# Gelfand-Dickey Realizations of
the supersymmetric classical W-algebras for
$\mathfrak{gl}(n+1|n)$ and $\mathfrak{gl}(n|n)$

Sylvain Carpentier† Department of Mathematical Sciences, Seoul National
University, Gwanak-ro 1, Gwanak-gu, Seoul 08826, Korea<EMAIL_ADDRESS>and
Uhi Rinn Suh⋆ Department of Mathematical Sciences and Research institute of
Mathematics, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul 08826,
Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper we realize the supersymmetric classical $W$-algebras
$\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ and
$\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ as differential algebras
generated by the coefficients of a monic superdifferential operator $L$. In
the case of $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ (resp.
$\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$) this operator is even (resp.
odd). We show that the supersymmetric Poisson vertex algebra bracket on these
supersymmetric W-algebras is the supersymmetric analogue of the quadratic
Gelfand-Dickey bracket associated to the operator $L$. Finally, we construct
integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A
key observation is that to construct these hierarchies on the algebra
$\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ one needs to introduce a new
concept of even supersymmetric Poisson vertex algebras.

†This work was supported by BK21 SNU Mathematical Sciences Division

⋆This work was supported by NRF Grant #2022R1C1C1008698 and Creative-
Pioneering Researchers Program by Seoul National University

##  1\. Introduction

Following a series of discoveries made on the Korteweg-de Vries equation
[GGKM67, Lax68, MGK68], the research on integrable nonlinear partial
differential equations took off in the 1970s. In particular, inspired by Peter
Lax, Gelfand and Dickey introduced the generalized KdV hierarchies [GD76].
These are infinite families of evolutionary PDEs defined on the differential
algebra $\mathcal{V}_{N}$ generated by the coefficients of a monic
differential operator of order $N\geq 2$

$L=\partial^{N}+u_{1}\partial^{N-1}+...+u_{N}\,,\,\,\,\mathcal{V}_{N}=\mathbb{C}[u_{i}^{(j)}\,|i=1,...,N\,,j\in\mathbb{Z}_{+}]\,.$

The total derivative $\partial$ acts on the generators of $\mathcal{V}_{N}$ by
$\partial(u_{i}^{(j)})=u_{i}^{(j+1)}$. They showed that the hierarchy of Lax
equations

$\frac{dL}{dt_{l}}=[L^{l/N}_{+},L]\,,\,\,l\geq 1$

is bi-Hamiltonian for the so-called first and second Gelfand-Dickey brackets.
The fractional powers of $L$ are elements in
$\mathcal{V}_{N}(\\!(\partial^{-1})\\!)$ and the residue of such a pseudo-
differential operator is the coefficient of $\partial^{-1}$. Finally the
subscript $+$ denotes the projection of a pseudo-differential operator onto
the subspace of differential operators. The Gelfand-Dickey brackets induce the
following two Lie algebra structures on the space of functionals
$\smallint\mathcal{V}_{N}:=\mathcal{V}_{N}/\partial\mathcal{V}_{N}$

(1.1) $\begin{split}&\big{[}\smallint a,\smallint
b\big{]}_{1}=\,\,\,\int\mathrm{Res}\,\,\Big{(}\,(L\frac{\delta a}{\delta
L})_{+}\frac{\delta b}{\delta L}-(\frac{\delta a}{\delta L}L)_{+}\frac{\delta
b}{\delta L}\,\Big{)},\\\ &\big{[}\smallint a,\smallint
b\big{]}_{2}=\,\,\,\int\mathrm{Res}\,\,\Big{(}\,(L\frac{\delta a}{\delta
L})_{+}L\frac{\delta b}{\delta L}-L(\frac{\delta a}{\delta
L}L)_{+}\frac{\delta b}{\delta L}\,\Big{)},\,\end{split}$

where we used the notation $\smallint$ for the projection map
$\mathcal{V}_{N}\rightarrow\smallint\mathcal{V}_{N}$. The variational
derivative of a functional $\smallint a\in\smallint\mathcal{V}_{N}$ with
respect to $L$ is the pseudo-differential operator

(1.2) $\frac{\delta a}{\delta L}:=\sum_{i=1}^{N}\partial^{i-N-1}\sum_{n\geq
0}(-1)^{n}(\frac{\partial a}{\partial
u_{i}^{(n)}})^{(n)}=\sum_{i=1}^{N}\partial^{i-N-1}\frac{\delta a}{\delta
u_{i}}\,.$

For a detailed review of the Hamiltonian PDEs formalism we refer to [KW81]. To
every functional $\smallint a\in\smallint\mathcal{V}_{N}$ one can associates
two derivations $Y_{a}$ and $Z_{a}$ of $\mathcal{V}_{N}$ commuting with the
total derivative $\partial$ and defined on the generators of the differential
algebra by

(1.3) $\begin{split}Y_{a}(L)&=(L\frac{\delta a}{\delta L}-\frac{\delta
a}{\delta L}L)_{+}\,,\\\ Z_{a}(L)&=(L\frac{\delta a}{\delta
L})_{+}L-L(\frac{\delta a}{\delta L}L)_{+}\,.\end{split}$

A series of independent works [GD77, Man78] has proved using lengthy
calculations that these assignments are Hamiltonian, i.e. for all $\smallint
a\,,\smallint b\in\smallint\mathcal{V}_{N}$

(1.4) $\begin{split}[Y_{a},Y_{b}]&=Y_{[a,b]_{1}}\,,\\\
[Z_{a},Z_{b}]&=Z_{[a,b]_{2}}\,.\end{split}$

It is clear that $Y_{a}$ and $Z_{a}$ are of the form
$\mathcal{M}((\frac{\delta a}{\delta u_{i}})_{i=1,...,N})$ for some $N\times
N$ matrix differential operator $\mathcal{M}$. Such a matrix is called a
Hamiltonian operator. This concept was recently reformulated using the
language of $\lambda$-brackets [BDSK09]. More precisely, one can construct a
map
$\\{\,{}_{\lambda}\,\\}:\,\,\mathcal{V}_{N}\otimes\mathcal{V}_{N}\rightarrow\mathcal{V}_{N}[\lambda]$
such that

(1.5)
$\begin{split}\\{u_{i}{}_{\lambda}u_{j}\\}&=\mathcal{M}_{ij}(\lambda)\,\\\
\end{split}$

for all $1\leq i,j\leq N$ and such that it satisfies all the axioms of a
Poisson vertex algebra (PVA). Above, the notation $P(\lambda)$ for a
differential operator $P$ simply means replacing $\partial^{l}$ by
$\lambda^{l}$. Conversely every PVA structure on $\mathcal{V}_{N}$ uniquely
defines an $N\times N$ Hamiltonian matrix differential operator. Even though
both notions are equivalent the language of $\lambda$-brackets and PVAs has
proven very efficient from a technical point of view and we will follow this
approach in our paper.

