# On free field realization of quantum affine W-algebras

Victor G. Kac and Minoru Wakimoto
Department of Mathematics, M.I.T., Cambridge, MA 02139, USA.
<EMAIL_ADDRESS>in part by Bert and Ann Kostant fund and by Simons
collaboration grant<EMAIL_ADDRESS>
###### Abstract

We find an explicit formula for the conformal vector of any quantum affine
$W$-algebra in its free field realization.

##  1 Introduction

The chiral part of a (super)conformal field theory is a vertex algebra which
admits a conformal vector $L$, for which the eigenvalues of the energy
operator $L_{0}$ lie in $\frac{1}{2}\mathbb{Z}_{\geq 0}$ and the multiplicity
of the 0 eigenvalue is 1. An important class of such vertex algebras are
quantum affine W-algebras $W^{k}(\mathfrak{g},x,f)$ [KRW] , [KW] (see also
[DSK]), attached to a “good” datum $(\mathfrak{g},x,f,k)$ , where
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ is a basic
Lie superalgebra, i. e. a simple finite-dimensional Lie superalgebra over an
algebraically closed field $\mathbb{F}$ of characteristic 0 with reductive
even part $\mathfrak{g}_{\bar{0}}$ and a fixed non-degenerate even invariant
supersymmetric bilinear form $(.\,|\,.)$, $x\in\mathfrak{g}_{\bar{0}}$ is such
that the eigenspace decomposition of $\mathfrak{g}$ with respect to
$\mathop{\rm ad}\,x$ defines a $\tfrac{1}{2}\mathbb{Z}$-grading

(1.1) $\mathfrak{g}\ =\ \underset{j\in\tfrac{1}{2}\mathbb{Z}}{\bigoplus}\
\mathfrak{g}_{j},$

$f\in\mathfrak{g}_{-1}$, and $k\in\mathbb{F}$.

A datum $(\mathfrak{g},x,f,k)$ is called good if $f\in\mathfrak{g}_{-1}$ is
such that

(1.2) $\mathfrak{g}^{f}\subset\mathfrak{g}_{\leq 0}.$

Hereafter $\mathfrak{g}^{f}$ (resp. $\mathfrak{g}_{j}^{f}$) denotes the
centralizer of $f$ in $\mathfrak{g}$ (resp. $\mathfrak{g}_{j}$), and we use
notation $\mathfrak{g}_{\leq m}=\underset{\rm j\leq m}{\oplus}\
\mathfrak{g}_{j}$, and similarly for $\geq m$, or $<m$, or $>m$. We also
denote by $p_{>0}$, $p_{j}$, etc., the projection of $\mathfrak{g}$ to
$\mathfrak{g}_{>0}$, $\mathfrak{g}_{j}$, etc., along (1.1). A special case of
a good datum is a Dynkin datum, defined by $x$ and $f$ from an
$s\ell_{2}$-triple $\\{e,x,f\\}$ in $\mathfrak{g}_{\bar{0}}$, where
$[x,e]=e,[e,f]=x,[x,f]=-f$.

Recall that a bilinear form $(.\,|\,.)$ on $\mathfrak{g}$ is called even if
$(\mathfrak{g}_{\bar{0}}|\mathfrak{g}_{\bar{1}})=0$, supersymmetric (resp.
superskewsymmetric) if $(a|b)=(-1)^{p(a)p(b)}(b|a)$ (resp.
$-(-1)^{p(a)p(b)}(b|a))$, and invariant if $([a,b]|c)=(a|[b,c])$.

In [KRW] for an arbitrary datum $(\mathfrak{g},x,f,k)$ a vertex algebra
homology complex

(1.3) $(V^{k}(\mathfrak{g})\otimes F^{{\rm ch}}\otimes F^{\mathop{\rm
ne}\,}\,,\,d_{(0)})\,$

was constructed, where $V^{k}(\mathfrak{g})$ is the universal affine vertex
algebra of level $k$ associated to $\mathfrak{g}$, and $F^{{\rm ch}}$ (resp.
$F^{\mathop{\rm ne}\,})$ is the vertex algebra of free charged fermions based
on $\mathfrak{g}_{>0}\oplus\mathfrak{g}^{*}_{>0}$ with reversed parity (resp.
of free neutral fermions based on $\mathfrak{g}_{1/2}$), and $d_{(0)}$ is an
explicitly constructed odd derivation of the vertex algebra
$\mathcal{C}^{k}(\mathfrak{g},x,f):=V^{k}(\mathfrak{g})\otimes F^{{\rm
ch}}\otimes F^{\mathop{\rm ne}\,}$.

Recall [K2] that for the construction of the vertex algebra of free fermions
based on a vector superspace $A$, one needs a superskewsymmetric bilinear form
on $A$. In the case of $F^{{\rm ch}}$ this bilinear form is defined via the
pairing of $\mathfrak{g}_{>0}$ and its dual $\mathfrak{g}^{*}_{>0}$, which is
identified with $\mathfrak{g}_{<0}$, using the bilinear form $(.|.)$; the
former is non-degenerate since the latter is. In the case of $F^{\mathop{\rm
ne}\,}$ this bilinear form is defined by the formula

(1.4) $<a,b>^{\mathop{\rm ne}\,}=(f|[a,b]),\ a,b\in\mathfrak{g}_{1/2}.$

The bilinear form (1.4) is non-degenerate if and only if

(1.5) $\mathop{\rm ad}\,f:\mathfrak{g}_{1/2}\to\mathfrak{g}_{-1/2}\,\,\hbox{is
a vector superspace isomorphism}.$

Since property (1.2) is equivalent to
$[f,\mathfrak{g}_{j}]=\mathfrak{g}_{j-1}$ if $j\leq 1/2$ [KW], property (1.5)
holds for any good datum.

The $\mathbb{Z}$-grading of the complex (1.3) is defined by

$\deg V^{k}(\mathfrak{g})=\deg F^{\mathop{\rm ne}\,}=0,\
\deg\mathfrak{g}_{>0}=\ -\deg\mathfrak{g}^{*}_{>0}=1.$

The homology of the complex (1.3) is called the _quantum affine W-algebra_ ,
attached to the datum $(\mathfrak{g},x,f,k)$, and is denoted by
$W^{k}(\mathfrak{g},x,f)$.

For a good datum, $[\mathfrak{g}_{0},f]=\mathfrak{g}_{-1}$, hence the orbit
$G_{0}(f)$ is Zariski open in $\mathfrak{g}_{-1}$, and therefore the vertex
algebra $W^{k}(\mathfrak{g},x,f)$ is independent, up to isomorphism, of the
choice of $f\in\mathfrak{g}_{-1}$, satisfying (1.2).

The main result of [KW] on the structure of the vertex algebra
$W^{k}(\mathfrak{g},x,f)$ is Theorem 4.1, which states that for a good datum
the $j$${}^{\scriptstyle\textrm{th}}$ homology of the complex (1.3) is zero if
$j\neq 0$, and the 0-th homology is the vertex algebra
$W^{k}(\mathfrak{g},x,f)$, which is a subalgebra of the vertex algebra
$\mathcal{C}^{k}(\mathfrak{g},x,f)$ freely generated by $d_{(0)}$-closed
elements $J^{\\{a_{i}\\}}$, where $a_{1},\ldots,a_{s}$ is a basis of
$\mathfrak{g}^{f}$ consisting of eigenvectors of $\mathop{\rm ad}\,x$. The
elements $J^{\\{a_{i}\\}}$ can be recursively computed, using equations (4.11)
and (4.12) from [KW]. The ”building blocks” for construction of elements
$J^{\\{a_{i}\\}}$ are the elements $J^{(a)}$, $a\in\mathfrak{g}^{f}$, defined
in [KRW], see formula (2.7) in Section 2 of the present paper. Theorem 4.1(a)
from [KW] states that for each $a\in\mathfrak{g}^{f}_{-j}\
(=\mathfrak{g}^{f}\cap\mathfrak{g}_{-j})$ the element $J^{\\{a\\}}-J^{(a)}$
lies in the subalgebra of the vertex a;gebra
$\mathcal{C}^{k}(\mathfrak{g},x,f)$, generated by elements $J^{(b)}$, where
$b\in\mathfrak{g}_{-s}$ with $0\leq s<j$ (recall that
$\mathfrak{g}^{f}_{j}\neq 0$ only for $j\leq 0$ by (1.2)), and by the neutral
fermions.

Consider the subalgebra $\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$ of the
vertex algebra $\mathcal{C}^{k}(\mathfrak{g},x,f)$ generated by the elements
$J^{(v)}$ with $v\in\mathfrak{g}_{\leq 0}$, and by the neutral fermions. It
follows from the above discussion that, for a good datum, all elements
$J^{\\{v\\}}$, $v\in\mathfrak{g}^{f}$, lie in
$\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$. It is easy to see that the
elements $J^{(v)}$ with $v\in\mathfrak{g}_{0}$ and neutral fermions generate a
subalgebra $\overline{\mathcal{C}}^{k}_{0}(\mathfrak{g},x,f)$ of the vertex
algebra $\mathcal{C}(\mathfrak{g},x,f)$ , and that the $J^{(v)}$ with
$v\in\mathfrak{g}_{<0}$ generate an ideal $U_{<0}$ of
$\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$ , such that
$\overline{\mathcal{C}}^{k}_{0}(\mathfrak{g},x,f)\ \cap\ U_{<0}=0$. Hence, the
canonical map
$\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)\to\overline{\mathcal{C}}^{k}_{0}(\mathfrak{g},x,f)$
induces a vertex algebra homomorphism

(1.6) $W^{k}(\mathfrak{g},x,f)\ \xrightarrow{\
}{\overline{\mathcal{C}}}^{k}_{0}(\mathfrak{g},x,f).$

Since the vertex algebra $\overline{\mathcal{C}}^{k}_{0}(\mathfrak{g},x,f)$ is
isomorphic to the tensor product of the universal affine vertex algebra
$V^{k^{\prime}}(\mathfrak{g}_{0})$ of ”shifted” level $k^{\prime}$ ([KW] ,
formula (2.5)), and the vertex algebra $F^{\mathop{\rm ne}\,}$, the map (1.6)
may be viewed as a free field realization (FFR) of the W-algebra
$W^{k}(\mathfrak{g},x,f)$.

In the case of a good datum, for the $a_{i}\in\mathfrak{g}^{f}_{j}$ with $j=0$
or $-1/2$ the elements $J^{\\{a_{i}\\}}$ are uniquely determined by the
$a_{i}$ and they are explicitly constructed in [KW], Section 2. The
construction of these elements is still valid for an arbitrary datum,
satisfying property (1.5). Furthermore, provided that $k\neq-h^{\vee}$ (i.e.
$k$ is not the critical level), we also constructed there an energy-momentum
element $L$, with respect to which the elements $J^{\\{a_{i}\\}}$ have
conformal weight $1-m_{i}$, where $[x,a_{i}]=m_{i}a_{i}$, and this
construction is again valid for an arbitrary datum satisfying property (1.5).

In [KW], Theorem 5.1(c), we found an explicit expression for $L$ in terms of
the elements $J^{(a_{i})}$ and the neutral fermions in the case of minimal
W-algebras, which allowed us to compute the FFR (1.6) for these W-algebras
explicitly (see [KW], Theorem 5.2).

The main results of the present paper, valid for an arbitrary datum
$(\mathfrak{g},x,f,k)$ satisfying property (1.5), are Theorem 3.1, which gives
an explicit expression of the element $J^{\\{f\\}}$ in terms of the elements
$J^{(f)}$, $J^{(a)}$ with $a\in\mathfrak{g}_{0}$ and $\mathfrak{g}_{-1/2}$,
and of neutral free fermions, and Theorem 3.2, which states that
$L=-\frac{1}{k+h^{\vee}}J^{\\{f\\}}$, provided that $k\neq-h^{\vee}$. This
leads to an explicit formula for the image of $L$ under the FFR (1.6) for an
arbitrary quantum affine W-algebra, attached to a good datum.

Throughout the paper the base field $\mathbb{F}$ is an algebraically closed
field of characteristic 0.

##  2  The complex $(\mathcal{C}^{k}(\mathfrak{g},x,f),\,d_{(0)})$ and the
W-algebra $W^{k}(\mathfrak{g},x,f)$.

First, recall the construction of vertex algebras $V^{k}(\mathfrak{g})$,
$F^{{\rm ch}}$ and $F^{\mathop{\rm ne}\,}$. We shall use the very convenient
language of non-linear Lie conformal superalgebras and $\lambda$-brackets
[DSK].

Given a Lie superalgebra $\mathfrak{g}$ with an invariant supersymmetric
bilinear form B, consider the $\mathbb{F}[\partial]$-module
$\mathbb{F}[\partial]\otimes\mathfrak{g}$ with the following non-linear
$\lambda$-bracket

(2.1) $[a_{\lambda}b]=[a,b]+\lambda B(a,b)1,\hskip
11.38109pta,b\in\mathfrak{g},$

and the universal enveloping vertex algebra $V^{B}(\mathfrak{g})$ of this non-
linear Lie conformal superalgebra. One often fixes such a bilinear form
$(.|.)$, lets $B(a,b)=k(a|b)$, $k\in\mathbb{F}$, and uses the notation
$V^{k}(\mathfrak{g})=V^{B}(\mathfrak{g})$. Then $V^{k}(\mathfrak{g})$ is
called the universal affine vertex algebra for $\mathfrak{g}$ of level $k$.

