathbb{C}$-algebras
$Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$ and
$Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$.

(1) For any $\mathbb{Z}_{2}$-homogeneous element $x\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$, we claim that

$[v_{\frac{\textsf{r}+1}{2}},x]\subseteq
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}.$

(i) Recall that $\Theta_{l+q+1}(1_{\chi})=v_{\frac{\textsf{r}+1}{2}}\otimes
1_{\chi}\in Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$. Since $x\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$, it follows from the Jacobi identity
that $[v_{\frac{\textsf{r}+1}{2}},x]\in Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$.

(ii) Since $v_{\frac{\textsf{r}+1}{2}}\in{\mathfrak{g}}(-1)_{\bar{1}}$, then
$[v_{\frac{\textsf{r}+1}{2}},v_{\frac{\textsf{r}+1}{2}}]\in{\mathfrak{g}}(-2)_{\bar{0}}\subseteq{\mathfrak{m}}_{\bar{0}}$,
and it follows from $x\in Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$ that
$[[v_{\frac{\textsf{r}+1}{2}},v_{\frac{\textsf{r}+1}{2}}],x]=0$. On the other
hand,

$\begin{array}[]{ccl}[[v_{\frac{\textsf{r}+1}{2}},v_{\frac{r+1}{2}}],x]&=&[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]]+(-1)^{|x|}[[v_{\frac{\textsf{r}+1}{2}},x],v_{\frac{\textsf{r}+1}{2}}]\\\
&=&[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]]-(-1)^{|x|}\cdot(-1)^{|x|+1}[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]]\\\
&=&[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]]+(-1)^{2|x|+2}[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]]\\\
&=&2[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]],\end{array}$

then $[v_{\frac{\textsf{r}+1}{2}},[v_{\frac{\textsf{r}+1}{2}},x]]=0$, i.e.
$[v_{\frac{\textsf{r}+1}{2}},x]\in
Q_{\chi}^{\text{ad}v_{\frac{\textsf{r}+1}{2}}}$.

Since
${\mathfrak{m}}^{\prime}={\mathfrak{m}}\,\oplus\,\mathbb{C}v_{\frac{\textsf{r}+1}{2}}$
as vector spaces, and (i) and (ii) show that
$[v_{\frac{\textsf{r}+1}{2}},x]\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$, then
$[v_{\frac{\textsf{r}+1}{2}},Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}]\subseteq
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$ follows by the variation of
$x$.

(2) We claim that the reverse of (1) is also true, i.e.
$[v_{\frac{\textsf{r}+1}{2}},Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}]\supseteq
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$.

As $v_{\frac{\textsf{r}+1}{2}}\in{\mathfrak{m}}^{\prime}$, and
$v_{\frac{\textsf{r}+1}{2}}\otimes 1_{\chi}\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$, then for any
$\mathbb{Z}_{2}$-homogeneous element $y\otimes 1_{\chi}\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}\subseteq
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$, we have
$[v_{\frac{\textsf{r}+1}{2}},y\otimes 1_{\chi}]=0$, and
$yv_{\frac{\textsf{r}+1}{2}}\otimes 1_{\chi}\in
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$ by the Jacobi identity. Since

$[v_{\frac{\textsf{r}+1}{2}},yv_{\frac{\textsf{r}+1}{2}}\otimes
1_{\chi}]=([v_{\frac{\textsf{r}+1}{2}},y]v_{\frac{\textsf{r}+1}{2}}+(-1)^{|y|}y[v_{\frac{\textsf{r}+1}{2}},v_{\frac{\textsf{r}+1}{2}}])\otimes
1_{\chi}=(-1)^{|y|}y\otimes 1_{\chi},$

i.e. $y\otimes
1_{\chi}=[v_{\frac{\textsf{r}+1}{2}},(-1)^{|y|}yv_{\frac{\textsf{r}+1}{2}}\otimes
1_{\chi}]$, it is straightforward that
$[v_{\frac{\textsf{r}+1}{2}},Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}]\supseteq
Q_{\chi}^{\text{ad}\,{\mathfrak{m}}^{\prime}}$ by the arbitrary of $y$.

Summing up, we complete the proof. ∎

Acknowledgements Most parts of the results in this article were established in
2012 when the first author was a student under the supervision of the second
author, which can be found in the first author’s PhD thesis [47] in East China
Normal University. The authors would like to thank Weiqiang Wang and Lei Zhao
whose work on super version of Kac-Weisfeiler property stimulated them to do
the present research. The authors got much help from the discussion with Hao
Chang and Weiqiang Wang. The authors express great thanks to them.

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