In [DS85], Drinfeld and Sokolov managed to connect these hierarchies of
Hamiltonian PDEs to the theory of Kac-Moody Lie algebras. First they
considered a so-called affine Hamiltonian structure
$\\{\,\,{}_{\lambda}\,\,\\}_{\text{aff}}$ on the differential algebra
$\mathcal{V}(\mathfrak{g})$ of differential polynomials on a semisimple Lie
algebra $\mathfrak{g}$ defined on its generators by

(1.6)
$\\{g{}_{\lambda}h\\}_{\text{aff}}=[g,h]+\lambda(g|h)\,,\,\,g,h\,\in\mathfrak{g}$

where $(\,|\,)$ is an invariant symmetric bilinear form. They also constructed
a hierarchy of compatible equations based on a linear matrix operator
$\mathcal{L}$ which are Hamiltonian for the affine structure. Their key result
is showing the existence of a differential subalgebra
$\mathcal{W}(\mathfrak{g})\subset\mathcal{V}(\mathfrak{g})$ such that

  1. $1$.

The PVA structure on $\mathcal{W}(\mathfrak{sl}(N+1))$ coincides with the
second Gelfand-Dickey Hamiltonian structure,

  2. $2$.

The restriction of the affine hierarchy to $\mathcal{W}(\mathfrak{sl}(N+1))$
is the $N$-th KdV hierarchy,

  3. $3$.

The algebra $\mathcal{W}(\mathfrak{g})$ coincides with the functions on
$\mathcal{L}$ invariant under a gauge action of a nilpotent subalgebra
$\mathfrak{n}\subset\mathfrak{g}$.

This process is now known as the Drinfeld-Sokolov reduction and the algebras
$\mathcal{W}(\mathfrak{g})$ as the principal classical W-algebras. Moreover,
Drinfeld and Sokolov showed in [DS85] that their hierarchies are bi-
Hamiltonian where the second PVA structure
$\\{\,\,{}_{\lambda}\,\,\\}_{\text{lin}}$ is defined on
$\mathcal{V}(\mathfrak{g})$ by

(1.7) $\\{g{}_{\lambda}h\\}_{\text{lin}}=(s|[g,h])$

for $g,h\in\mathfrak{g}$. Here $s\in\mathfrak{g}$ is an element of highest
principal grading. After reduction to the subalgebra
$\mathcal{W}(\mathfrak{g})$ this linear bracket coincides with the first
Gelfand-Dickey bracket in the case where $\mathfrak{g}=\mathfrak{sl}(N+1)$.

Later, the notion of W-algebra was further generalized and the classical
W-algebra $\mathcal{W}(\mathfrak{g},F)$ was constructed for any given basic
simple Lie superalgebra $\mathfrak{g}$ and even nilpotent element
$F\in\mathfrak{g}$ [KRW04]. Note that when $\mathfrak{g}$ is a Lie algebra and
$F$ is principal, the corresponding W-algebra $\mathcal{W}(\mathfrak{g},F)$ is
just $\mathcal{W}(\mathfrak{g})$. Unfortunately, one can only construct
integrable hierarchies using the method of Drinfeld-Sokolov for a few
classical W-algebras. De Sole-Kac-Valeri successfully overcame that difficulty
for arbitrary simple Lie algebra $\mathfrak{g}$ by introducing matrix valued
Adler operators [DSKV15, DSKV16a, DSKV16b, DSKV18]. When $\mathfrak{g}$ is a
Lie superalgebra, there are only partial results as far as integrable
hierarchies are concerned [Suh18, CLS24].

Our overarching goal is to generalize this picture to classical supersymmetric
(SUSY) W-algebras. SUSY W-algebras were introduced by Madsen-Ragoucy [MR94]
and later reinterpretated by Molev-Ragoucy-Suh [MRS21] using the language of
SUSY vertex algebras [K]. Roughly speaking a SUSY vertex algebra is a vertex
algebra together with an odd derivation $D$ whose square is the even
derivation $\partial$ of the vertex algebra. The OPE relation of SUSY vertex
algebras can be expressed using a SUSY analogue of the $\lambda$-bracket
[HK07] where the even indeterminate $\lambda$ is replaced by a formal odd
indeterminate $\chi$ such that $D\chi+\chi D=-2\chi^{2}\,.$

As a fundamental example, the OPE product of the affine SUSY vertex algebra
$V^{k}(\bar{\mathfrak{g}})$ associated with a Lie superalgebra $\mathfrak{g}$
is determined by the odd bracket