The vertex algebra $F(A)$ of fermions based on the vector superspace $A$ with
a skewsupersymmetric bilinear form $<.,.>$ is defined as the universal
enveloping vertex algebra of the $\mathbb{F}[\partial]$-module
$\mathbb{F}[\partial]\otimes\mathfrak{g}$ with the non-linear
$\lambda$-bracket

(2.2) $[a_{\lambda}b]=<a,b>1,\hskip 11.38109pta,b\in A.$

Given a datum $(\mathfrak{g},x,f,k)$ as described in the introduction, the
associated homology complex $(\mathcal{C}^{k}(\mathfrak{g},x,f),d_{(0)})$ is
constructed as follows. Let $A^{{\rm
ch}}=\Pi(\mathfrak{g}_{>0}\oplus\mathfrak{g}^{*}_{>0})$, where $\Pi$ stands
for the reversal of parity, and define on it a skewsupersymmetric bilinear
form $<.,.>^{{\rm ch}}$ by the pairing of the vector superspace
$\Pi\mathfrak{g}_{>0}$ and its dual $\Pi\mathfrak{g}^{*}_{>0}$, and let
$A^{\mathop{\rm ne}\,}=\mathfrak{g}_{1/2}$ with the bilinear form
$<a,b>^{\mathop{\rm ne}\,}$ defined by (1.4). Then
$\mathcal{C}^{k}(\mathfrak{g},x,f)$ is the universal enveloping vertex algebra
of the non-linear Lie confirmal superalgebra
$\mathbb{F}[\partial](\mathfrak{g}\oplus A^{{\rm ch}}\oplus A^{\mathop{\rm
ne}\,})$ with the $\lambda$-brackets defined by (2.1) and (2.2) on the
summands and zero between the distinct summands. The vertex algebra
$\mathcal{C}^{k}(\mathfrak{g},x,f)$ is isomorphic to
$V^{k}(\mathfrak{g})\otimes F^{{\rm ch}}\otimes F^{\mathop{\rm ne}\,}$, where
$F^{{\rm ch}}=F(A^{{\rm ch}})$ and $F^{\mathop{\rm ne}\,}=F(A^{\mathop{\rm
ne}\,})$. Letting

(2.3) $\deg V^{k}(\mathfrak{g})=\deg F^{\mathop{\rm ne}\,}=0,\hskip
11.38109pt\deg\mathfrak{g}_{>0}=-\deg\mathfrak{g}^{*}_{>0}=1\,\,\hbox{on}\,\,F^{{\rm
ch}},$

defines a $\mathbb{Z}$-grading of this vertex algebra :

(2.4)
$\mathcal{C}^{k}(\mathfrak{g},x,f)=\underset{j\in\mathbb{Z}}{\oplus}\,\mathcal{C}^{k}_{j}\,.$

In order to define the differential $d_{(0)}$ choose a basis
$\\{u_{i}\\}_{i\in S}$ of $\mathfrak{g}$, compatible with parity and the
$\frac{1}{2}\mathbb{Z}$-grading (1.1), let $\\{u^{i}\\}_{i\in S}$ be its dual
basis of $\mathfrak{g}$ with respect to the bilinear form $(.|.)$, i. e.
$(u_{i}|u^{j})=\delta_{i,j}$, and denote by $\\{u_{i}\\}_{i\in S_{>0}}$ (resp.
$\\{u_{i}\\}_{i\in S_{j}}$) the part of $\\{u_{i}\\}_{i\in S}$, which is a
basis of $\mathfrak{g}_{>0}$ (resp. $\mathfrak{g}_{j}$). Let
$\\{\varphi_{i}\\}_{i\in S_{>0}}$ be the corresponding to $\\{u_{i}\\}_{i\in
S_{>0}}$ basis of $\Pi\mathfrak{g}_{>0}$, and let $\\{\varphi^{i}\\}_{i\in
S_{>0}}$ be the dual basis of $\Pi\mathfrak{g}^{*}_{>0}$. Let
$\\{\Phi_{i}\\}_{{i\in S}_{1/2}}$ be the corresponding to $\\{u_{i}\\}_{i\in
S_{1/2}}$ basis of $A^{\mathop{\rm ne}\,}$. For $u\in\mathfrak{g}$ let
$\Phi_{u}=\underset{i\in S_{1/2}}{\sum}$ $\gamma_{i}\Phi_{i}$ (resp.
$\varphi_{u}=\underset{i\in S_{>0}}{\sum}\gamma_{i}\varphi_{i}$) if $p_{1/2}u$
(resp. $p_{>0}u$) $=\underset{i}{\sum}\gamma_{i}u_{i}$, where
$\gamma_{i}\in\mathbb{F}$.

Introduce the following element of the vertex algebra
$\mathcal{C}^{k}(\mathfrak{g},x,f)$ :

(2.5) $\begin{array}[]{l}d=\underset{i\in
S_{>0}}{\sum}\big{(}(-1)^{p(i)}u_{i}\otimes\varphi^{i}\otimes
1+(f|u_{i})\otimes\varphi^{i}\otimes 1\big{)}+\underset{i\in
S_{1/2}}{\sum}1\otimes\varphi^{i}\otimes\Phi_{i}\\\
+\frac{1}{2}\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}1\otimes:\varphi^{i}\varphi^{j}\varphi_{[u_{j},u_{i}]}:\otimes
1,\end{array}$

where $p(i)$ stands for the parity $p(u_{i})$ in the Lie superalgebra
$\mathfrak{g}$. The element $d$ is independent of the choice of the basis of
$\mathfrak{g}$. One checks that $[d_{\lambda}d]=0$ ([KRW], Theorem 2.1),
therefore $[d_{(0)},d_{(0)}]=0$ and $d^{2}_{(0)}=0$. Thus, $d_{(0)}$ is a
homology differential of the vertex algebra
$\mathcal{C}^{k}(\mathfrak{g},x,f)$. The homology of the complex
$(\mathcal{C}^{k}(\mathfrak{g},x,f),d_{(0)})$ is the quantum affine W-algebra
$W^{k}(\mathfrak{g},x,f)$.

One has the following formulas for the action of $d_{(0)}$ of the generators
of the vertex algebra $\mathcal{C}^{k}(\mathfrak{g},x,f)$ (cf. [KRW],
formula(2.4)), where $a\in\mathfrak{g}$, and thereafter we skip the tensor
product signs:

(2.6) $\begin{array}[]{lll}&d_{(0)}a=&\underset{j\in
S_{>0}}{\sum}\big{(}(-1)^{p(j)}:\varphi^{j}[u_{j},a]:+k(a|u_{j})\partial\varphi^{j}\big{)};\\\\[4.30554pt]
&d_{(0)}\varphi_{a}=&p_{>0}(a)+(a|f)+(-1)^{p(a)}\Phi_{a}+\underset{j\in
S_{>0}}{\sum}:\varphi^{j}\varphi_{[u_{j},p_{>0}a]}:;\\\\[4.30554pt]
&d_{(0)}\varphi^{i}=&-\frac{1}{2}\underset{j,s\in
S_{>0}}{\sum}(-1)^{p(i)p(j)}c_{j,s}^{i}:\varphi^{j}\varphi^{s}:,\,\,\hbox{where}\,\,[u_{j},u_{s}]=\sum_{i}c_{j,s}^{i}u_{i}\
;\\\\[4.30554pt] &d_{(0)}\Phi_{a}=&\underset{j\in
S_{1/2}}{\sum}(u_{j}|[a,f])\varphi^{j}.\end{array}$

Recall that the ”building blocks” for elements of the W-algebra
$W^{k}(\mathfrak{g},x,f)$ are the following elements of
$\mathcal{C}^{k}(\mathfrak{g},x,f)$ for $v\in\mathfrak{g}$ :

(2.7) $J^{(v)}=v+\sum_{\begin{subarray}{c}j\in
S_{>0}\end{subarray}}(-1)^{p(j)}:\varphi_{[v,u_{j}]}\varphi^{j}:.$

Denote by $\mathcal{C}^{k}_{-}(\mathfrak{g},x,f)$ the subalgebra of the vertex
algebra $\mathcal{C}^{k}(\mathfrak{g},x,f)$ , generated by the elements
$J^{(v)}$ ($v\in S_{\leq 0}$), $\varphi^{i}$ ($i\in S_{>0}$), and $\Phi_{i}$
($i\in S_{1/2}$). By (2.6), this subalgebra is $d_{(0)}$-invariant. Let, as
above, $\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$ be the subalgebra of
$\mathcal{C}^{k}_{-}(\mathfrak{g},x,f)$, generated by the $J^{(v)}$ ($v\in
S_{\leq 0}$), and the $\Phi_{i}$ ($i\in S_{1/2}$). Then, by (2.6), we have

(2.8) $\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)\cap
d_{(0)}\mathcal{C}^{k}_{-}(\mathfrak{g},x,f)=0.$

Let $\kappa(a,b)=str_{\mathfrak{g}}(\mathop{\rm ad}\,a)(\mathop{\rm ad}\,b)$
be the Killing form on $\mathfrak{g}$. Recall that

(2.9) $\kappa(a,b)=2h^{\vee}(a|b).$

For the projection $p_{j}:\mathfrak{g}\to\mathfrak{g}_{j}$ (resp
$p_{>0}:\mathfrak{g}\to\mathfrak{g}_{>0}$) along the grading (1.1), define the
”partial” Killing forms

(2.10) $\kappa_{j}(a,b)={\rm str}_{\mathfrak{g}}(p_{j}(\mathop{\rm
ad}\,a)(\mathop{\rm ad}\,b))\,(\hbox{resp.}\,\kappa_{>0}(a,b)={\rm
str}_{\mathfrak{g}}(p_{>0}(\mathop{\rm ad}\,a)(\mathop{\rm ad}\,b)).$

Elements $J^{(v)}$ for $v\in\mathfrak{g}_{0}$ obey $\lambda$-brackets of the
universal affine vertex algebra $V^{B_{0}}(\mathfrak{g}_{0})$ [KRW , Theorem
2.4(c)]:

(2.11) $[{J^{(a)}}_{\lambda}J^{(b)}]=J^{(a,b)}+\lambda B_{0}(a,b),\
a,b\in\mathfrak{g}_{0},$

where

(2.12) $B_{0}(a,b)=k(a|b)+\frac{1}{2}(\kappa(a,b)-\kappa_{0}(a,b)).$

In fact, (2.11) holds for $a\in\mathfrak{g}_{i},\,b\in\mathfrak{g}_{j}$ with
$ij\geq 0$ (of course $B_{0}(a,b)=0$ if $ij\geq 0$ and $i$ or $j$ is non-
zero). Hence, we have the following

######  Corollary 2.1.

The subalgebra $V^{B_{0}}(\mathfrak{g}_{<0})$ of the vertex algebra
$V^{B_{0}}(\mathfrak{g}_{\leq 0})$ is an ideal, the factor algebra being
$V^{B_{0}}(\mathfrak{g}_{0})$.

The proof of the following formula from [KW], formula (2.6), uses formulas
(2.6) :

$\displaystyle\,\,\,\,\,\,\,d_{(0)}(J^{(v)})=$ $\displaystyle\sum_{j\in
S_{>0}}([f,v]|u_{j})\varphi^{j}+\sum_{j\in
S_{>0}}(-1)^{p(v)(p(j)+1)}:\varphi^{j}\Phi_{p_{1/2}[v,u_{j}]}:$
$\displaystyle-\sum_{\begin{subarray}{c}j\in S_{>0}\\\
[v,u_{j}]\in\mathfrak{g}_{\leq
0}\end{subarray}}(-1)^{p(j)(p(v)+1)}:\varphi^{j}J^{([v,u_{j}])}:+\sum_{j\in
S_{>0}}(k(v|u_{j})+\kappa_{>0}(v,u_{j}))\partial\varphi^{j}.$

From now on we shall assume that condition (1.5) holds, so that we can define
the basis $\\{\Phi^{i}\\}_{i\in S_{1/2}}$ of $A^{\mathop{\rm ne}\,}$, dual to
$\\{\Phi_{i}\\}_{i\in S_{1/2}}$ with respect to the bilinear form (1.4). Then
we have: $d_{(0)}\Phi^{i}=\varphi^{i}$.

As has been mentioned in the introduction, for a good grading the
$d_{(0)}$-closed elements $J^{\\{a\\}}$ are uniquely determined for
$a\in\mathfrak{g}^{f}_{j}$ for $j=0$ or $-1/2$. The $d_{(0)}$-closed elements
$J^{\\{a\\}}$ for $a\in\mathfrak{g}^{f}_{0}$ can be constructed, provided that
(1.5) holds, and they are as follows (see [KRW], Theorem 2.4(a)) :

(2.14) $J^{\\{a\\}}=J^{(a)}\
+\frac{(-1)^{p(a)}}{2}\sum_{\begin{subarray}{c}j\in
S_{1/2}\end{subarray}}:\Phi^{j}\Phi_{[u_{j},a]}:.$

These elements obey $\lambda$-brackets of the universal affine vertex algebra
$V^{B_{1/2}}(\mathfrak{g}^{f}_{0})$:

(2.15) $[{J^{\\{a\\}}}_{\lambda}J^{\\{b\\}}]=J^{\\{[a,b]\\}}+\lambda
B_{1/2}(a,b),$

where

(2.16) $B_{1/2}(a,b)=k(a|b)+\kappa_{>0}(a,b)-\frac{1}{2}\kappa_{1/2}(a,b).$

The $d_{(0)}$-closed elements $J^{\\{v\\}}$ for $v\in\mathfrak{g}^{f}_{-1/2}$
are as follows (see [KW] , Theorem 2.1(d)]):

(2.17)
$J^{\\{v\\}}=J^{(v)}-\frac{(-1)^{p(v)}}{3}\sum_{\begin{subarray}{c}i,j\in
S_{1/2}\end{subarray}}:\Phi^{i}\Phi^{j}\Phi_{[u_{j},[u_{i},v]]}:+\underset{i\in
S_{1/2}}{\sum}\big{(}:J^{([v,u_{i}])}\Phi^{i}:-(k(v|u_{i})+\kappa_{>0}(v,u_{i}))\partial\Phi^{i}\big{)},$

and one has ([KW], Theorem 2.1(e)) :

(2.18) $[{J^{\\{a\\}}}_{\lambda}J^{\\{v\\}}]=J^{\\{[a,v]\\}}\ \ \hbox{if}\ \
a\in\mathfrak{g}^{f}_{0},\,v\in\mathfrak{g}^{f}_{-1/2}.$

######  Remark 2.1.

The elements $\varphi^{i}$ coincide with the elements, denoted by
$\varphi_{i}^{*}$ in [KRW] and [KW], but are different from the elements,
denoted by $\varphi^{i}$ in [DSK]. The advantage of this less natural choice
is that then the construction of the $W$-algebra $W^{k}(\mathfrak{g},x,f)$
works for an arbitrary finite-dimensional Lie superalgebra $\mathfrak{g}$ with
an arbitrary supersymmetric (possibly degenerate) invariant bilinear form
$(.|.)$. (The simplicity of $\mathfrak{g}$ and the non-degeneracy of $(.|.)$
are needed in the next sections.)

##  3  A formula for $J^{\\{f\\}}$ and the energy-momentum element $L$ of
$W^{k}(\mathfrak{g},x,f)$

Choose a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}_{0}$, containing a
Cartan subalgebra of $\mathfrak{g}_{0}^{f}$. It is a Cartan subalgebra of
$\mathfrak{g}$. Choose a set of positive roots of $\mathfrak{g}$, compatible
with the grading (1.1). Recall that the dual Coxeter number $h^{\vee}$ of the
simple Lie superalgebra $\mathfrak{g}$ with the given invariant bilinear form
$(.\ |\ .)$ is the half of the eigenvalue of the Casimir operator $\sum_{j\in
S}u^{j}u_{j}$ on $\mathfrak{g}$, and it is given by the formula

(3.1) $h^{\vee}\ =\ (\rho|\theta)\ +\ \frac{1}{2}\ (\theta|\theta)\ ,$

where $\theta$ is the highest root and $\rho$ is the half of the difference
between sums of positive even roots and positive odd roots. Provided that
$k\neq-h^{\vee}$, the energy-momentum (or Virasaro) element of the vertex
algebra $\mathcal{C}^{k}(\mathfrak{g},x,f)$ is defined by [KRW] for an
arbitrary datum, satisfying (1.5) :

(3.2) $L\ =\ L^{\mathfrak{g}}\ +\ \partial x\ +\ L^{{\rm ch}}\ +\
L^{\mathop{\rm ne}\,},$

where

$\displaystyle L^{\mathfrak{g}}=$
$\displaystyle\frac{1}{2(k+h^{\vee})}\underset{j\in S}{\sum}\ :u^{j}u_{j}:,$
$\displaystyle L^{{\rm ch}}=$ $\displaystyle\underset{j\in
S_{>0}}{\sum}(1-m_{j}):(\partial\varphi^{j})\varphi_{j}:\ -\
m_{j}:\varphi^{j}\partial\varphi_{j}:,$ $\displaystyle L^{ne}=$
$\displaystyle\frac{1}{2}\underset{j\in S_{1/2}}{\sum}\
:(\partial\Phi^{j})\Phi_{j}:,$

and the $m_{j}$ are defined by $[x,u_{j}]=m_{j}u_{j}$.