(1.8) $[\bar{g}{}_{\chi}\bar{h}]=\overline{[g,h]}+k\chi(g|h)$

for $g,h\in\mathfrak{g}$ and $k\in\mathbb{C}$. Here $\bar{\mathfrak{g}}$ is
the parity reversed superspace of $\mathfrak{g}.$ The classical SUSY W-algebra
$\mathcal{W}^{k}(\bar{\mathfrak{g}},f)$ of level $k$ associated with a basic
Lie superalgebra $\mathfrak{g}$ and an odd nilpotent $f$ is defined as a
Hamiltonian reduction of the quasi-classical limit of
$V^{k}(\bar{\mathfrak{g}}).$ They are isomorphic whenever $k\neq 0$ and we
will simply write
$\mathcal{W}(\bar{\mathfrak{g}},f):=\mathcal{W}^{1}(\bar{\mathfrak{g}},f)$.
The SUSY PVA structure of $\mathcal{W}(\bar{\mathfrak{g}},f)$ that we denote
by $\\{\cdot{}_{\chi}\cdot\\}^{q}$ is well-known especially when $f$ is a odd
principal nilpotent. In this paper we start our project of constructing
integrable systems for each of these algebras in the case where
$\mathfrak{g}=\mathfrak{gl}(m|n)$ for $m=n$ or $n+1$ and the odd nilpotent
element $f$ is principal [MRS21, RSS23, LSS23, Suh20]. In the spirit of
Drinfeld and Sokolov, we show that for these types the reduction of the affine
brackets leads to SUSY analogues of the Gelfand-Dickey brackets [HN91].

Let $N$ be an integer greater than $2$. Consider the differential superalgebra
$\mathcal{W}_{N}$ generated by the coefficients of the super-differential
operator

$L=D^{N}+u_{1}D^{N-1}+...+u_{N}\,,\quad\,p(u_{i})\equiv i\ (\text{mod}2)\,.$

The odd derivation $D$ acts on the generators of $\mathcal{W}_{N}$ by
$D(u_{i}^{(j)})=u_{i}^{(j+1)}$. The following hierarchy of equations

(1.9)
$\frac{dL}{dt_{l}}=[L^{2l/N}_{+}\,,\,L]\,,\quad\,[d/dt_{l}\,,\,D]=0\,,\,\,l\geq
1,$

defines pairwise commuting even evolutionary derivations of $\mathcal{W}_{N}$.
However, there is a fundamental difference depending on the parity of $N$:

  1. 1\. If $N=2n$ is even, the conserved densities $(\text{res }L^{\frac{q}{n}})_{q\geq 1}$ of the hierarchy (1.9) are odd.

  2. 2\. If $N=2n+1$ is odd, the conserved densities $(\text{res }L^{\frac{2q-1}{2n+1}})_{q\geq 1}$ of the hierarchy (1.9) are even.

We note that in the first case the $2n$-th root of $L$ is not defined whilst
in the second case the residues of even powers of $L^{\frac{1}{2n+1}}$ are
trivial. We want to discuss the Hamiltonian formalism underneath these
hierarchies. These Hamiltonian structures should map conserved densities to
even derivations. As we just noted it must be of different nature depending on
the parity of $N$. In this paper, we construct these and relate them to the
SUSY Gelfand-Dickey brackets as well as some SUSY classical W-algebras.

What is commonly understood by SUSY Hamiltonian operator? An evolutionary
derivation of $\mathcal{W}_{N}$ is a derivation which supercommutes with the
total derivative $D$ hence which is completly determined by its values on the
generators $u_{i}$. We denote such an object by $X_{F}$ where $F$ is the
characteristic vector $X_{F}(u_{i})=F_{i}$. We send the reader to the main
text (3.11) for the definition of the variational derivative $\delta/\delta
L:\smallint\mathcal{W}_{N}\rightarrow(\mathcal{W}_{N})^{N}$. An $N\times N$
matrix differential operator $H$ is called Hamiltonian if the subspace
$\\{X_{H(\frac{\delta a}{\delta L})}\,|\,\smallint
a\in\smallint\mathcal{W}_{N}\\}$ is a sub Lie superalgebra of the superalgebra
of evolutionary derivations of $\mathcal{W}_{N}$. Essentially, this matrix
maps functionals to evolutionary derivations. If the map is odd (resp. even)
we say that it is an odd (resp. even) Hamiltonian operator.

Taking an odd Hamiltonian matrix differential operator $H$ and replacing $D$
by the odd indeterminate $\chi$, one obtains a $\chi$-bracket
$\\{\,\cdot\,{}_{\chi}\,\cdot\,\\}:\mathcal{W}_{N}\otimes\mathcal{W}_{N}\rightarrow\mathbb{C}[\chi]\otimes\mathcal{W}_{N}$
satisfying the axioms of a SUSY PVA [HK07]. This bracket is odd, i.e. for all
$a,b\in\mathcal{W}_{N}$ we have

$p(\\{a\,{}_{\chi}\,b\\})=p(a)+p(b)+1.$

Moreover, the Hamiltonian derivation can be rewritten as

$X_{H(\frac{\delta a}{\delta
L})}=\\{\,a\,{}_{\chi}\,\cdot\,\\}\big{|}_{\chi=0}\,.$

######  Theorem 1.1 (Theorem 3.12).

Let $m$ and $n$ be positive integers such that $m=n$ or $m=n+1$ and $N=m+n.$
Consider the Lie superalgebra $\mathfrak{g}=\mathfrak{gl}(m|n)$ and the
principal nilpotent element $f$. There is a differential algebra isomorphism
$\phi$ from the SUSY W-algebra $\mathcal{W}(\bar{\mathfrak{g}},f)$ to
$\mathcal{W}_{N}$, such that the SUSY PVA structure of
$\mathcal{W}(\bar{\mathfrak{g}},f)$ induces the so-called SUSY quadratic
Gelfand-Dickey bracket on $\int\mathcal{W}_{N}$. More precisely, for any
$a,b\in\mathcal{W}_{N}$,

(1.10)
$\int\phi\\{\phi^{-1}(a)\,{}_{\chi}\,\phi^{-1}(b)\\}^{q}|_{\chi=0}=(-1)^{N+p(a)}\int\mathrm{Res}\,\,\,((L\frac{\delta
a}{\delta L})_{+}L\frac{\delta b}{\delta L}-L(\frac{\delta a}{\delta
L}L)_{+}\frac{\delta b}{\delta L}).$

It is known [HK07] that a SUSY PVA bracket on a differential superalgebra
$\mathcal{P}$ induces a degree $1$ Lie superalgebra bracket
$\int\\{\cdot{}_{\chi}{}\cdot\\}|_{\chi=0}$ on the quotient
$\mathcal{P}/D\mathcal{P}.$ Hence the following corollary directly follows.

######  Corollary 1.2.

The SUSY quadratic Gelfand-Dickey bracket in Definition 3.14 is a degree $1$
Lie superalgebra bracket on the space of functionals $\int\mathcal{W}_{N}.$

This bracket was introduced in [HN91] where it is claimed to be a Poisson
bracket based on the work [STS83], but we have not found a explicit proof of
the fact that it satisfies the Jacobi identity.

######  Theorem 1.3 (Theorem 3.17).

The hierarchy (1.9) is Hamiltonian for the SUSY quadratic Gelfand-Dickey
structure when $N=2n$ is even with conserved densities $\text{ Res }L^{q/n}$
for $q\geq 1$.

However, when $N=2n+1$ is odd, we must construct an even Hamiltonian operator
$K$ to connect the even conserved densities to the even evolutionary
derivations (1.9). The first example of even Hamiltonian operator appeared in
[Pop09] in the study of the supersymmetric Sawada-Kotera equation. We note
that in [Pop09] this operator is called odd since it induces a odd Poisson
bracket in a different sense. This leads us to define in section 4 the concept
of even SUSY PVAs. This structure is given by a even bracket
$\mathcal{W}_{2n+1}\otimes\mathcal{W}_{2n+1}\rightarrow\mathbb{C}[\chi]\otimes\mathcal{W}_{2n+1}$
such that for all $a,b\in\mathcal{W}_{2n+1}$,

$p(\\{a\,{}_{\chi}\,b\\})=p(a)+p(b).$

In particular, we have for all $a\in\mathcal{W}_{2n+1}$

$X_{K(\frac{\delta a}{\delta L})}=\\{a\,{}_{\chi}\,\cdot\\}|_{\chi=0}\,.$

We then construct even SUSY PVA brackets on the differential algebras
$\mathcal{V}^{k}(\overline{\mathfrak{gl}}(n+1|n))$. We show that the SUSY
W-algebra $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n),f)$ is stable for this
even SUSY PVA bracket $\\{\cdot\,{}_{\chi}\,\cdot\\}^{e}$ and moreover prove
the following theorem.

######  Theorem 1.4 (Theorem 4.17).

Recall the map $\phi$ in Theorem 1.1. The image by $\phi$ of the reduction of
the even SUSY PVA structure to the space of functionals
$\smallint\mathcal{W}_{2n+1}$ is a Lie superalgebra bracket of Gelfand-Dickey
type. More precisely, for any $a,b\in\mathcal{W}_{2n+1}$,

(1.11)
$\int\phi\\{\phi^{-1}(a)\,{}_{\chi}\,\phi^{-1}(b)\\}^{e}=\int\,{\\{a\,{}_{\chi}\,b\\}^{e}}|_{\chi=0}=\int\text{
Res }(L\frac{\delta a}{\delta L}\frac{\delta b}{\delta
L}+(-1)^{p(a)+1}\frac{\delta a}{\delta L}L\frac{\delta b}{\delta L}).$

Finally, we conclude section 4 by

######  Theorem 1.5 (Theorem 4.22).

The hierarchy (1.9) for odd $N=2n+1$ is Hamiltonian for the even linear SUSY
Gelfand-Dickey structure in Definition 4.19 with conserved densities $\text{
Res }L^{\frac{2q-1}{2n+1}}$ for $q\geq 1$.

We suggest that the structures now referred to as SUSY PVAs in the literature
should be called odd SUSY PVAs when the context is ambiguous, to distinguish
them from the even SUSY PVAs introduced in this paper.

##  2\. supersymmetric classical W-algebras

###  2.1. SUSY Poisson vertex algebra

Throughout this paper, we consider vector superspaces defined over
$\mathbb{C}$. For such a vector superspace $V$, we denote its even (resp. odd)
subspace by $V_{\bar{0}}$ (resp. $V_{\bar{1}}$) and the parity of a
homogeneous element $a\in V$ by $p(a).$

Let $\mathcal{R}$ be a $\mathbb{C}[D]$-module, where $D$ is an odd operator
acting on a superspace $\mathcal{R}$. For an odd indeterminate $\chi$, we
extend the $\mathbb{C}[D]$-module structure to $\mathbb{C}[\chi]\otimes R$ by
letting $D\chi+\chi D=-2\chi^{2}.$ A linear map

(2.1)
$[\cdot\,{}_{\chi}\,\cdot]:\mathcal{R}\otimes\mathcal{R}\to\mathbb{C}[\chi]\otimes\mathcal{R}$

is called a (odd) $\chi$-bracket on $\mathcal{R}$ if for all $a,b\in R$

  1. (odd parity) $p([a{}_{\chi}b])=p(a)+p(b)+1$,

  2. (sesquilinearity) $[Da{}_{\chi}b]=\chi[a{}_{\chi}b],$ $[a{}_{\chi}Db]=(-1)^{p(a)+1}(D+\chi)[a_{\chi}b]\,.$

Moreover, if a $\chi$-bracket satisfies the properties:

  1. (skew-symmetry) $[b{}_{\chi}a]=(-1)^{p(a)p(b)}[a{}_{-\chi-D}b]_{\leftarrow},$

  2. (Jacobi identity) $[a{}_{\chi}[b{}_{\gamma}c]]=(-1)^{p(a)+1}[[a{}_{\chi}b]_{\chi+\gamma}c]+(-1)^{(p(a)+1)(p(b)+1)}[b_{\gamma}[a_{\chi}c]]$

for all $a,b,c\in\mathcal{R}$, then $\mathcal{R}$ is called a supersymmetric
Lie conformal algebra (SUSY LCA). We denote by $a_{(n)}b\in\mathcal{R}$ the
coefficient of $\chi^{n}$ in $[a{}_{\chi}b]$, so that
$[a{}_{\chi}b]=\sum_{n\in\mathbb{Z}_{+}}\chi^{n}a_{(n)}b.$ In the RHS of the
skew-symmetry axiom,
$[a_{-\chi-D}b]_{\leftarrow}:=\sum_{n\in\mathbb{Z}_{+}}(-D-\chi)^{n}a_{(n)}b$
and the Jacobi identity holds in $\mathbb{C}[\chi,\gamma]\otimes\mathcal{R}$
for an odd indeterminate $\gamma$ such that $\chi\gamma+\gamma\chi=0$ and
$D\gamma+\gamma D=-2\gamma^{2}.$ More precisely, the Jacobi identity is
computed by the following formula.