The central charge of this Virasoro element is equal to (see [KRW], Remark
2.2)

(3.3) $c(\mathfrak{g},x,k)\ =\mathop{\rm
sdim\,}\mathfrak{g}_{0}-\frac{1}{2}\mathop{\rm
sdim\,}\mathfrak{g}_{1/2}-\frac{12}{k+h^{\vee}}|\rho-(k+h^{\vee})x|^{2}.$

With respect to this $L$ the elements $\varphi_{j}$ (resp. $\varphi^{j}$) are
primary of conformal weight $1-m_{j}$ (resp. $m_{j}$), the $\Phi_{j}$ are
primary of conformal weight $1/2$, and $a\in\mathfrak{g}_{j}$ has conformal
weight $1-j$ and is primary, unless $j=0$ and $(a|x)\neq 0$. Actually one has:

(3.4)
$[L_{\lambda}a]=(\partial+\lambda)a-\lambda^{2}k(a|x)\,\,\hbox{for}\,\,a\in\mathfrak{g}_{0}.$

Furthermore, it was shown in [KRW] that the element $d$, defined by $(2.5)$ is
primary of conformal weight 1, hence $[d_{\lambda}L]=\lambda d$ and
$d_{(0)}L=[d_{\lambda}L]|_{\lambda=0}=0$. Hence, the homology class of $L$
defines an energy-momentum element of the vertex algebra
$W^{k}(\mathfrak{g},x,f)$ , which is denoted again by $L$. Note that though,
for a good datum, the $W$-algebra $W^{k}(\mathfrak{g},x,f)$ is independent, up
to isomorphism, of the choice of $x$ with given $f$ [AKM], the element $L$
does depend on $x$.

As has been mentioned in the introduction, the explicit expressions of the
elements $J^{\\{a\\}}$ which generate the subalgebra $W^{k}(\mathfrak{g},x,f)$
of the vertex algebra ${\overline{\mathcal{C}}}^{k}(\mathfrak{g},x,f)$,
associated to a good datum, are known only for $a\in\mathfrak{g}_{-j}$, where
$j=0$ and $\frac{1}{2}$. In view of (1.6) it is important to find an explicit
expression for $J^{\\{f\\}}$. This is the first main result of the paper. The
second main result is the formula $L=-\frac{1}{k+h^{\vee}}J^{\\{f\\}}$ in
$W^{k}(\mathfrak{g},x,f)$. Both results hold for an arbitrary datum
$(\mathfrak{g},x,f,k)$, satisfying (1.5).

Let

$\Omega_{0}\ =\ \underset{j\in S_{0}}{\sum}\ (\mathop{\rm
ad}\,u^{j})(\mathop{\rm ad}\,u_{j}).$

######  Proposition 3.1.

The operator $\Omega_{0}$ is diagonalizable on $\mathfrak{g}_{j}$ for each
$j>0$.

###### Proof.

Choose a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}_{0}$ ; it is a
Cartan subalgebra of $\mathfrak{g}$, containing $x$. Choose a set of positive
roots in $\mathfrak{h}^{*}$, compatible with the
$\frac{1}{2}\mathbb{Z}$-grading (1.1), and let $e_{i},f_{i}$ be the Chevalley
generators of $\mathfrak{g}$. Then for each $j>0$, the $\mathfrak{g}$-module
$\mathfrak{g}_{j}$ is the sum of lowest weight modules with the lowest weight
vectors that are commutators of the $e_{i}$, such that
$e_{i}\in\mathfrak{g}_{>0}$. Since the restriction of $\Omega_{0}$ to each of
these summands is diagonalizable, proposition follows. ∎

Let $\rho_{>0}$ (resp. $\rho_{j}$)$\in\mathfrak{h}^{*}=\mathfrak{h}$ be the
half of the difference between the sums of positive even and positive odd
roots of $\mathfrak{h}$ in $\mathfrak{g}_{>0}$ (resp. $\mathfrak{g}_{j}$). (We
idenitfy $\mathfrak{g}$ with $\mathfrak{g}^{*}$ using $(.|.)$).

######  Proposition 3.2.

The element $\rho_{>0}$ lies in the center of $\mathfrak{g}_{0}$ if (1.5)
holds.

###### Proof.

By the Jacobi identity for $J^{(a)}$, $J^{(b)}$, with
$a,b\in\mathfrak{g}_{0}$, and $L$, we have

$[{J^{(a)}}_{\lambda}[{J^{(b)}}_{\mu}L]]-(-1)^{p(a)p(b)}[{J^{(b)}}_{\mu}[{J^{(a)}}_{\lambda}L]]=[[{J^{a)}}_{\lambda}J^{(b)}]_{\lambda+\mu}L].$

Using (3.4) and the skewsymmetry of the $\lambda$-bracket, this gives
$0=-(\lambda+\mu)^{2}(\rho_{>0}|[a,b])$. Hence $([\rho_{>0},a]|b)=0$ for all
$b\in\mathfrak{g}_{0}$. It follows that $[\rho_{>0},\mathfrak{g}_{0}]=0$. ∎

######  Theorem 3.1.

For the datum $(\mathfrak{g},x,f,k)$, satisfying (1.5), the following element
of $\mathcal{C}^{k}(\mathfrak{g},x,f)$ is $d_{(0)}$-closed :

$J^{\\{f\\}}=J^{(f)}+\underset{j\in
S_{1/2}}{\sum}(-1)^{p(j)}:\Phi^{j}J^{([f,u_{j}])}:-\frac{1}{2}\underset{j\in
S_{0}}{\sum}:J^{(u^{j})}J^{(u_{j})}:$ $-(k+h^{\vee})\partial J^{(x)}+\partial
J^{(\rho_{>0})}+\frac{k+h^{\vee}}{2}\underset{j\in S_{1/2}}{\sum}\ :\Phi^{j}\
\partial\Phi_{j}:.$

######  Theorem 3.2.

Assuming that the datum $(\mathfrak{g},x,f,k)$ satisfies (1.5), and that
$k\neq-h^{\vee}$, the element $L+\frac{1}{k+h^{\vee}}J^{\\{f\\}}$ of
$\mathcal{C}(\mathfrak{g},x,f)$ is $d_{(0)}$-exact. Consequently
$L=-\frac{1}{k+h^{\vee}}J^{\\{f\\}}$ in $W^{k}(\mathfrak{g},x,f)$.

As an immediate consequence of (2.11), (2.12), (2.14)-(2.17) and Theorems 3.1
and 3.2, we obtain the following corollary.

######  Corollary 3.1.

Provided that the datum $(\mathfrak{g},x,f,k)$ satisfies (1.5) and
$k\neq-h^{\vee}$, one has a homomorphism of the vertex algebra
$W^{k}(\mathfrak{g},x,f)$ to the vertex algebra
$V^{B_{0}}(\mathfrak{g}_{0})\otimes F^{\mathop{\rm
ne}\,}(\mathfrak{g}_{1/2})$, such that

$J^{\\{a\\}}\ \mapsto\ a\ +\ \frac{(-1)^{p(a)}}{2}\underset{j\in
S_{1/2}}{\sum}:\Phi^{j}\Phi_{[u_{j},a]}:\,\,\,\hbox{if}\,\,\,a\in\mathfrak{g}^{f}_{0},$
$J^{\\{v\\}}\ \mapsto\underset{i\in
S_{1/2}}{\sum}:[v,u_{i}]\Phi^{i}:-\frac{(-1)^{p(v)}}{3}\underset{i,j\in
S_{1/2}}{\sum}:\Phi^{i}\Phi^{j}\Phi_{[u_{j},[u_{i},v]]}:$ $-\underset{i\in
S_{1/2}}{\sum}\big{(}k(v|u_{i})+\kappa_{>0}(v,u_{i})\big{)}\partial\Phi^{i}\,\,\,\hbox{if}\,\,\,v\in\mathfrak{g}^{f}_{-1/2},$
$L\ \mapsto\ \frac{1}{k+h^{\vee}}\big{(}\frac{1}{2}\underset{j\in
S_{0}}{\sum}:u^{j}u_{j}:\ +\
\partial((k+h^{\vee})x-\rho_{>0})\big{)}-\frac{1}{2}\underset{j\in
S_{1/2}}{\sum}:\Phi^{j}\partial\Phi_{j}:.$

##  4  Proof of Theorem 3.1

Let $U$ and $V$ be finite-dimensional vector spaces over $\mathbb{F}$ with a
non-degenerate even pairing $<.,.>:U\times V\xrightarrow{}\mathbb{F}$. Choose
dual bases $\\{u_{i}\\}_{i\in I}$ and $\\{u^{i}\\}_{i\in I}$ of $U$ and $V$
respectively, i.e. $<u_{i},u^{j}>=\delta_{i,j}$. Then for any $A\in\mathop{\rm
End}U$ and $B\in\mathop{\rm End}V$ we have :

(4.1) ${\rm str}_{U}A=\underset{i\in
I}{\sum}(-1)^{p(i)}<Au_{i},u^{i}>,\,\,{\rm str}_{V}B=\underset{i\in
I}{\sum}(-1)^{p(i)}<u_{i},Bu^{i}>,$

where, as before, $p(i)$ stands for $p(u_{i})(=p(u^{i}))$. This will be used
to prove the following lemma.

######  Lemma 4.1.

For $u,v\in\mathfrak{g}$ we have

(4.2) $\kappa_{<0}(u,v)=\kappa_{>0}(u,v)-{\rm
str}_{\mathfrak{g}{>0}}\,p_{>0}\mathop{\rm ad}\,[u,v],$ (4.3)
$\kappa_{>0}(u,v)=\frac{1}{2}\big{(}\kappa(u,v)-\kappa_{0}(u,v)\ +\ {\rm
str}_{\mathfrak{g}>0}\,p_{>0}\mathop{\rm ad}\,[u,v]\big{)},$ (4.4)
$\kappa_{0}(u,v)=(\Omega_{0}u|v)=(u|\Omega_{0}v).$

###### Proof.

We may assume that $(\mathop{\rm ad}\,u)(\mathop{\rm ad}\,v)$ preserves the
$\frac{1}{2}Z$-grading (1.1) and that $p(u)=p(v)$. In order to prove (4.2), we
use (4.1) with

$U\ =\ \mathfrak{g}_{>0},\,V=\mathfrak{g}_{<0},\,<.,.>=(.|.),\,A=(\mathop{\rm
ad}\,v)(\mathop{\rm ad}\,u),\,B=(\mathop{\rm ad}\,u)(\mathop{\rm ad}\,v).$

We have by the second and then the first formula in (4.1):

$\kappa_{<0}(u,v)\ =\ {\rm str}_{\mathfrak{g}_{<0}}B=\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}(u_{i}|[u,[v,u^{i}]])\ =\ \underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}([[u_{i},u],v]|u^{i})$ $=\
(-1)^{p(u)p(v)}\underset{i\in S_{>0}}{\sum}(-1)^{p(i)}\
([v,[u,u_{i}]]|u^{i})=(-1)^{p(u)p(v)}{\rm str}_{\mathfrak{g}_{>0}}\ A$ $=\
(-1)^{p(u)p(v)}{\rm str}_{\mathfrak{g}_{>0}}\ \mathop{\rm ad}\,[v,u]\ +\ {\rm
str}_{\mathfrak{g}_{>0}}\ (\mathop{\rm ad}\,u)(\mathop{\rm ad}\,v)=-{\rm
str}_{\mathfrak{g}_{>0}}\mathop{\rm ad}\,[u,v]\ +\ \kappa_{>0}(u,v).$

Formula (4.3) follows from (4.2) since
$\kappa(u,v)=\kappa_{>0}(u,v)+\kappa_{0}(u,v)+\kappa_{<0}(u,v)$. The proof of
(4.4) is similar, by letting $U=V=\mathfrak{g}_{0}$, $A=(\mathop{\rm
ad}\,u)(\mathop{\rm ad}\,v)$. ∎

######  Lemma 4.2.

Let, as before, $\\{u_{i}\\}_{i\in S_{>0}}$ be a basis of $\mathfrak{g}_{>0}$
and $\\{u^{i}\\}_{i\in S_{>0}}$ the dual basis of $\mathfrak{g}_{<0}$, i.e.
$(u_{i}|u^{j})=\delta_{i,j}$, and let $v\in\mathfrak{g}_{0}$. Then

(4.5) $\underset{i\in S_{>0}}{\sum}(-1)^{p(i)}[u_{i},u^{i}]=2\rho_{>0},$ (4.6)
${\rm str}_{\mathfrak{g}_{>0}}\mathop{\rm ad}\,v=2(\rho_{>0}|v).$

###### Proof.

Since the LHS is independent on the choice of dual bases, we may take for
$\\{u_{i}\\}_{i\in S_{>0}}$ the basis
$\\{e_{\alpha}\\}_{\alpha\in\Delta_{>0}}$ of $\mathfrak{g}_{>0}$, so that the
dual basis of $\mathfrak{g}_{<0}$ is
$\\{e_{-\alpha}\\}_{\alpha\in\Delta_{>0}}$ with $(e_{\alpha}|e_{-\alpha})=1$,
and hence $[e_{\alpha},e_{-\alpha}]=\alpha$. Then (4.5) follows.

For (4.6) we have :

${\rm str}_{\mathfrak{g}_{>0}}\ \mathop{\rm ad}\,v\ =\ \underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}([v,u_{i}]|u^{i})\ =\ \underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}(v|[u_{i},u^{i}])=2(v|\rho_{>0})$

by (4.5). ∎

Denote by I, II, …, IV the operator $d_{(0)}$, applied to each of the six
terms in the RHS of the formula for $J^{\\{f\\}}$ in Theorem 3.1. We have to
prove that the sum of these six elements of
$\mathcal{C}^{k}(\mathfrak{g},x,f)$ is equal to $0$.