(2.2)
$\displaystyle[a{}_{\chi}\gamma^{n}b]=(-1)^{n}\gamma^{n}[a{}_{\chi}b],\quad[a{}_{\gamma}\chi^{n}b]=(-1)^{n}\chi^{n}[a{}_{\chi}b],$
$\displaystyle[\chi^{n}a{}_{\chi+\gamma}b]=(-1)^{n}\chi^{n}[a{}_{\chi+\gamma}b].$

######  Definition 2.1.

Let $\mathcal{P}$ be a unital supersymmetric differential algebra with an odd
derivation D. A SUSY Lie conformal algebra $(\mathcal{P},D,\\{\ {}_{\chi}\
\\})$ is called an odd SUSY Poisson vertex algebra (odd SUSY PVA) if it
satisfies the Leibniz rule:

(2.3)
$\\{a{}_{\chi}bc\\}=\\{a{}_{\chi}b\\}c+(-1)^{(p(a)+1)p(b)}b\\{a{}_{\chi}c\\}\quad\text{
for }\quad a,b,c\in\mathcal{P}.$

Note that this definition of an odd SUSY PVA corresponds to the $N_{k}=1$ SUSY
PVA in [HK07] In this paper, we are interested in SUSY PVAs which are algebras
of differential polynomials, i.e. of the form

(2.4) $\mathcal{P}=\mathbb{C}[u_{i}^{(m)}|i\in I,m\in\mathbb{Z}_{+}],$

where $I$ is a finite set and the odd derivation $D$ acts on the generators of
$P$ by $D(u_{i}^{(m)}):=u_{i}^{(m+1)}$.

######  Proposition 2.2.

[CS2020] Let $\mathcal{P}$ be a differential algebra of polynomials (2.4)
endowed with a $\chi$-bracket $\\{\ {}_{\chi}\ \\}$. Suppose that the skew-
symmetry and Jacobi identity axioms hold for a set of generators
$(u_{i})_{i\in I}$ of $\mathcal{P}$, i.e.,

$\displaystyle\\{u_{i}\,{}_{\chi}\,u_{j}\\}=(-1)^{\tilde{\imath}\tilde{\jmath}}\\{u_{j}\,{}_{-\chi-D}\,u_{i}\\}_{\leftarrow};$
$\displaystyle\\{u_{i}\,{}_{\chi}\,\\{u_{j}\,{}_{\gamma}u_{k}\,\\}\\}=(-1)^{\tilde{\imath}+1}\\{\\{u_{i}\,{}_{\chi}\,u_{j}\\}{}_{\chi+\gamma}u_{k}\\}+(-1)^{(\tilde{\imath}+1)(\tilde{\jmath}+1)}\\{u_{j}\,{}_{\gamma}\\{u_{i}\,{}_{\chi}\,u_{k}\\}\\}$

for $i,j,k\in I$. Here $\tilde{\imath}$ and $\tilde{\jmath}$ are parities of
$u_{i}$ and $u_{j}$, respectively. If the bracket $\\{\,{}_{\chi}\,\\}$ also
satisfies the Leibniz rule, then $P$ is a SUSY PVA.

###### Proof.

See [CS2020] Theorem 2.15. ∎

Thanks to the Leibniz rule, sesquilinearity and skew-symmetry axioms, an odd
SUSY PVA bracket on a algebra of differential polynomials is completely
determined by its values on pairs of generators $(u_{i},u_{j}),\,i,j\in I.$ We
will now try to find a formula expressing the bracket $\\{a\,{}_{\chi}\,b\\}$
for all $a,b\in\mathcal{P}$ in terms of the brackets
$\\{u_{i}\,{}_{\chi}\,u_{j}\\}$. For any $a,b\in\mathcal{P}$, we define the
super-differential operator

$\\{a\,{}_{D}\,b\\}\,:=\sum_{n\in\mathbb{Z}_{+}}(-1)^{n(p(a)+p(b))+\frac{n(n-1)}{2}}a_{(n)}b\,D^{n}\,\,.$

Conversely, we can retrieve the $\chi$-bracket of $a$ and $b$ from the
operator $\\{a\,{}_{D}\,b\\}$ by the formula

$\\{a\,{}_{D+\chi}\,b\\}(1)=\\{a\,{}_{\chi}\,b\\}\,.$

For any $a,b,c\in\mathcal{P}$ we will make the slight abuse of notation

$\\{a\,{}_{D+\chi}\,b\\}\,\,c:=\sum_{n\in\mathbb{Z}_{+}}(-1)^{n(p(a)+p(b))+\frac{n(n-1)}{2}}a_{(n)}b\,(D+\chi)^{n}(c)\,.$

By the skew-symmetry and Leibniz rule, we have

(2.5)
$\\{ab{}_{\chi}c\\}=(-1)^{p(b)p(c)}\\{a{}_{\chi+D}c\\}b+(-1)^{p(a)(p(b)+p(c))}\\{b{}_{\chi+D}c\\}a$

and hence for all $a,b\in\mathcal{P}$,

(2.6) $\displaystyle\\{\,a\,{}_{\chi}\,b\,\\}$ $\displaystyle=\sum_{i\in
I,m\in\mathbb{Z_{+}}}(-1)^{(p(a)+\tilde{\imath}+m)p(b)}\\{u_{i}^{(m)}{}_{\chi+D}\,b\\}\frac{\partial
a}{\partial u_{i}^{(m)}}$ $\displaystyle=\sum_{i\in
I,\,m\in\mathbb{Z}_{+}}(-1)^{(p(a)+\tilde{\imath}+m)p(b)+m(\tilde{\imath}+p(b)+1)+\frac{m(m-1)}{2}}\\{u_{i}\,{}_{\chi+D}\,b\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}.$

Finally we obtain what we will call the master formula:

(2.7) $\displaystyle\\{\,a\,{}_{\chi}\,b\,\\}$ $\displaystyle=\sum_{i,j\in
I,\,m,n\in\mathbb{Z}_{+}}s_{i,j}^{m,n}\frac{\partial b}{\partial
u_{j}^{(n)}}\\{\,u_{i}^{(m)}\,{}_{\chi+D}\,u_{j}^{(n)}\\}\frac{\partial
a}{\partial u_{i}^{(m)}}$ $\displaystyle=\sum_{i,j\in
I,\,m,n\in\mathbb{Z}_{+}}s_{i,j}^{m,n}\,t_{i,j}^{m,n}\frac{\partial
b}{\partial
u_{j}^{(n)}}(\chi+D)^{n}\\{\,u_{i}\,{}_{\chi+D}\,u_{j}\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}},$

where
$s_{i,j}^{m,n}=(-1)^{p(b)(p(a)+\tilde{\jmath}+n+1)+(\tilde{\jmath}+n)(\tilde{\imath}+m)}$
and
$t_{i,j}^{m,n}=(-1)^{n(\tilde{\imath}+m+1)+m(\tilde{\imath}+\tilde{\jmath}+1)+\frac{m(m-1)}{2}}$
for $\tilde{\imath}:=p(u_{i})$ and $\tilde{\jmath}:=p(u_{j})$.

######  Definition 2.3.

Let $\mathcal{P}$ be the algebra of differential polynomials (2.4). The
variational derivative of an element $a\in\mathcal{P}$ with respect to $u_{i}$
is given by

(2.8) $\frac{\delta a}{\delta
u_{i}}=\sum_{m\in\mathbb{Z}_{+}}(-1)^{m\tilde{\imath}+\frac{m(m+1)}{2}}D^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}.$

Let us consider the projection map
$\int:\mathcal{P}\to\smallint\mathcal{P}:=\mathcal{P}/D(\mathcal{P})$. The
notation is justified by its ‘integration by parts’ property $\int
D(a)b=-\int(-1)^{p(a)}aD(b)$. We call an element of $\smallint\mathcal{P}$ a
functional. By sesquilinearity, given $\smallint a\in\smallint\mathcal{P}$ the
following linear map is well-defined

(2.9) $\mathcal{P}\to\mathcal{P},\quad v\mapsto\\{\smallint
a\,,\,v\\}:=\\{\,a\,{}_{\chi}\,v\,\\}|_{\chi=0}.$

By the Leibniz rule, this map is a derivation of $\mathcal{P}$ of parity
$p(a)+1.$ It follows from the Jacobi identity that the superspace of
functionals is endowed with a degree 1 Lie superalgebra bracket defined by

(2.10) $\\{\,\smallint a\,,{}\,\smallint
b\,\\}:=\smallint\\{\,a\,{}_{\chi}\,b\,\\}|_{\chi=0}.$

Comparing (2.7) and (2.8), we have

(2.11) $\begin{split}\\{\smallint a,\smallint b\\}&=\sum_{i,j\in
I}(-1)^{p(a)p(b)+p(b)\tilde{\jmath}+p(b)+\tilde{\imath}\tilde{\jmath}}\int\frac{\delta
b}{\delta u_{j}}\\{u_{i}{}_{D}u_{j}\\}(\frac{\delta a}{\delta u_{i}})\\\
&=\sum_{i,j\in
I}(-1)^{p(a)\tilde{\jmath}+\tilde{\imath}\tilde{\jmath}}\int\\{u_{i}{}_{D}u_{j}\\}(\frac{\delta
a}{\delta u_{i}})\,\,\frac{\delta b}{\delta u_{j}}\,.\end{split}$

###  2.2. SUSY classical W-algebras

Let $\mathfrak{g}$ be a finite basic Lie superalgebra with a
$\mathbb{Z}/2$-grading
$\mathfrak{g}=\bigoplus_{i\in\frac{\mathbb{Z}}{2}}\mathfrak{g}(i)$ and denote

(2.12)
$\mathfrak{n}=\bigoplus_{i>0}\mathfrak{g}(i),\quad\mathfrak{b}=\bigoplus_{i\leq
0}\mathfrak{g}(i),\quad\mathfrak{b}_{+}=\bigoplus_{i\geq 0}\mathfrak{g}(i).$

Suppose that there is an odd element $f\in\mathfrak{g}(-\frac{1}{2})$
satisfying the properties:

(2.13) $\text{$\text{ad}f|_{\mathfrak{n}}$ : injective\ ,
\quad$\text{ad}f|_{\mathfrak{b}}$ : surjective}.$

For example, (2.13) holds when $f$ is an odd nilpotent element in a subalgebra
isomorphic to $\mathfrak{osp}(1|2)$ and one takes the Dynkin grading induced
from the $\mathfrak{sl}_{2}$-triple $(E,H,F=\frac{1}{2}[f,f])$.