By formula (2) for $v=f$, element I is equal to the sum of the following four
elements :

(4.7) $I_{A}=\underset{j\in
S_{3/2}}{\sum}:\varphi^{j}\Phi_{[f,u_{i}]}:,\,\,I_{B}=-\underset{j\in
S_{1}}{\sum}(-1)^{p(j)}:\varphi^{j}J^{([f,u_{j}])}:,$ (4.8)
$I_{C}=-\underset{j\in
S_{1/2}}{\sum}(-1)^{p(j)}:\varphi^{j}J^{([f,u_{j}])}:,\,\,I_{D}=\underset{j\in
S_{1}}{\sum}\big{(}k(f|u_{j})+\kappa_{>0}(f,u_{j})\big{)}\partial\varphi^{j}.$

By (2) for $v=f$ and the last formula in (2.6), using that $d_{(0)}$ is an odd
derivation of the vertex algebra $\mathcal{C}^{k}(\mathfrak{g},x,f)$, one
obtains that element II is equal to the sum of the following five elements:

(4.9) $II_{A}=\underset{j\in
S_{1/2}}{\sum}(-1)^{p(j)}:\varphi^{j}J^{([f,u_{j}])}:,\,\,II_{B}=\underset{j\in
S_{1/2}}{\sum}\ \underset{i\in
S_{3/2}}{\sum}(f|[[f,u_{j}],u_{i}]):\Phi^{j}\varphi^{i}:,$ (4.10)
$II_{C}=\underset{j\in S_{1/2}}{\sum}\ \underset{i\in
S_{1}}{\sum}(-1)^{p(j)(p(i)+1)}:\Phi^{j}\varphi^{i}\Phi_{[[f,u_{j}],u_{i}]}:,$
(4.11) $II_{D}=-\underset{i,j\in
S_{1/2}}{\sum}(-1)^{p(i)(p(j)+1)}:\Phi^{j}\varphi^{i}J^{([[f,u_{j}],u_{i}])}:,$
(4.12) $II_{E}=\underset{i,j\in
S_{1/2}}{\sum}\big{(}k([f,u_{j}]|u_{i})+\kappa_{>0}([f,u_{j}],u_{i})\big{)}:\Phi^{j}\partial\varphi^{i}:.$

It is easy to see that $I_{A}+II_{B}=0$ , and since also $I_{C}+II_{A}=0$ , we
obtain

(4.13) $I+II=I_{B}+I_{D}+II_{C}+II_{D}+II_{E}.$

######  Lemma 4.3.

. One has

$(a)II_{C}=0.$

$(b)II_{D}=\underset{i\in S_{1/2}}{\sum}\ \underset{k\in
S_{0}}{\sum}:\Phi_{[u^{k},u_{i}]}\varphi^{i}\ J^{(u_{k})}:.$

###### Proof.

By (4.10), using that

$\Phi_{[[f,u_{j}],u_{k}]}=\underset{i\in S_{1/2}}{\sum}\langle
u_{i},[[f,u_{j}],u_{k}]\rangle^{\mathop{\rm ne}\,}\Phi^{i},$

we obtain :

$II_{C}=\underset{i,j\in S_{1/2}}{\sum}\ \underset{k\in
S_{1}}{\sum}(-1)^{p(j)(p(k)+1)}\langle
u_{i},[[f,u_{j}],u_{k}]\rangle^{\mathop{\rm
ne}\,}:\Phi^{j}\varphi^{k}\Phi^{i}:$ $=\underset{i,j\in
S_{1/2}}{\sum}\underset{k\in S_{1}}{\sum}(-1)^{(p(i)+p(j)(p(k)+1)}\langle
u_{i},[[f,u_{j}],u_{k}]\rangle^{\mathop{\rm
ne}\,}:\Phi^{j}\Phi^{i}\varphi^{k}:$ $=\underset{i,j\in S_{1/2}}{\sum}\
\underset{k\in S_{1}}{\sum}\
([[f,u_{i}],[f,u_{j}]]|u_{k}):\Phi^{j}\Phi^{i}\varphi^{k}:.$

If one exchanges $i$ and $j$ in the summation of the last expression, $II_{C}$
doesn’t change. On the other hand, looking at each summand in this expression,
we see that it changes the sign, hence $II_{C}=-II_{C}$, proving (a).

By (4.11), using that, for $i,j\in S_{1/2}$,

$[[f,u_{j}],u_{i}]=\underset{k\in S_{0}}{\sum}([[f,u_{j}],u_{i}]|u^{k})u_{k},$

we obtain :

$II_{D}=-\underset{i\in S_{1/2}}{\sum}\ \underset{k\in
S_{0}}{\sum}(-1)^{p(i)p(k)}:\Phi_{[u_{i},u^{k}]}\varphi^{i}J^{(u_{k})}:,$

proving (b). ∎

Next, we treat the term III. For that introduce structure constants
$c^{k}_{ij}$ and $c^{k}_{j}(v)$ for $i,j,k\in S_{>0}$ and
$v\in\mathfrak{g}_{0}$ by

$[u_{i},u_{j}]=\underset{k}{\sum}c^{k}_{ij}u_{k},\,\,[v,u_{j}]=\underset{k}{\sum}c^{k}_{j}(v)u_{k}.$

######  Lemma 4.4.

(a) For $v\in\mathfrak{g}_{0}$ and $k\in S_{>0}$ one has :

(4.14) $[{\varphi^{k}}_{\lambda}J^{(v)}]=\underset{j\in
S_{>0}}{\sum}c^{k}_{j}(v)\varphi^{j},$ (4.15)
$:\varphi^{k}J^{(v)}:-(-1)^{p(v)(p(k)+1)}:J^{(v)}\varphi^{k}:\,=\underset{j\in
S_{>0}}{\sum}c^{k}_{j}(v)\partial\varphi^{j}.$

(b) For $u\in\mathfrak{g}_{0},v\in\mathfrak{g}_{1/2}$ and $j\in S_{>0}$ one
has

(4.16)
$::\Phi_{v}\varphi^{j}:J^{(u)}:=:\Phi_{v}:\varphi^{j}J^{(u)}::+\underset{k\in
S_{>0}}{\sum}c^{j}_{k}(u):(\partial\Phi_{v})\varphi^{k}:,$ (4.17)
$::\varphi^{j}\Phi_{v}:J^{(u)}:=:\varphi^{j}:\Phi_{v}J^{(u)}::+\underset{k\in
S_{>0}}{\sum}(-1)^{p(u)p(v)}c^{j}_{k}(u):\varphi^{k}\partial\Phi_{v}:.$

###### Proof.

It uses the $\lambda$-bracket calculus, see [K2], [DSK]. Formula (4.14)
follows by the non-commutative Wick formula, (4.15) by quasicommutativity, and
(4.16) by quasiassociativity of a vertex algebra. As an example, we prove here
(4.16). By quasiassociativity we have

$::\Phi_{v}\varphi^{j}:J^{(u)}:-:\Phi_{v}:\varphi^{j}J^{(u)}::$
$=(-1)^{p(u)(p(j)+1)}(\int^{-\partial}_{0}:\Phi_{v}[{J^{(u)}}_{\lambda}\varphi^{j}]d\lambda:-:\Phi_{v}\int^{-\partial}_{0}[{J^{(u)}}_{\lambda}\varphi^{j}]d\lambda:).$

Using (4.14), we obtain that the RHS is equal to

$-\underset{k\in
S_{>0}}{\sum}c^{j}_{k}(u)(\int^{-\partial}_{0}:\Phi_{v}\varphi^{k}:d\lambda-:\Phi_{v}\int^{-\partial}_{0}\varphi^{k}d\lambda:)$
$=\underset{k\in
S_{>0}}{\sum}c^{j}_{k}(u)(\partial:\Phi_{v}\varphi^{k}:-:\Phi_{v}\partial\varphi^{k}:)=\underset{k\in
S_{>0}}{\sum}c^{j}_{k}(u):(\partial\Phi_{v})\varphi^{k}:,$

proving (4.16) ∎

We have, by formula (2), for $i\in S_{0}$ :

$d_{(0)}J^{(u_{i})}\ =\underset{j\in
S_{1}}{\sum}(f|[u_{i},u_{j}])\varphi^{j}+\underset{j\in
S_{1/2}}{\sum}(-1)^{p(i)(p(j)+1)}:\varphi^{j}\Phi_{[u_{i},u_{j}]}:.$

It follows that

(4.18) $d_{(0)}\underset{i\in S_{0}}{\sum}:J^{(u^{i})}\
J^{(u_{i})}:=A_{1}+A_{2}+A_{3}+A_{4},$

where

(4.19) $A_{1}=\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1}}{\sum}(f|[u^{i},u_{j}]):\varphi^{j}J^{(u_{i})}:,$ (4.20)
$A_{2}=\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1/2}}{\sum}(-1)^{p(i)(p(j)+1)}::\varphi^{j}\Phi_{[u^{i},u_{j}]}:J^{(u_{i})}:,$
(4.21) $A_{3}=\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1}}{\sum}(-1)^{p(i)}(f|[u_{i},u_{j}]):J^{(u^{i})}\varphi^{j}:,$ (4.22)
$A_{4}=\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1/2}}{\sum}(-1)^{p(i)}(-1)^{p(i)(p(j)+1)}:J^{(u^{i})}\varphi^{j}\Phi_{[u_{i},u_{j}]}:.$

In order to simplify expressions for those elements, recall the operator
$\Omega_{0}$, defined by $(\ref{eq:3.4})$. By Proposition 3.1, this operator
is diagonalizable in $\mathfrak{g}_{j}$. Hence we can choose
$u_{i}\in\mathfrak{g}_{1/2}$ (resp. $\mathfrak{g}_{1})$ to be eigenvectors of
$\Omega_{0}$ ; denote by $a_{i}$ (resp. $b_{i}$) the corresponding
eigenvalues.

We have by (4.16) :

$A_{2}=\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1/2}}{\sum}:\varphi^{j}\Phi_{[u^{i},u_{j}]}J^{(u_{i})}:+\underset{k\in
S_{1/2}}{\sum}\ \underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1/2}}{\sum}c^{j}_{i,k}:\varphi^{k}\partial\Phi_{[u^{i},u_{j}]}:.$

The first sum in this expression is equal to

(4.23) $A^{{}^{\prime}}_{2}=\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1/2}}{\sum}:\Phi_{[u^{i},u_{j}]}\varphi^{j}J^{(u_{i})}:,$

while the second sum is equal to

$\underset{k\in S_{1/2}}{\sum}:\varphi^{k}\underset{i\in S_{0}}{\sum}\ \
\partial\Phi_{[u^{i},[u_{i},u_{k}]]}:.$

Hence we obtain

(4.24) $A_{2}=A^{{}^{\prime}}_{2}+\underset{k\in
S_{1/2}}{\sum}a_{k}:\varphi^{k}\partial\Phi_{k}:.$

Next, we obtain, using (2.6), (4.14) and (4.21),

$A_{3}=A_{1}-\underset{k\in S_{1}}{\sum}\underset{i\in
S_{0}}{\sum}\underset{j\in
S_{1}}{\sum}c_{i,k}^{j}(f|[u^{i},u_{j}])\partial\varphi^{k},$

hence

(4.25) $A_{3}=A_{1}-\underset{k\in
S_{1}}{\sum}b_{k}(f|u_{k})\partial\varphi^{k},$

since

$\underset{j\in
S_{1}}{\sum}c_{i,k}^{j}(f|[u^{i},u_{j}])=(f|\Omega_{0}(u_{k})).$

Finally, for $A_{4}$, given by (4.22), we have, using (4.15) and (4.23) :

(4.26) $A_{4}=A^{{}^{\prime}}_{2}-\underset{k\in
S_{1/2}}{\sum}a_{k}:(\partial\varphi^{k})\Phi_{k}:.$

From (4.18) - (4.26) we obtain that the element III is equal to the sum of the
four elements

(4.27) $III_{A}=-\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1}}{\sum}(f|[u^{i},u_{j}]):\varphi^{j}J^{(u_{i})}:,$ (4.28)
$III_{B}=-\underset{i\in S_{0}}{\sum}\ \underset{j\in
S_{1/2}}{\sum}:\Phi_{[u^{i},u_{j}]}\varphi^{j}J^{(u_{i})}:,$ (4.29)
$III_{C}=\frac{1}{2}\underset{j\in
S_{1}}{\sum}b_{j}(f|u_{j})\partial\varphi^{j},$ (4.30)
$III_{D}=\frac{1}{2}\underset{j\in
S_{1/2}}{\sum}a_{j}\big{(}:(\partial\varphi^{j})\Phi_{j}:-:\varphi^{j}\partial\Phi_{j}:\big{)}.$

We have :

$III_{A}=\underset{j\in S_{1}}{\sum}(-1)^{p(j)}:\varphi^{j}\underset{i\in
S_{0}}{\sum}(u^{i}|[f,u_{j}])J^{(u_{i})}:,$

hence, by $(\ref{eq:4.7})$,

(4.31) $III_{A}=-I_{B}.$

Using Lemma 4.3(a), we obtain

(4.32) $II_{D}=-III_{B}.$

Hence, by $(\ref{eq:4.13})$ and Lemma 4.3, we have

(4.33) $I+II+III=I_{D}+II_{E}+III_{C}+III_{D}.$

Next, we have

(4.34) $I_{D}+III_{C}=\underset{j\in
S_{1}}{\sum}\big{(}(k+h^{\vee})(f|u_{j})+(\rho_{>0}|[f,u_{j}])\big{)}\partial\varphi^{j},$
(4.35) $II_{E}+III_{D}=(k+h^{\vee})\underset{i\in
S_{1/2}}{\sum}:\Phi_{i}\partial\varphi^{i}:\ -\frac{1}{2}\underset{i\in
S_{1/2}}{\sum}:(\partial\Phi_{\Omega_{0}u_{i}})\varphi^{i}:\ +\underset{i\in
S_{1/2}}{\sum}:\Phi_{[u_{i},\rho_{>0}]}\partial\varphi^{i}:.$

Indeed, by (4.8) and (4.29), we have, using (4.3) and (2.9):

$I_{D}+III_{C}\ =\ \underset{j\in
S_{1}}{\sum}\big{(}(k+\frac{1}{2}b_{j})(f|u_{j})+h^{\vee}(f|u_{j})-\frac{1}{2}\kappa_{0}(f,u_{j})+\frac{1}{2}{\rm
str}_{\mathfrak{g}_{>0}}\mathop{\rm ad}\,[f,u_{j}]\big{)}\partial\varphi^{j}.$

Applying (4.4) and (4.6) to the RHS, we obtain (4.34).

In order to prove (4.35), we rewrite (4.30) as follows :

$III_{D}\ =\ \frac{1}{2}\underset{k\in
S_{1/2}}{\sum}\big{(}:(\partial\varphi^{k})\Phi_{\Omega_{0}u_{k}}:-:\varphi^{k}\partial\Phi_{\Omega_{0}u_{k}}:\big{)}=\frac{1}{2}\underset{k\in
S_{1/2}}{\sum}\big{(}:\Phi_{\Omega_{0}u_{k}}\partial\varphi^{k}:-:(\partial\Phi_{\Omega_{0}u_{k}})\varphi^{k}:\big{)}.$

We also rewrite (4.12), using (4.3), (4.4) and (4.6), as follows :

$II_{E}\ =\ (k+h^{\vee})\ \underset{i\in
S_{1/2}}{\sum}:\Phi_{i}\partial\varphi^{i}:-\frac{1}{2}\underset{i\in
S_{1/2}}{\sum}:\Phi_{\Omega_{0}u_{i}}\partial\varphi^{i}:+\underset{i\in
S_{1/2}}{\sum}:\Phi_{[u_{i},\rho_{>0}]}\partial\varphi^{i}:.$

Adding up these two expressions, we get (4.35).