Let $\bar{\mathfrak{g}}$ be the parity reversed superspace of $\mathfrak{g}.$
The affine SUSY PVA of level $k\in\mathbb{C}$, denoted by
$\mathcal{V}^{k}(\bar{\mathfrak{g}})$, is isomorphic as a differential algebra
to the supersymmetric algebra of the superspace
$\mathbb{C}[D]\otimes\bar{\mathfrak{g}}$. The SUSY PVA $\chi$-bracket [HK07]
on $\mathcal{V}^{k}(\bar{\mathfrak{g}})$ is defined on its generators by

(2.14)
$\\{\bar{g}{}_{\chi}\bar{h}\\}_{\text{aff}}=(-1)^{p(g)}(\,\overline{[g,h]}+k\chi(g|h)\,)$

for $g,h\in\mathfrak{g}$. The classical SUSY W-algebra associated with
$\mathfrak{g}$ and $f$ is defined by

(2.15)
$\mathcal{W}^{k}(\bar{\mathfrak{g}},f)=\\{w\in\mathcal{V}^{k}(\bar{\mathfrak{g}})\,|\,\rho(\\{\bar{v}{}_{\chi}w\\}_{\text{aff}})=0\text{
for all }v\in\mathfrak{n}\\},$

where
$\rho:\mathbb{C}[\chi]\otimes\mathcal{V}^{k}(\bar{\mathfrak{g}})\to\mathbb{C}[\chi]\otimes
S(\mathbb{C}[D]\otimes\bar{\mathfrak{b}})$ is the algebra homomorphism defined
by

(2.16) $D^{m}\bar{g}\mapsto\left\\{\begin{array}[]{ll}D^{m}\bar{g}&\text{ if
}g\in\mathfrak{b},\\\ -D^{m}(f|g)&\text{ if
}g\in\mathfrak{n};\end{array}\right.\quad\chi\mapsto\chi\,,$

for $m\in\mathbb{Z}_{+}$. One can check that the $\chi$-bracket on
$\mathcal{W}^{k}(\bar{\mathfrak{g}},f)$ defined by
$\\{w_{1}{}_{\chi}w_{2}\\}:=\rho(\\{w_{1}{}_{\chi}w_{2}\\}_{\text{aff}})$ for
$w_{1},w_{2}\in\mathcal{W}^{k}(\bar{\mathfrak{g}},f)$ induces a SUSY PVA
structure [Suh20]. They are isomorphic whenever $k\neq 0$ and we will simply
denote
$\mathcal{W}(\bar{\mathfrak{g}},f):=\mathcal{W}^{1}(\bar{\mathfrak{g}},f)$. By
the same reason, we denote
$\mathcal{V}(\bar{\mathfrak{g}}):=\mathcal{V}^{1}(\bar{\mathfrak{g}}).$

Here we remark that in other references, for example [MRS21], a SUSY W-algebra
is defined under the assumption that $f$ is an odd nilpotent element in an
$\mathfrak{osp}(1|2)$ subalgebra but (2.13) is enough for the properties that
are used in this paper.

######  Example 2.4.

Let $\mathfrak{g}=\mathfrak{gl}(n+1|n)$. Take a basis
$\\{\,e_{ij}\,|\,i,j=1,2,\cdots,2n+1\,\\}$ of $\mathfrak{g}$, where the parity
of $e_{ij}\in\mathfrak{g}$ is $p(e_{ij})\equiv i+j\ (\text{mod }2)$ and
$e_{ij}\in\mathfrak{g}(\frac{j-i}{2})$. One can check that the odd element
$f=\sum_{i=1}^{2n}e_{i+1\,i}\in\mathfrak{g}(-\frac{1}{2})$ is in a subalgebra
of $\mathfrak{g}$ isomorphic to $\mathfrak{osp}(1|2)$. Hence
$\mathcal{W}(\bar{\mathfrak{g}},f)$ is well-defined.

######  Example 2.5.

Let $\mathfrak{g}=\mathfrak{gl}(n|n)$ and take a basis
$\\{\,e_{ij}\,|\,i,j=1,2,\cdots,2n\,\\}$, where the parity of
$e_{ij}\in\mathfrak{g}$ is $p(e_{ij})\equiv i+j\ (\text{mod }2)$. Consider the
grading on $\mathfrak{g}$ such that $e_{ij}\in\mathfrak{g}(\frac{j-i}{2})$ and
take an odd element
$f=\sum_{i=1}^{2n-1}e_{i+1\,i}\in\mathfrak{g}(-\frac{1}{2})$. Then $f$
satisfies the properties (2.13) and hence $\mathcal{W}(\bar{\mathfrak{g}},f)$
is well-defined.

###  2.3. Lax matrix operators

Let $\bar{\mathfrak{b}}$ be the parity reversed superspace of $\mathfrak{b}$
and $\mathcal{V}(\bar{\mathfrak{b}})$ be the differential superalgebra of
polynomials generated by $\bar{\mathfrak{b}}$. Let $\\{q^{i}|i\in S\\}$ and
$\\{{q}_{i}|i\in S\\}$ be bases of $\mathfrak{b}_{+}$ and ${\mathfrak{b}}$
such that $(q^{i}|q_{j})=\delta_{ij}$. We introduce the universal Lax operator

(2.17) $\mathcal{L}^{u}=D+\sum_{i\in S}q^{i}\otimes\bar{q}_{i}-f\otimes
1\in(\mathbb{C}D\oplus\mathbb{C}\partial)\ltimes\left(\mathfrak{g}\otimes\mathcal{V}(\bar{\mathfrak{b}})\right),$

where $\mathbb{C}[D]$ acts on
$\mathfrak{g}\otimes\mathcal{V}(\bar{\mathfrak{b}})$ via $D(g\otimes
X)=(-1)^{p(g)}g\otimes D(X)$. Note that
$\mathfrak{g}\otimes\mathcal{V}(\bar{\mathfrak{b}})$ is a Lie superalgebra for
the bracket

(2.18) $[g\otimes X,h\otimes Y]=(-1)^{(p(X))p(h)}[g,h]\otimes XY,$

and can be naturally extended to the semi-direct product
$\mathbb{C}[D]\ltimes\left(\mathfrak{g}\otimes\mathcal{V}(\bar{\mathfrak{b}})\right)$.

######  Proposition 2.6 ([CS2020]).

Let $V$ be a subspace of $\mathfrak{g}$ such that $[\mathfrak{n},f]\bigoplus
V=\mathfrak{b}_{+}$.

  1. (1)

There exists a unique even element
$N_{c}\in\mathfrak{n}\otimes\mathcal{V}(\mathfrak{\bar{b}})$ such that
$e^{\text{ad}N_{c}}\mathcal{L}^{u}=\mathcal{L}^{c}$ and

$Q^{c}:=\mathcal{L}^{c}-D+f\otimes 1\in
V\otimes\mathcal{V}(\bar{\mathfrak{b}}).$

The operator $\mathcal{L}^{c}$ is called the canonical Lax operator associated
with our choice of $f$ and $V$.