######  Lemma 4.5.

One has

$\Omega_{0}(u)\ =\
2[\rho_{>0},u]\,\,\,\hbox{for}\,\,\,u\in\mathfrak{g}_{1/2}.$

###### Proof.

Since elements I, II, III lie in the image of $d_{(0)}$ and $d^{2}_{(0)}=0$ ,
we obtain, using (4.33) :

$0\ =\ d_{(0)}(I_{D}\ +\ III_{C})\ +\ d_{(0)}(III_{E}\ +\ III_{D}).$

Substituting here (4.34) and (4.35) and using formulas (2.6) for the action of
$d_{(0)}$ , we obtain :

$0\ =\underset{i,j\in
S_{1/2}}{\sum}(-1)^{p(i)}<u_{j},\frac{1}{2}\Omega_{0}(u_{i})-[\rho_{>0},u_{i}]>^{\mathop{\rm
ne}\,}:\varphi^{i}\partial\varphi^{j}:.$

Due to the non-degeneracy of the bilinear form $<.,.>^{\mathop{\rm ne}\,}$,
the lemma follows. ∎

Finally, we treat the remaining three elements IV, V, and VI. Using (2), we
obtain :

(4.36) $IV=-(k+h^{\vee})\underset{j\in S_{1}}{\sum}\
(f|u_{j})\partial\varphi^{j}\ -\frac{k+h^{\vee}}{2}\underset{j\in
S_{1/2}}{\sum}\partial:\Phi_{j}\varphi^{j}:,$

(4.37) $V=-\underset{j\in
S_{1}}{\sum}(\rho_{>0}|[f,u_{j}])\partial\varphi^{j}+\underset{j\in
S_{1/2}}{\sum}\partial:\Phi_{[\rho_{>0},u_{j}]}\varphi^{j}:.$

Using (2.6), we obtain

(4.38) $VI=\frac{k+h^{\vee}}{2}\underset{j\in
S_{1/2}}{\sum}(:\varphi^{j}\partial\Phi_{j}:-:\Phi_{j}\partial\varphi^{j}:).$

Adding up (4.36) - (4.38), we obtain

(4.39) $\begin{array}[]{l}IV+V+VI=-(k+h^{\vee})\underset{j\in
S_{1}}{\sum}(f|u_{j})\partial\varphi^{j}-\underset{j\in
S_{1}}{\sum}(\rho_{>0}|[f,u_{j}])\partial\varphi^{j}\\\
-(k+h^{\vee})\underset{j\in
S_{1/2}}{\sum}:\Phi_{j}\partial\varphi^{j}:+\underset{j\in
S_{1/2}}{\sum}\partial:\Phi_{[\rho_{>0},u_{j}]}\varphi^{j}:.\end{array}$

Adding up (4.33) and (4.39), and using (4.34), (4.35) and Lemma 4.5, we
conclude that $d_{(0)}J^{\\{f\\}}\ =\ 0$, completing the proof of Theorem 3.1
.

##  5 Proof of Theorem 3.2

First, introduce the following convenient notation. Let $\mathfrak{a}$ (resp.
$\mathfrak{a}$’) be the sum of some $\mathfrak{g}_{j}$’s (resp. the remaining
$\mathfrak{g}_{j}$’s) in (1.1). Then we let $\delta_{u,\mathfrak{a}}=1$ (resp.
0 ) if $u\in\mathfrak{a}$ (resp. $\mathfrak{a}$’). Then we have for $u$ ,
$v\in\mathfrak{g}$ :

(5.1) $\underset{i\in S_{>0}}{\sum}(u_{i}|v)u^{i}\ =\
\delta_{v,\mathfrak{g}_{<0}}v\text{;}\,\,\underset{i\in
S_{>0}}{\sum}(v|u^{i})u_{i}\ =\ \delta_{v,\mathfrak{g}_{>0}}v;$ (5.2)
$\underset{i\in S_{>0}}{\sum}(u|u^{i})(u_{i}|v)\ =\
\delta_{u,\mathfrak{g}_{>0}}(u|v)\ =\ \delta_{v,\mathfrak{g}_{<0}}(u|v);$
(5.3) $\underset{i\in S_{>0}}{\sum}(u|u^{i})(v|u_{i})\ =\
\delta_{u,\mathfrak{g}_{>0}}(v|u)\ =\ \delta_{v,\mathfrak{g}_{<0}}(v|u).$

Similar formulas hold if we replace $S_{>0}$ by $S_{0}$ , and
$\mathfrak{g}_{>0}$ and $\mathfrak{g}_{<0}$ by $\mathfrak{g}_{0}$ ; these
formulas will be denoted by (5.1)’ , (5.2)’ and (5.3)’ .

Next, let $v^{{\rm ch}}$ denote the second summand on the right in (2.7). Then

(5.4) $v^{{\rm ch}}\ =\ \underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}([v,u_{j}]|u^{i}):\varphi_{i}\varphi^{j}:\ .$

Next, by condition (1.5), we have

(5.5) $u^{i}\ =\ [u^{(i)},f],\,\,i\in S_{1/2}\ ,$

where the $\\{u^{(i)}\\}_{i\in S_{1/2}}$ is a basis of $\mathfrak{g}_{1/2}$ ,
dual to $\\{u_{i}\\}_{i\in S_{1/2}}$ with respect to the bilinear form (1.4).

Next, by the quasiassociativity of the normally ordered product, we have for
$i,j,k,l\in S_{>0}$

(5.6) $::\varphi_{i}\varphi^{j}:\varphi_{k}:\ =\
:\varphi_{i}\varphi^{j}\varphi_{k}:\ +\
(-1)^{p(j)}\delta_{j,k}\partial\varphi_{i}\ ;$ (5.7)
$::\varphi_{i}\varphi^{j}:\varphi_{k}:\ =\
:\varphi_{i}\varphi^{j}\varphi^{k}:\ +\
(-1)^{(p(i)+1)(p(j)+1)}\delta_{i,k}\partial\varphi^{j}\ ;$ (5.8)
$\begin{split}::\varphi_{i}\varphi^{j}&::\varphi_{k}\varphi^{l}::\ =\
:\varphi_{i}\varphi^{j}\varphi_{k}\varphi^{l}:\ +\
(-1)^{p(k)}\delta_{j,k}:(\partial\varphi_{i})\varphi^{l}:\\\ &-\
(-1)^{p(j)p(k)}(-1)^{p(i)(p(j)+p(k))}\delta_{i,l}:\varphi_{k}\partial\varphi^{j}:\
.\end{split}$

######  Lemma 5.1.

We have, using (5.4) :

(5.9) $\underset{i\in S_{0}}{\sum}:u^{i}(u_{i})^{\rm ch}:=\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}:p_{0}([u_{j},u^{i}])\varphi_{i}\varphi^{j}:=\underset{i\in
S_{0}}{\sum}:(u^{i})^{\rm ch}u_{i}:;$ (5.10) $\begin{split}\underset{i\in
S_{0}}{\sum}:&(u^{i})^{\rm ch}(u_{i})^{\rm ch}:=\underset{i,j,k,l\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}([u_{l},u^{k}]|p_{0}[u_{j},u^{i}]):\varphi_{i}\varphi^{j}\varphi_{k}\varphi^{l}:\\\
&+\underset{\underset{[u_{j},u^{i}]\in\mathfrak{g}_{0}}{i,j,k\in
S_{>0}}}{\sum}\
(-1)^{p(i)}(u_{j}|[u^{k},[u_{k},u^{i}]])(:(\partial\varphi_{i})\varphi^{j}:-:\varphi_{i}\partial\varphi^{j}:).\end{split}$

###### Proof.

Using the invariance of the bilinear form $(.|.)$ and (5.1)’, we obtain

$\underset{i\in S_{0}}{\sum}:u^{i}(u_{i})^{\rm ch}:=\underset{j,k\in
S_{>0}}{\sum}(-1)^{p(j)}:p_{0}([u_{k},u^{j}])\varphi_{j}\varphi^{k}:\ ,$

which is the first equality in (5.9) after replacing indices $j,k$ by $i,j$ .
The proof of the second equality in (5.9) is the same.

Using the invariance of the bilinear form $(.|.)$ and (5.2)’, we obtain

$\underset{i\in S_{0}}{\sum}:(u^{i})^{\rm ch}(u_{i})^{\rm
ch}:=\underset{j,k,r,s\in
S_{>0}}{\sum}(-1)^{p(j)+p(r)}([u_{s},u^{r}]|p_{0}[u_{k},u^{j}])::\varphi_{j}\varphi^{k}::\varphi_{r}\varphi^{s}::\
.$

Using (5.8), we see that this is equal to

$=\underset{j,k,r,s\in S_{>0}}{\sum}(-1)^{p(j)+p(r)}([u_{s},u^{r}]\ |\
p_{0}[u_{k},u^{j}]):\varphi_{j}\varphi^{k}\varphi_{r}\varphi^{s}:$
$+\underset{j,k,s\in
S_{>0}}{\sum}(-1)^{p(j)}([u_{s},u^{k}]|p_{0}[u_{k},u^{j}]):(\partial\varphi_{j})\varphi^{s}:$
$-\underset{j,k,s\in S_{>0}}{\sum}(-1)^{p(j)}(u_{s}|[u^{k},p_{0}[u_{k},u^{j}]\
]):\varphi_{j}\partial\varphi^{s}:\ .$

In the last term we used the invariance of $(.|.)$ and relabeling of indices;
we also used that $(a|b)\neq 0$ implies that $p(a)=p(b)$ in order to simplify
the sign. Now (5.10) easily follows. ∎

######  Lemma 5.2.

Recalling that $[u_{i},u_{j}]=\underset{k}{\sum}c^{k}_{ij}u_{k}$ for $i,j,k\in
S_{>0}$ , we have

(5.11) $\underset{i,j,k,\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}c^{k}_{ij}:\varphi_{k}\varphi^{j}u^{i}:=\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}:p_{<0}([u_{j},u^{i}])\varphi_{i}\varphi^{j}:\ ,$
(5.12) $\underset{i,k\in
S_{>0}}{\sum}(-1)^{p(i)}:[u_{k},u^{i}]\varphi_{i}\varphi^{k}:-\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}c^{k}_{ij}:\varphi_{k}\varphi^{j}u^{i}:$
$=\underset{i,j\in S_{>0}}{\sum}(-1)^{p(i)}:p_{\geq
0}([u_{j},u^{i}])\varphi_{i}\varphi^{j}:\ .$

###### Proof.

Using that $c^{k}_{ij}=([u_{i},u_{j}]|u^{k})$, that the the bilinear form $(.\
|\ .)$ is invariant, equation (5.1), and that $p(i)+p(k)\ =\ p(j)$ if
$c^{k}_{ij}\neq\ 0$, we obtain :

$\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}c^{k}_{ij}:\varphi_{k}\varphi^{j}u^{i}:=\underset{j,k\in
S_{>0}}{\sum}(-1)^{p(j)}:\varphi_{k}\varphi^{j}p_{<0}[u_{j},u^{k}]:,$

from which (5.11) follows.

By (5.11), the LHS of (5.12) is equal to

$(\underset{i,j\in S_{>0}}{\sum}\ -\
\underset{\underset{[u_{j},u^{i}]\in\mathfrak{g}_{<0}}{i,j\in
S_{>0}}}{\sum})(-1)^{p(i)}:[u_{j},u^{i}]\varphi_{i}\varphi^{j}:.$

Formula (5.12) follows. ∎

######  Lemma 5.3.

The expression

$A_{<0}\ =\ \underset{i,j,k,l\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}([u_{l},u^{k}]|p_{<0}[u_{j},u^{i}]):\varphi_{i}\varphi^{j}\varphi_{k}\varphi^{l}:$

is equal to $\frac{1}{2}A_{\neq 0}$, where $A_{\neq 0}$ is obtained from
$A_{<0}$ by replacing $p_{<0}$ by $p_{\neq 0}$ .

###### Proof.

Exchanging $i$ with $k$ and $j$ with $l$ in $A_{<0}$ , we obtain

$A_{<0}\ =\ \underset{i,j,k,l}{\sum}(-1)^{p(i)+p(k)}([u_{l},u^{k}]\ |\
p_{>0}[u_{j},u^{i}]):\varphi_{i}\varphi^{j}\varphi_{k}\varphi^{l}:.$

Adding the two expressions for $A_{<0}$, we obtain $A_{\neq 0}$. ∎

From (5.4) we obtain

(5.13) $f^{{\rm ch}}\ =\ \underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}(f|[u_{j},u^{i}]):\varphi_{i}\varphi^{j}:.$

######  Lemma 5.4.

We have

$\underset{i\in S_{1/2}}{\sum}(-1)^{p(i)}:\Phi^{i}[f,u_{i}]^{{\rm
ch}}:=\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(j)}:\Phi_{[u_{j},u^{i}]}\varphi_{i}\varphi^{j}:.$

###### Proof.

Substituting in the LHS the expression (5.4) for $v=[f,u_{i}]$, we obtain, by
invariance of $(.|.)$ and (1.4) ,

$\underset{i\in S_{1/2}}{\sum}\underset{j,k\in S_{>0}}{\sum}(-1)^{p(i)+p(k)}\
\langle u_{i},[u_{j},u^{k}]\rangle^{\mathop{\rm
ne}\,}:\Phi^{i}\varphi_{k}\varphi^{j}:$ $=\underset{j,k,\in
S_{>0}}{\sum}(-1)^{p(j)}\underset{i\in S_{1/2}}{\sum}\langle
u_{i},[u_{j},u^{k}]\rangle^{\mathop{\rm
ne}\,}:\Phi^{i}\varphi_{k}\varphi^{j}:$ $=\underset{j,k\in
S_{>0}}{\sum}(-1)^{p(j)}:\Phi_{[u_{j},u^{k}]}\varphi_{k}\varphi^{j}:,$

proving the lemma. ∎

######  Lemma 5.5.

Let

$P_{0}\ =\ -f\ -\underset{i\in
S_{1/2}}{\sum}(-1)^{p(i)}:\Phi^{i}[f,u_{i}]:+1/2\underset{i\in
S_{0}}{\sum}:u^{i}u_{i}:$ $-\partial\rho_{>0}+(k+h^{\vee})\partial
J^{(x)}-\frac{k+h^{\vee}}{2}\underset{i\in
S_{1/2}}{\sum}:\Phi^{i}\partial\Phi_{i}:$ $-h^{\vee}\underset{i,\in
S_{>0}}{\sum}(-1)^{p(i)}:\varphi_{i}\partial\varphi^{i}:-\underset{i,j,k,\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}c^{k}_{ij}:\varphi_{k}\varphi^{j}u^{i}:$
$+\underset{i,k\in
S_{>0}}{\sum}(-1)^{p(i)}:[u_{k},u^{i}]\varphi_{i}\varphi^{k}:.$

Then

(5.14) $(k+h^{\vee})L=d_{(0)}(\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:\varphi_{i}u^{i}:)+P_{0}.$

###### Proof.

By (3.2) we have

(5.15) $(k+h^{\vee})L=\frac{1}{2}\underset{j\in
S}{\sum}:u^{j}u_{j}:+(k+h^{\vee})\partial x+(k+h^{\vee})L^{{\rm
ch}}+(k+h^{\vee})L^{\mathop{\rm ne}\,}.$

Choosing, as usual, dual bases $\\{h_{i}\\}$ and $\\{h^{i}\\}$,
$i=1,\cdots,l$, of $\mathfrak{h}$ and root vectors
$\\{e_{\alpha}\\}_{\alpha\in\Delta_{+}},\\{e_{-\alpha}\\}_{\alpha\in\Delta_{+}}$
of $\mathfrak{g}$, where $(e_{\alpha}|e_{-\alpha})=1$, we obtain, using
quasicommutativity of the normally ordered product, that the first term in the
RHS of (5.15) is

(5.16)
$\underset{\alpha\in\Delta_{+}}{\sum}(-1)^{p(\alpha)}:e_{\alpha}e_{-\alpha}:+\frac{1}{2}\sum^{l}_{i=1}h^{i}h_{i}-\partial\rho=\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:u_{i}u^{i}:+\frac{1}{2}\underset{i\in
S_{0}}{\sum}:u^{i}u_{i}:-\partial\rho_{>0}.$

We also have

(5.17) $\partial x^{{\rm ch}}=\underset{i\in S>0}{\sum}(-1)^{p(i)}\ m_{i}\
\partial(:\varphi_{i}\varphi^{i}:)\ .$

Using (5.16) and (5.17), equation (5.15) can be rewritten as follows:

(5.18) $\begin{split}&(k+h^{\vee})L=\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:u_{i}u^{i}:+\frac{1}{2}\underset{i\in
S_{0}}{\sum}:u^{i}u_{i}:-\partial\rho_{>0}\\\ &+(k+h^{\vee})\partial
J^{(x)}-(k+h^{\vee})\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:\varphi_{i}\partial\varphi^{i}:+\frac{k+h^{\vee}}{2}\underset{i\in
S_{1/2}}{\sum}:(\partial\Phi^{i})\Phi_{i}:\ .\end{split}$

Next, we compute $d_{(0)}(:\varphi_{i}u^{i}:)$, $i\in S_{>0}$, using (2.6) and
that $d_{(0)}$ is an odd derivation of the normally ordered product:

$d_{(0)}(:\varphi_{i}u^{i}:)=:u_{i}u^{i}:+\underset{j,k\in
S_{>0}}{\sum}(-1)^{p(k)}c^{k}_{ij}:\varphi_{k}\varphi^{j}u^{i}:+(f|u_{i})u^{i}$
$+(-1)^{p(i)}:\Phi_{u_{i}}[u^{(i)},f]:-k:\varphi_{i}\partial\varphi^{i}:-\underset{k\in
S_{>0}}{\sum}:[u_{k},u^{i}]\varphi_{i}\varphi^{k}:\ .$

We have used for the 3-rd term in the RHS that $(f|u_{i})=0$ if $p(i)\neq 0$,
and formula (5.5) for the 4-th term. It follows that

(5.19) $\begin{split}&\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}d_{(0)}(:\varphi_{i}u^{i}:)=\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:u_{i}u^{i}:+\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}c^{k}_{i,j}:\varphi_{k}\varphi^{j}u^{i}:\\\ &+f\
-\underset{i\in S_{1/2}}{\sum}(-1)^{p(i)}:\Phi^{i}[u_{i},f]:\\\
&-\underset{i,k\in
S_{>0}}{\sum}(-1)^{p(i)}:[u_{k},u^{i}]\varphi_{i}\varphi^{k}:-k\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:\varphi_{i}\partial\varphi^{i}:\ .\end{split}$

We have used for the 3-rd term in the RHS that $f=\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}(f|u_{i})u^{i}$ , and for the 4-th term that

$\underset{i\in S_{1/2}}{\sum}:\Phi_{i}[u^{(i)},f]:=\underset{i\in
S_{1/2}}{\sum}(-1)^{p(i)}d_{(0)}(:\varphi_{i}u^{i}:).$

Therefore $(k+h^{\vee})L-\sum_{i\in
S_{>0}}(-1)^{p(i)}d_{(0)}(:\varphi_{i}u^{i}:)$ is the difference of the right
hand sides of equations (5.18) and (5.19), which is $P_{0}$ . ∎

######  Lemma 5.6.

We have

$P_{0}\ =-J^{\\{f\\}}+P_{1}\ ,$

where

$P_{1}\ =\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}(f|[u_{j},u^{i}]):\varphi_{i}\varphi^{j}:+\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(j)}:\Phi_{[u_{j},u^{i}]}\varphi_{i}\varphi^{j}:$
$-\frac{1}{2}\underset{i,j,k.\ell\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}([u_{\ell},u^{k}]|p_{0}[u_{j},u^{i}]):\varphi_{i}\varphi^{j}\varphi_{k}\varphi^{\ell}:$
$-\frac{1}{2}\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)}(u_{j}|[u^{k},p_{0}[u_{k},u^{i}]])(:\partial\varphi_{i}\varphi^{j}:-:\varphi_{i}\partial\varphi^{j}:)$
$+\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}(\rho_{>0}|[u_{j},u^{i}])\partial:\varphi_{i}\varphi^{j}:$
$-h^{\vee}\underset{i\in S_{>0}}{\sum}(-1)^{p(i)}\
:\varphi_{i}\partial\varphi^{i}\ +\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}:p_{>0}([u_{j},u^{i}])\varphi_{i}\varphi^{j}:.$

Consequently, by (5.14), we have

(5.20) $(k+h^{\vee})L+J^{\\{f\\}}\ \equiv\ P_{1}\mod\mathop{\rm Im\,}d_{(0)}.$

###### Proof.

First, we compute, using Lemma 5.1,

(5.21) $\begin{split}&\underset{i\in S_{0}}{\sum}:J^{(u^{i})}J^{(u_{i})}:\
=\underset{i\in S_{0}}{\sum}:u^{i}u_{i}:+2\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}:p_{0}([u_{j},u^{i}])\varphi_{i}\varphi^{j}:\\\
&+\underset{i,j,k,\ell\in
S_{>0}}{\sum}(-1)^{p(i)+p(k)}([u_{\ell},u^{k}]|p_{0}[u_{j},u^{i}]):\varphi_{i}\varphi^{j}\varphi_{k}\varphi^{\ell}:\\\
&+\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)}(u_{j}|[u^{k},p_{0}[u_{k},u^{i}]])(:\partial\varphi_{i}\varphi^{j}:-:\varphi_{i}\partial\varphi^{j}:).\end{split}$

Hence, for $P_{0}$, defined in Lemma (5.5), and $J^{\\{f\\}}$, defined in
Theorem 3.1, we have

(5.22) $P_{0}+J^{\\{f\\}}\ =A,$

where

(5.23) $\begin{split}&A\ =(f^{{\rm ch}}+\underset{i\in
S_{1/2}}{\sum}(-1)^{p(i)}:\Phi^{i}[f,u_{i}]^{{\rm
ch}}:)+\frac{1}{2}\underset{i\in
S_{0}}{\sum}(:u^{i}u_{i}:-:J^{(u^{i})}J^{(u_{i})}:)\\\ &+\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}(\rho_{>0}|[u_{j},u^{i}])\partial:\varphi_{i}\varphi^{j}:-h^{\vee}\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:\varphi_{i}\partial\varphi^{i}:\\\ &+\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}:p_{\geq
0}([u_{j},u^{i}])\varphi_{i}\varphi^{j}:.\end{split}$

Here we used Lemma 5.4 for the first term, formula (5.21) for the second term
and formula (5.12) for the last term.

From (5.23) it is straightforward to deduce that $A=P_{1}$. This completes the
proof of Lemma 5.6. ∎

######  Lemma 5.7.

Let

$P_{2}\ =-\frac{1}{2}\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)}(u_{j}|[u^{k},p_{0}[u_{k},u^{i}]])(:(\partial\varphi_{i})\varphi^{j}:-:\varphi_{i}\partial\varphi^{j}:)$
$+\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(i)}(\rho_{>0}|[u_{j},u^{i}])\partial(:\varphi_{i}\varphi^{j}:)-h^{\vee}\underset{i\in
S_{>0}}{\sum}(-1)^{p(i)}:\varphi_{i}\partial\varphi^{i}:$
$-\frac{1}{2}\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)}(u_{j}|[u^{k},p_{<0}[u_{k},u^{i}]]):(\partial\varphi_{i})\varphi^{j}:$
$+\underset{i,j,k\in
S_{>0}}{\sum}(-1)^{p(i)}(u_{j}|[u^{k},p_{>0}[u_{k},u^{i}]]):\varphi_{i}\partial\varphi^{j}:.$

Then

$P_{2}\ =P_{1}-\frac{1}{2}d_{(0)}\underset{i,j\in
S_{>0}}{\sum}(-1)^{p(j)}:\varphi_{[u_{i},u^{j}]}\varphi_{j}\varphi^{i}:.$

Consequently, by (5.20) , we have

$(k+h^{\vee})L+J^{\\{f\\}}\ \equiv\ P_{2}\mod\mathop{\rm Im\,}d_{(0)}.$

###### Proof.

It is similar to that of Lemma 5.6, and therefore is omitted . ∎

######  Lemma 5.8.

Let

$\varphi=\underset{i,j\in S_{>0}}{\sum}(a_{ij}:(\partial\varphi_{i})\
\varphi^{j}:\ +b_{ij}\ :\varphi_{i}\ \partial\varphi^{j}:),$

where $a_{ij},b_{ij}\in\mathbb{F}$. Then $d_{(0)}\varphi=0$ implies that
$\varphi=0$.

###### Proof.

It is clear from (2.6). ∎

Now it is easy to complete the proof of Theorem 3.2. By Lemma 5.7,
$(k+h^{\vee})L+J^{\\{f\\}}\ \equiv\ P_{2}\mod\mathop{\rm Im\,}d_{(0)}$ , where
$d_{(0)}P_{2}=0$ since $d_{(0)}L=0=d_{(0)}J^{\\{f\\}}$. But $P_{2}$ has the
form of $\varphi$ in Lemma 5.8, hence $P_{2}\ =0$.

##  6  Examples

6.1 Minimal W-algebras. Let $\theta\in\mathfrak{h}^{*}=\mathfrak{h}$ be the
highest root for some ordering of roots of the Lie superalgebra
$\mathfrak{g}$. The W-algebra $W^{k}(\mathfrak{g},\theta)$ is called a minimal
W-algebra [KRW], [KW] if the $\frac{1}{2}\mathbb{Z}$-grading (1.1) has the
form

$\mathfrak{g}=\oplus_{j=-1}^{1}\mathfrak{g}_{j},\,\text{where}\
\mathfrak{g}_{-1}=\mathbb{F}e_{-\theta}.$

In this case $f=e_{-\theta}$ lies in the non-zero nilpotent orbit of minimal
dimension in one of the simple components of $\mathfrak{g}_{\overline{0}}$.
Conversely, if $f$ lies in the non-zero orbit of minimal dimension in a simple
component of $\mathfrak{g}_{\overline{0}}$, then the corresponding $W$-algebra
is a minimal $W$-algebra in all cases, except when $\mathfrak{g}=osp(3|n)$ and
the simple component of $\mathfrak{g}_{\overline{0}}$ is $so_{3}$. Minimal
$W$-algebras were studied in detail in [KRW] and [KW].

Obviously, for a minimal $W$-algebra, $\rho_{1}=x$, and it follows from [KW],
formulas (5.6), (5.11), that $\rho_{1/2}=(h^{\vee}-2)x$. Hence,

(6.1) $\rho_{>0}\ =\ (h^{\vee}-1)x.$

Therefore, $\rho_{>0}-(k+h^{\vee})x\ =\ -(k+1)x$, and the FFR, given by
Corollary 3.1, coincides with that, given by [KW], Theorem 5.3.

6.2 Principal W-algebras. Let $\\{e_{*},\rho^{\vee},f_{*}\\}$ be a principal
$sl_{2}$-triple, where $x=\rho^{\vee}$ is the half of the sum of positive
coroots of $\mathfrak{g}_{\overline{0}}$. Then the datum
$(\mathfrak{g},\rho^{\vee},f_{*},k)$ is a Dynkin datum. The corresponding
W-algebra $W^{k}(\mathfrak{g},\rho^{\vee},f_{*})$ is called the principal
W-algebra, associated to $\mathfrak{g}$.

If $\mathfrak{g}$ is a Lie algebra, then $\mathfrak{g}_{\pm 1/2}=0$ and
$\mathfrak{g}_{0}=\mathfrak{h}$, and therefore

(6.2)
$\rho_{>0}=\rho\,(\in\mathfrak{h}^{*}=\mathfrak{h}),\,\,\hbox{and}\,\,\mathfrak{g}^{f_{*}}_{0}=0,$

where $\rho$ is the half of the sum of positive roots of $\mathfrak{g}$. Hence
the FFR in this case is a homomorphism
$W^{k}(\mathfrak{g},\rho^{\vee},f_{*})\to V^{B_{0}}(\mathfrak{h})$, for which

(6.3) $L\ \mapsto\ \frac{1}{2(k+h^{\vee})}\underset{j\in
S_{0}}{\sum}:u^{j}u_{j}:+\partial\rho^{\vee}-\frac{1}{k+h^{\vee}}\partial\rho.$

The principal $W$-algebras for arbitrary simple Lie algebras were first
constructed in [FF].

The element $2x$ is determined by its Dynkin labels
$2\alpha_{i}(x),\,i=1,...,{\rm rank}\,\mathfrak{g}$, which are known to take
values $0,1$, and $2$. In the case when $\mathfrak{g}$ is a simple Lie algebra
all the Dynkin labels of $2\rho^{\vee}$ are equal to 2.

Let now $\mathfrak{g}$ be a basic Lie superalgebra, which is not a Lie
algebra. Then $\mathfrak{g}$ may have several non-isomorphic sets of simple
roots, and the Dynkin diagrams, corresponding to the choices of positive
roots, compatible with the grading (1.1) may be different. Below we list the
Dynkin labels $2\alpha_{i}(\rho^{\vee}),\,i=1,...,{\rm rank}\,\mathfrak{g}$,
for all basic Lie superalgebras $\mathfrak{g}$, which are not Lie algebras.
For exceptional Lie superalgebras $\mathfrak{g}$ they can be found in [H2]. We
use notation for basic Lie superalgebras and their Dynkin diagrams from [K1].

$I$. $A(m,n),\,m>n\geq 0$, $m-n=2k+1$, $k\in\mathbb{Z}_{\geq 0}$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(5.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(5.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(5.5,1.0){\line(1,0){1.0}} \par\put(7.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(7.05,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(7.6,1.0){\line(1,0){1.0}}
\par\put(9.4,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(10.0,1.0){\line(1,0){1.0}}
\par\put(11.52,2.0){\makebox(0.0,0.0)[c]{2}}
\put(11.52,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(12.1,1.0){\line(1,0){1.0}}
\par\put(13.75,2.0){\makebox(0.0,0.0)[c]{1}}
\put(13.75,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(14.3,1.0){\line(1,0){1.0}}
\par\put(15.95,2.0){\makebox(0.0,0.0)[c]{1}}
\put(15.95,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(16.5,1.0){\line(1,0){1.0}}
\par\put(18.25,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(18.9,1.0){\line(1,0){1.2}}
\par\put(20.75,2.0){\makebox(0.0,0.0)[c]{1}}
\put(20.75,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(21.35,1.0){\line(1,0){1.0}}
\par\put(23.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(23.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(23.5,1.0){\line(1,0){1.0}}
\par\put(25.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(25.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(25.5,1.0){\line(1,0){1.0}}
\par\put(27.25,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(28.0,1.0){\line(1,0){1.0}}
\par\put(29.5,2.0){\makebox(0.0,0.0)[c]{2}}
\put(29.5,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}} \par\end{picture}\end{array}$

where the number of white nodes at the beginning and the end is equal to $k$,
and the number of grey nodes is $2(n+1)$.

$II$. $A(m,n),\,m>n\geq 0$, $m-n=2k$, $k\in\mathbb{Z}_{\geq 1}$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(5.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(5.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(5.5,1.0){\line(1,0){1.0}} \par\put(7.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(7.05,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(7.6,1.0){\line(1,0){1.0}}
\par\put(9.4,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(10.0,1.0){\line(1,0){1.0}}
\par\put(11.52,2.0){\makebox(0.0,0.0)[c]{2}}
\put(11.52,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(12.1,1.0){\line(1,0){1.0}}
\par\put(13.75,2.0){\makebox(0.0,0.0)[c]{0}}
\put(13.75,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(14.3,1.0){\line(1,0){1.0}}
\par\put(15.95,2.0){\makebox(0.0,0.0)[c]{2}}
\put(15.95,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(16.5,1.0){\line(1,0){1.0}}
\par\put(18.25,2.0){\makebox(0.0,0.0)[c]{0}}
\put(18.25,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(18.9,1.0){\line(1,0){1.2}}
\par\put(20.75,2.0){\makebox(0.0,0.0)[c]{2}}
\put(20.75,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(21.35,1.0){\line(1,0){1.0}}
\par\put(23.0,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(23.6,1.0){\line(1,0){1.0}}
\par\put(25.25,2.0){\makebox(0.0,0.0)[c]{0}}
\put(25.25,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(25.85,1.0){\line(1,0){1.0}}
\par\put(27.5,2.0){\makebox(0.0,0.0)[c]{2}}
\put(27.5,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(28.1,1.0){\line(1,0){1.0}}
\par\put(29.75,2.0){\makebox(0.0,0.0)[c]{2}}
\put(29.75,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(30.25,1.0){\line(1,0){1.0}}
\par\put(31.75,2.0){\makebox(0.0,0.0)[c]{2}}
\put(31.75,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(32.25,1.0){\line(1,0){1.0}}
\par\put(34.0,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(34.75,1.0){\line(1,0){1.0}}
\par\put(36.3,2.0){\makebox(0.0,0.0)[c]{2}}
\put(36.3,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}} \par\end{picture}\end{array}$

where the number of white nodes at the beginning (resp. end) is equal to $k$
(resp. $k-1$), and the number of grey nodes is $2(n+1)$ .

$III$. $A(m,m),\,m\geq 1$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(5.0,2.0){\makebox(0.0,0.0)[c]{0}}
\put(5.0,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(5.5,1.0){\line(1,0){1.0}} \par\put(7.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(7.05,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(7.6,1.0){\line(1,0){1.0}} \par\put(9.25,2.0){\makebox(0.0,0.0)[c]{0}}
\put(9.25,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(9.8,1.0){\line(1,0){1.0}} \par\put(11.4,2.0){\makebox(0.0,0.0)[c]{2}}
\put(11.4,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(11.9,1.0){\line(1,0){1.0}}
\par\put(13.75,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(14.3,1.0){\line(1,0){1.0}}
\par\put(15.9,2.0){\makebox(0.0,0.0)[c]{0}}
\put(15.9,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(16.4,1.0){\line(1,0){1.2}}
\par\put(18.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(18.0,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(18.6,1.0){\line(1,0){1.0}}
\par\put(20.2,2.0){\makebox(0.0,0.0)[c]{0}}
\put(20.2,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\end{picture}\end{array}$

In all cases I - III the total number of nodes is $m+n+1$ .

$IV$. $B(m,n),\,m\geq 0,\,n\geq 1$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(5.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(5.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(5.5,1.0){\line(1,0){1.0}} \par\put(7.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(7.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(7.6,1.0){\line(1,0){1.0}}
\par\put(9.4,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(10.0,1.0){\line(1,0){1.0}}
\par\put(11.52,2.0){\makebox(0.0,0.0)[c]{2}}
\put(11.52,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(12.1,1.0){\line(1,0){1.0}}
\par\put(13.75,2.0){\makebox(0.0,0.0)[c]{1}}
\put(13.75,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(14.3,1.0){\line(1,0){1.0}}
\par\put(15.95,2.0){\makebox(0.0,0.0)[c]{1}}
\put(15.95,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(16.5,1.0){\line(1,0){1.0}}
\par\put(18.25,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(18.9,1.0){\line(1,0){1.2}}
\par\put(20.75,2.0){\makebox(0.0,0.0)[c]{1}}
\put(20.75,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(22.0,1.0){\makebox(0.0,0.0)[c]{$\Longrightarrow$}}
\par\put(23.2,2.0){\makebox(0.0,0.0)[c]{1}}
\put(23.2,1.0){\makebox(0.0,0.0)[c]{${\raisebox{-3.22916pt}{\Huge{\textbullet}}}$}}
\par\end{picture}\end{array}$

where the number of white nodes is $m-n$ if $m\geq n$, and is $n-m-1$ if
$m\geq n-1$; the total number of nodes is $m+n$.

$V$. $C(n)\,,\,n\,\geq\,3$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(5.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(5.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(5.5,1.0){\line(1,0){1.0}} \par\put(7.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(7.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(7.6,1.0){\line(1,0){1.0}}
\par\put(9.4,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(10.0,1.0){\line(1,0){1.0}}
\par\put(11.52,2.0){\makebox(0.0,0.0)[c]{2}}
\put(11.52,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(12.1,1.0){\line(1,0){1.0}}
\par\put(13.75,2.0){\makebox(0.0,0.0)[c]{2}}
\put(13.75,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(14.3,1.0){\line(1,0){1.0}}
\par\put(15.95,2.0){\makebox(0.0,0.0)[c]{1}}
\put(15.95,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(16.95,-1.0){\makebox(0.0,0.0)[c]{1}}
\put(15.95,-1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(13.9,0.7){\line(1,-1){1.6}} \par\put(15.95,0.5){\line(0,-1){1.0}}
\par\end{picture}\end{array}$

where the number of white nodes is $n-2$.

$VI$. $D(m,n),\,\,m\geq 2,\,n\geq 1$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(5.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(5.0,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(5.5,1.0){\line(1,0){1.0}} \par\put(7.0,2.0){\makebox(0.0,0.0)[c]{2}}
\put(7.05,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(7.6,1.0){\line(1,0){1.0}}
\par\put(9.4,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(10.0,1.0){\line(1,0){1.0}}
\par\put(11.52,2.0){\makebox(0.0,0.0)[c]{2}}
\put(11.52,1.0){\makebox(0.0,0.0)[c]{$\bigcirc$}}
\par\put(12.0,1.0){\line(1,0){1.0}}
\par\put(13.5,2.0){\makebox(0.0,0.0)[c]{1}}
\put(13.5,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(14.0,1.0){\line(1,0){1.0}}
\par\put(15.5,2.0){\makebox(0.0,0.0)[c]{1}}
\put(15.5,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(16.1,1.0){\line(1,0){1.0}}
\par\put(18.0,1.0){\makebox(0.0,0.0)[c]{$\cdots$}}
\par\put(18.7,1.0){\line(1,0){1.0}}
\par\put(20.3,2.0){\makebox(0.0,0.0)[c]{1}}
\put(20.3,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(20.9,1.0){\line(1,0){1.0}}
\par\put(22.55,2.0){\makebox(0.0,0.0)[c]{1}}
\put(22.55,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(23.55,-1.0){\makebox(0.0,0.0)[c]{1}}
\put(22.55,-1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(20.4,0.6){\line(1,-1){1.6}} \par\put(22.55,0.5){\line(0,-1){1.0}}
\par\end{picture}\end{array}$

where the number of white nodes is $m-n-1$, if $m\geq n+1$, and is $n-m$ if
$m\leq n$; the total number of nodes is $m+n$

$VII$. $D(2,1;a)$ :

$\begin{array}[]{c
c}\begin{picture}(20.0,8.0)\par\put(6.0,2.0){\makebox(0.0,0.0)[c]{1}}
\put(6.0,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(6.4,1.0){\line(1,0){1.0}} \par\put(7.95,2.0){\makebox(0.0,0.0)[c]{1}}
\put(7.95,1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(8.95,-1.0){\makebox(0.0,0.0)[c]{1}}
\put(7.95,-1.0){\makebox(0.0,0.0)[c]{$\bigotimes$}}
\par\put(6.4,0.6){\line(1,-1){1.6}} \par\put(7.95,0.5){\line(0,-1){1.0}}
\par\end{picture}\end{array}$

where the simple roots are
$\\{\epsilon_{1}+\epsilon_{2}+\epsilon_{3}\,,\,\epsilon_{1}-\epsilon_{2}-\epsilon_{3}\,,\,-\epsilon_{1}-\epsilon_{2}+\epsilon_{3}\\}$.

$VIII$. $F(4)$ :

1$\bigotimes$1$\bigotimes$$\Longleftarrow$2$\bigcirc$1$\bigotimes$

where the simple roots are
$\\{\frac{1}{2}(\delta+\epsilon_{1}-\epsilon_{2}-\epsilon_{3}),\frac{1}{2}(\delta-\epsilon_{1}+\epsilon_{2}+\epsilon_{3}),\,\frac{1}{2}(-\delta+\epsilon_{1}-\epsilon_{2}+\epsilon_{3}),\,\epsilon_{2}-\epsilon_{3}\\}$.

$IX$. $G(3)$ :

1•$\Longleftarrow$1$\bigotimes$2$\bigcirc$

where the simple roots are
$\\{\delta,-\delta+\epsilon_{1},\,\epsilon_{2}-\epsilon_{1}\\}$.

Looking at these diagrams, we see that for all basic Lie superalgebras
$\mathfrak{g}$, except for $A(m,n)$ with $m-n$ even, which we shall exclude
from consideration, the $\frac{1}{2}\mathbb{Z}$-grading (1.1), corresponding
to the principal nilpotent element, is defined by

$\alpha_{i}(\rho^{\vee})=1\,\,(\hbox{resp}=\frac{1}{2})\,\,\hbox{if}\,\,\alpha_{i}\,\,\hbox{is
even (resp. odd)},$

where $\alpha_{i}$, $i=1,...,{\rm rank}\,\mathfrak{g}$, are simple roots..
Hence this grading is compatible with the parity and
$\mathfrak{g}_{0}=\mathfrak{h}$. It follows that (6.2) still holds.
Furthermore, $\mathfrak{g}_{1/2}$ (resp. $\mathfrak{g}_{-1/2}$) is a purely
odd space, spanned by the $e_{\alpha_{i}}$ (resp. $e_{-\alpha_{i}}$), where
the $\alpha_{i}$ are all odd simple roots. The element $f\in\mathfrak{g}_{-1}$
can be chosen as follows. Let $f^{0}$ (resp. $f^{1}$) be the sum of all
$e_{-\alpha_{i}}$ with $\alpha_{i}$ even (resp. odd); then

$f=f^{0}+[f^{1},f^{1}].$

######  Remark 6.1.

It is probably impossible to write down a complete FFR of an arbitrary
$W$-algebra, with explicit expressions for all elements beyond those of
conformal weight $1$, $\frac{3}{2}$, and $L$. However, in many cases
(including the minimal one) the $W$-algebra $W^{k}(\mathfrak{g},x,f)$ is
generated by elements of conformal weight $1$ and $\frac{3}{2}$ (this happens,
for example, when $\mathfrak{g}^{f}_{-1/2}$ generates
$\mathfrak{g}^{f}_{<0}$). In such cases formulas (2.14) and (2.17) can be
extended to the complete FFR of this $W$-algebra.

##  7  Appendix A. A more conceptual proof of Theorem 3.2 for Dynkin datum

We give a proof under the following assumptions :

(i) the restriction of the bilinear form $(.\ |\ .)$ to $\mathfrak{g}^{f}_{0}$
is non-degenerate;

(ii) (even part of $\mathfrak{g}_{-1}$) $\cap$ (center of $\mathfrak{g}^{f}$)
$=\mathbb{F}f$.

Property (i) holds for any basic Lie superalgebra $\mathfrak{g}$ and its
Dynkin grading (1.1). Property (ii) holds for a Dynkin grading by the
Brylinski-Kostant Theorem [BK], [P] in the Lie algebra case, and its analogue
in the Lie superalgebra case, which follows from [H1] and [H2].

######  Lemma 7.1.

Let L be the element of $\mathcal{C}^{k}(\mathfrak{g},x,f)$, given by (3.1),
and let $L^{{}^{\prime}}=-\frac{1}{k+h^{\vee}}\ J^{\\{f\\}}$ , where
$J^{\\{f\\}}$ is an element of $\mathcal{C}^{k}(\mathfrak{g},x,f)$ given by
Theorem 3.1. Let $a\in\mathfrak{g}^{f}_{0}$ and
$J^{\\{a\\}}\in\mathcal{C}^{k}(\mathfrak{g},x,f)$ be defined by (2.14). Then

(a)
$[L_{\lambda}J^{\\{a\\}}]=(\partial+\lambda)J^{\\{a\\}}+\lambda^{2}(\rho_{>0}|a)$.

(b)
$[L^{{}^{\prime}}_{\lambda}J^{\\{a\\}}]=(\partial+\lambda)J^{\\{a\\}}+\lambda^{2}(\rho_{>0}|a)$.

###### Proof.

From (2.11) and (2.14) we deduce

(7.1) $[{J^{\\{a\\}}}_{\lambda}J^{(v)}]=J^{([a,v])}+\lambda
B_{0}(a,b),\,a\in\mathfrak{g}^{f}_{0},\,v\in\mathfrak{g}_{\leq 0}.$

Using (2.14) we obtain

(7.2)
$[{J^{\\{a\\}}}_{\lambda}\Phi_{u}]=(-1)^{p(a)}\Phi_{[a,u]},\,a\in\mathfrak{g}^{f}_{0},\,u\in\mathfrak{g}_{1/2}.$

From (7.1) and (7.2) we deduce, by making use of the non-commutative Wick
formula, for $a\in\mathfrak{g}^{f}_{0}$, $i\in S_{0}$, $j\in S_{1/2}$ :

$[{J^{\\{a\\}}}_{\lambda}:J^{(u^{i})}J^{(u_{i})}:]=\,:J^{([a,u^{i}])}J^{(u_{i})}:+(-1)^{p(a)p(i)}:J^{(u^{i})}J^{([a,u_{i}])}:$
$+(-1)^{p(i)}\lambda
J^{([u_{i},[u^{i},a]])}+\lambda(B_{0}(a,u^{i})J^{(u_{i})}+(-1)^{p(i)}B_{0}(a,u_{i})J^{(u^{i})})+\frac{1}{2}\lambda^{2}B_{0}(a,[u^{i},u_{i}]),$
$[{J^{\\{a\\}}}_{\lambda}:\Phi^{j}\partial\Phi_{j}:]\ =\ (-1)^{p(a)(p(j)\ +\
1)}\lambda:\Phi^{j}\Phi_{[a,u_{j}]}:\ +\ \frac{1}{2}(-1)^{p(a)}\lambda^{2}\
\langle[a,u^{j}],u_{j}\rangle^{\mathop{\rm ne}\,},$
$[{J^{\\{a\\}}}_{\lambda}:\Phi^{j}J^{([f,u_{j}])}:]=(-1)^{p(a)}\
:\Phi_{[a,u^{j}]}J^{([f,u_{j}])}:\ +(-1)^{p(a)p(j)}\
:\Phi^{j}J^{([a,[f,u_{j}]])}:.$

Summing up both sides of the first formula over $i\in S_{0}$, the second and
third formula over $j\in S_{1/2}$, we obtain the following three formulas for
$a\in\mathfrak{g}^{f}_{0}$ :

(7.3) $\underset{i\in
S_{0}}{\sum}[{J^{\\{a\\}}}_{\lambda}:J^{(u^{i})}J^{(u_{i})}:]=2(k+h^{\vee})\lambda
J^{(a)},$ (7.4) $\underset{j\in S_{1/2}}{\sum}[{a^{\mathop{\rm
ne}\,}}_{\lambda}:\Phi^{j}\partial\Phi_{j}:]=-2\lambda a^{\mathop{\rm ne}\,},$

where $a^{\mathop{\rm ne}\,}$ is the secind term on the right in (2.14) ,

(7.5) $\underset{j\in
S_{1/2}}{\sum}[{J^{\\{a\\}}}_{\lambda}(-1)^{p(j)}:\Phi^{j}J^{([f,u_{j}])}:]=0.$

Using (7.1), we obtain

(7.6) $[{J^{\\{a\\}}}_{\lambda}J^{(f)}]=0=[{J^{\\{a\\}}}_{\lambda}J^{(x)}]\
\text{for}\,a\in\mathfrak{g}^{f}_{0}.$

Using Proposition 3.2, we obtain from (7.1) :

(7.7)
$[{J^{\\{a\\}}}_{\lambda}J^{(\rho_{>0})}]=\lambda(k+h^{\vee})(a|\rho_{>0}).$

Now we can complete the proof of the lemma. Formula (a) is straightforward by
the discussion in Section 3, cf. (3.4). Below we shall prove (b). We have for
$a\in\mathfrak{g}^{f}_{0}$ :

$[{J^{\\{a\\}}}_{\lambda}J^{\\{f\\}}]=[{J^{\\{a\\}}}_{\lambda}J^{(f)}]+\
\underset{i\in
S_{1/2}}{\sum}(-1)^{p(i)}[{J^{\\{a\\}}}_{\lambda}:\Phi^{i}J^{([f,u_{i}])}:]$
$-\frac{1}{2}\underset{i\in
S_{0}}{\sum}[{J^{\\{a\\}}}_{\lambda}:J^{(u^{i})}J^{(u_{i})}:]-(k+h^{\vee})[{J^{\\{a\\}}}_{\lambda}\partial
J^{(x)}]$ $+[{J^{\\{a\\}}}_{\lambda}\ \partial
J^{(\rho_{>0})}]+\frac{1}{2}(k+h^{\vee})\underset{i\in
S_{1/2}}{\sum}[{J^{\\{a\\}}}_{\lambda}:\Phi^{i}\partial\Phi_{i}:].$

The first and the fourth terms on the right are equal to $0$ by (7.6), and the
second term equals $0$ by (7.5). The third term equals $-(k+h^{\vee})\lambda
J^{(a)}$ by (7.3). The fifth term equals
$\lambda^{2}(k+h^{\vee})(a|\rho_{>0})$ by (7.7). The sixth term equals
$-(k+h^{\vee})\lambda a^{\mathop{\rm ne}\,}$ by (7.4). Thus, we have :

$[{J^{{\\{f\\}}}}_{\lambda}J^{\\{a\\}}]=-[{J^{\\{a\\}}}_{-\partial-\lambda}J^{\\{f\\}}]=-(k+h^{\vee})(\partial+\lambda)J^{\\{a\\}}-\lambda^{2}(k+h^{\vee})(a|\rho_{>0})$
$=\ -(k+h^{\vee})((\partial+\lambda)J^{\\{a\\}}+\lambda^{2}(a|\rho_{>0})),$

proving formula (b). ∎

######  Remark 7.1.

The same proof shows that Lemma 7.1 holds for arbitrary grading (1.1),
satisfying (1.5), if we replace the coefficient of $\lambda^{2}$ by
$(\rho_{>0}|a)-(k+h^{\vee})(x|a)$.

######  Lemma 7.2.

Assume that condition (i) holds and let $M$ be an element of confirmal weight
2 (with respect to $L$), which lies in the subalgebra of
$\mathcal{C}^{k}(\mathfrak{g},x,f)$, generated by the elements $J^{\\{a\\}}$,
$a\in\mathfrak{g}^{f}_{0}$. Suppose that

(7.8) $[{J^{\\{a\\}}}_{\lambda}M]=0\,\,\hbox{for
all}\,\,\,a\in\mathfrak{g}^{f}_{0}.$

Then $M=0$.

###### Proof.

By the condition (i), the subalgebra of $\mathcal{C}^{k}(\mathfrak{g},x,f)$
generated by the elements $J^{\\{a\\}},a\in\mathfrak{g}^{f}_{0}$, is the
affine vertex algebra $V^{B_{1/2}}(\mathfrak{g}^{f}_{0})$ (see (2.15),
(2.16)), associated to the Lie superalgebra $\mathfrak{g}^{f}_{0}$ and a non-
degenerative bilinear form on it, for all, but finitely many values of $k$. It
follows that $\mathfrak{g}_{0}^{f}$ is a direct sum of its center, some basic
Lie superalgebras and Lie superalgebras $gl(n|n),n\geq 1$. Therefore, for all,
but finitely many values of $k$, this vertex algebra carries the Sugawara
element, with respect to which all elements $J^{\\{a\\}}$,
$a\in\mathfrak{g}^{f}_{0}$, have conformal weight 1 (see e.g. [K2]); for
$gl(n|n)$ we use the modification of the Sugawara construction, discussed in
Appendix B. Hence, the center of $V^{B_{1/2}}(\mathfrak{g}^{f}_{0})$ consists
of the multiples of the vacuum vector. Since condition (7.8) implies that M is
a central element of $V^{B_{1/2}}(\mathfrak{g}^{f}_{0})$, we conclude that
$M=0$ for all, but finitely many $k$, hence for all $k$. ∎

Now we can complete the proof of Theorem 3.2. Since L has conformed weight 2
(hereafter conformal weight is meant with respect to L), by [KW], Theorem
4.1(a), L is $d_{(0)}$-equivalent to an element of the form

(7.9) $J^{\\{f^{{}^{\prime}}\\}}\ =\ J^{(f^{{}^{\prime}})}\ +\ M_{1},$

where $f^{{}^{\prime}}\in\mathfrak{g}^{f}_{-1}$ and $M_{1}$ lies in the
subalgebra of $\mathcal{C}^{k}(\mathfrak{g},x,f)$ generated by the $J^{(a)}$
with $a\in\mathfrak{g}_{0}+\mathfrak{g}_{-1/2}$ and the $\Phi_{i},i\in
S_{1/2}$ . Let $v\in\mathfrak{g}^{f}$ and let
$J^{\\{v\\}}\in\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$ be an element,
given by [KW], Theorem 4.1(a). Since $J^{\\{f^{{}^{\prime}}\\}}$ is
$d_{(0)}$-equivalent to L, we have in $W^{k}(\mathfrak{g},x,f)$

(7.10) $[{J^{\\{f^{{}^{\prime}}\\}}}_{\lambda}J^{\\{v\\}}]\ =\ \partial
J^{\\{v\\}}\ +\ O(\lambda).$

But, due to (2.8), equation (7.10) is an equality in
$\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$. It is easy to conclude from
(7.9), using (2.11) (which can be used since
$f^{{}^{\prime}}\in\mathfrak{g}_{-1}$ and $v\in\mathfrak{g}_{\leq 0}$), that
we have the following equality in
$\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$ :

(7.11) $[{J^{\\{f^{{}^{\prime}}\\}}}_{\lambda}J^{\\{v\\}}]\ =\
J^{\\{[f^{{}^{\prime}},v]\\}}+A+B+O(\lambda),$

where $A$ is a linear combination of elements of the form $\sum\
:\partial^{i_{1}}J^{(v_{1})}\dots\partial^{i_{n}}J^{(v_{n})}:$ with $n\geq 2$
or $i_{1}\ +\dots+i_{n}\geq 1$, and $B$ is a linear combination of normally
ordered products, involving the $\Phi_{i}$. Comparing (7.10) and (7.11), we
conclude that

$[f^{{}^{\prime}},\mathfrak{g}^{f}]=0.$

Hence, by condition (ii), we have

$f^{{}^{\prime}}=\gamma f\ \text{for some}\ \gamma\in\mathbb{F}.$

Therefore, L - $\gamma J^{(f)}$ is a sum of normally ordered products of
elements of $\overline{\mathcal{C}}^{k}(\mathfrak{g},x,f)$ of conformal weight
$<2$. It follows that, for $J^{\\{f\\}}$ from Theorem 3.1, we have

(7.12) $L\ -\ \gamma J^{\\{f\\}}\ =\ M,$

where M is a $d_{(0)}$-closed element of conformal weight 1. Hence, by [KW],
Theorem 4.1(b) , M is a linear combination of elements of the form :
$J^{\\{a\\}}J^{\\{b\\}}$ : and $\partial J^{\\{c\\}}$ , where $a,\ b,\ c\ \in\
\mathfrak{g}^{f}_{0}$. By Lemma 7.1, M satisfies (7.8), hence, by Lemma 7.2,
$M=0$, and we have

(7.13) $L=\gamma J^{\\{f\\}}.$

In order to complete the proof of Theorem 3.2, it remains to show that

(7.14) $\gamma\ =\ -\frac{1}{k+h^{\vee}}.$

For that we use the following formula, which can be deduced from the
discussion of properties of L in Section 3 :

(7.15) $[L_{\lambda}J^{(v)}]\ =\ (\partial+\Delta_{v}\lambda)J^{(v)}\ +\
\lambda^{2}(\rho_{>0}|v),\,\,v\in\mathfrak{g},$

where $\Delta_{v}$ is the conformal weight of $J^{(v)}$. Using formula (7.15),
we obtain

(7.16) $[L_{\lambda}\gamma J^{\\{f\\}}]\ =\ (\partial+2\lambda)\gamma
J^{\\{f\\}}\ +\
\lambda^{3}\big{(}-\gamma(k+h^{\vee})\frac{c(\mathfrak{g},x,k)}{12}+\gamma\beta\big{)}.$

where

$\beta=\
(\rho_{>0}|\rho_{>0})-(\rho|\rho)+\frac{1}{6}(\rho_{1/2}|\rho_{>0})+\frac{1}{24}{\rm
str}_{\mathfrak{g}_{0}\oplus\mathfrak{g}_{1/2}}\Omega_{0}.$

Using (7.13) and comparing (7.16) with

$[L_{\lambda}L]\ =\ (\partial+2\lambda)L\ +\ \frac{\lambda^{3}}{12}\
c(\mathfrak{g},x,k),$

where $c(\mathfrak{g},x,k)$, given by (3.3), is non-zero for generic k, we
obtain (7.14) (and also that $\beta=0$). This completes the proof of Theorem
3.2.

##  8  Appendix B. Theorem 3.2 for $\mathfrak{g}=gl(n|n)$.

Let $\mathfrak{g}$ be a finite-dimensional Lie superalgebra over $\mathbb{F}$
with an even invariant supersymmetric bilinear form $(.|.)$. In order to apply
the Sugawara construction, we need two properties:

(i) the bilinear form $(.|.)$ is non-degenerate, so that we can choose dual
bases $\\{u_{i}\\}$ and $\\{u^{i}\\}$ of $\mathfrak{g}$ with respect to this
form and construct the Casimir operator

$\Omega\ =\ \underset{i}{\sum}\ u^{i}u_{i}\ \in U(\mathfrak{g});$

(ii) the Casimir operator $\Omega$ acts on $\mathfrak{g}$ as a scalar (which
we denoted by $2h^{\vee}$).

In this Appendix we consider $\mathfrak{g}=gl(n|n)$ with the bilinear form
$(a|b)={\rm str}\,ab$. The property (i) holds, but (ii) fails. However, we
will show that the Sugawara operator $L^{\mathfrak{g}}$, appearing in the
formula (3.2) for L can be modified, so that the resulting modified $L$ is a
Virasoro vector, which satisfies a modified Theorem 3.2 with the modified
$J^{\\{f\\}}$.

Let $I$ be the identity matrix in $\mathfrak{g}$ and let

(8.1) $\omega\ =\ \underset{i}{\sum}\ :u^{i}u_{i}:\ \in V^{k}(\mathfrak{g}).$

The following formulas are obtained by straightforward computations, where
$a\in\mathfrak{g}$ :

(8.2) $\Omega(a)\ =\ -2(a|I)I;$ (8.3) $[a_{\lambda}\omega]\ =\ 2\lambda
ka-2\lambda(a|I)I;$ (8.4) $[a_{\lambda}I]\ =\ \lambda k(a|I);$ (8.5)
$[a_{\lambda}:I^{2}:]\ =\ 2\lambda k(a|I)I.$

Introduce the modified Sugawara operator

(8.6) $L^{\mathfrak{g}}\ =\ \frac{1}{2k}\omega\ +\ \frac{1}{2k^{2}}\ :I^{2}:.$

It is straightforward to deduce from (8.3) - (8.5) the following two formulas:

(8.7) $[a_{\lambda}L^{\mathfrak{g}}]\ =\ \lambda
a,\,\,\text{hence}\,\,[{L^{\mathfrak{g}}}_{\lambda}a]\ =\
(\partial+\lambda)a;$ (8.8) $[{L^{\mathfrak{g}}}_{\lambda}L^{\mathfrak{g}}]\
=\ (\partial+2\lambda)L^{\mathfrak{g}}.$

Hence, we have the following proposition.

######  Proposition 8.1.

. The element $L^{\mathfrak{g}}$ defined by (8.6) is a Virasoro vector of
$V^{k}(\mathfrak{g})$ with central charge 0, for which $a\in\mathfrak{g}$ have
conformal weight 1. Hence, $V^{k}(\mathfrak{g})$ is a conformal vertex algebra
of CFT type for all $k\neq 0$.

Finally, we have the following version of Theorem 3.2 for
$\mathfrak{g}=gl(n|n)$.

######  Theorem 8.1.

Let $L^{\mathfrak{g}}$ be defined by (8.6), let $L$ be defined by (3.2) with
this $L^{\mathfrak{g}}$, and let $\tilde{J}^{\\{f\\}}$ be the element, defined
in Theorem 3.1 for $h^{\vee}=0$. Then the element

$J^{\\{f\\}}\ =\ \tilde{J}^{\\{f\\}}-\frac{1}{2k}:I^{2}:$

is $d_{(0)}$-closed and $L=-\frac{1}{k}J^{\\{f\\}}$ is an energy-momentum
vector of $W^{k}(\mathfrak{g},x,f)$ with central charge given by formula (3.3)
with $h^{\vee}=0$.

###### Proof.

By the same proof as that of Theorem 3.1, the element $\tilde{J}^{\\{f\\}}$ is
$d_{(0)}$-closed, and also, by (2.6), the element $I$ is $d_{(0)}$\- closed,
hence the same holds for $:I^{2}:$. Hence the element $J^{\\{f\\}}$ is
$d_{(0)}$-closed.

Let $\tilde{L}\ =\frac{1}{2k}\omega+\partial x+L^{ch}+L^{ne}$ (cf. (3.2)). By
the same proof as that of Theorem 3.2, we have:
$\tilde{L}+\frac{1}{k}\tilde{J}^{\\{f\\}}$ is $d_{(0)}$-exact. Since this
element coincides with $L+\frac{1}{k}J^{\\{f\\}}$, the theorem is proved. ∎

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