  2. (2)

Let $\\{v_{i}\,|\,i\in J\\}$ be a basis of $V$ and let $Q^{c}=\sum_{i\in
J}v_{i}\otimes w_{i}$. Then the W-algebra $\mathcal{W}(\bar{\mathfrak{g}},f)$
is freely generated by the elements $w_{i}$’s as a differential algebra.

###### Proof.

See Theorem 3.14 in [CS2020]. ∎

There is another realization of the classical SUSY W-algebras. We identify an
element in $\mathcal{V}(\bar{\mathfrak{b}})$ with a linear function from the
odd space
$\mathcal{F}:=(\mathfrak{b}_{+}\otimes\mathcal{V}(\bar{\mathfrak{b}}))_{\bar{1}}$
to $\mathcal{V}(\bar{\mathfrak{b}})$ defined by

(2.19) $\bar{a}^{(n)}(b\otimes X):=(b|a)X^{(n)},\quad AB(b\otimes
X)=A(b\otimes X)B(b\otimes X)$

for $a\in\mathfrak{b},$ $b\in\mathfrak{b}_{+}$, and
$A,B,X\in\mathcal{V}(\bar{\mathfrak{b}})$. In particular, for
$Q:=\bigoplus_{i\in S}q^{i}\otimes\bar{q}_{i}$ and
$A\in\mathcal{V}(\bar{\mathfrak{b}})$, we have

(2.20) $A(Q)=A.$

For $P\in\mathcal{F}$, we consider the corresponding operator
$\mathcal{L}_{P}:=D+P-f\otimes 1$. Two elements $P$ and $P^{\prime}$ in
$\mathcal{F}$ are called gauge equivalent if there is an element
$N\in\mathfrak{n}\otimes\mathcal{V}(\bar{\mathfrak{b}})$ such that
$e^{\text{ad\,}N}\mathcal{L}_{P}=\mathcal{L}_{P^{\prime}}.$ On the other hand,
since $\mathcal{W}(\bar{\mathfrak{g}},f)$ is a subalgebra of
$\mathcal{V}(\bar{\mathfrak{b}})$, it can also be realized as an algebra of
functions on $\mathcal{F}.$ As such, it turns out that this subalgebra
$\mathcal{W}(\bar{\mathfrak{g}},f)$ identifies with the gauge invariant
functions in the following sense.

######  Proposition 2.7.

[CS2020] The differential algebra $\mathcal{W}(\bar{\mathfrak{g}},f)$ is the
set of gauge invariant functions $w$, that is
$w\in\mathcal{W}(\bar{\mathfrak{g}},f)$ if and only if $w(P)=w(P^{\prime})$
whenever two elements $P$ and $P^{\prime}$ in $\mathcal{F}$ are gauge
equivalent.

For $a\in\mathcal{V}(\bar{\mathfrak{b}})$, we denote its variational
derivative with respect to $\bar{\mathfrak{b}}$ by

(2.21) $\frac{\delta a}{\delta q}:=\sum_{i\in S}q_{i}\otimes\frac{\delta
a}{\delta\bar{q}_{i}}.$

The following theorem holds.

######  Theorem 2.8.

[CS2020] The degree 1 Lie superalgebra bracket on
$\int\mathcal{W}(\bar{\mathfrak{g}},f)$ can be realized via the universal Lax
operator and variational derivatives. More precisely, for
$w_{1},w_{2}\in\mathcal{W}(\bar{\mathfrak{g}},f)$,

(2.22) $\\{\smallint w_{1}\,,\ \smallint
w_{2}\\}=-\int\left(\left.\frac{\delta w_{1}}{\delta
q}\right|\left[\mathcal{L}^{u},\frac{\delta w_{2}}{\delta q}\right]\right).$

##  3\. SUSY quadratic Gelfand-Dickey brackets and SUSY W-algebras

###  3.1. SUSY pseudo-differential operators

Let $N$ be an integer greater than $2$. Let
$\mathcal{W}_{N}:=\mathbb{C}[u_{i}^{(n)}|i=1,\cdots,N,\,n\in\mathbb{Z}_{+}]$
be the differential superalgebra where the generators have parity
$p(u_{i})\equiv i$ (mod $2$) and the odd derivation $D$ is defined on the
generators by $D(u_{i}^{(n)})=u_{i}^{(n+1)}$. The algebra of pseudo-
differential operators on $\mathcal{W}_{N}$ is the superspace of Laurent
series
$\mathcal{W}_{N}(\\!(D^{-1})\\!):=\mathcal{W}_{N}\otimes\mathbb{C}(\\!(D^{-1})\\!)$
endowed with the product

(3.1) $\displaystyle D\cdot aD^{m}:=a^{\prime}D^{m}+(-1)^{p(a)}aD^{m+1},$
$\displaystyle D^{-1}\cdot
aD^{m}=\sum_{k\in\mathbb{Z}_{+}}(-1)^{\frac{k(k-1)}{2}+(k-1)p(a)}a^{(k)}D^{m-k-1},$

where $a\in\mathcal{W}_{N}$, $m\in\mathbb{Z}$ and
$a^{\prime}:=D(a),\,a^{(n)}:=D^{n}(a)$. In general, for $n\in\mathbb{Z}$, we
have

(3.2) $\displaystyle D^{2n}\cdot
aD^{m}=\sum_{j\in\mathbb{Z}_{+}}{\binom{n}{j}}a^{(2j)}D^{m+2n-2j},$
$\displaystyle D^{2n+1}\cdot
aD^{m}=\sum_{j\in\mathbb{Z}_{+}}{\binom{n}{j}}\Big{(}(-1)^{p(a)}a^{(2j)}D^{m+1+2n-2j}+a^{(2j+1)}D^{m+2n-2j}\Big{)}.$

Elements in the subalgebra
$\mathcal{W}_{N}[D]:=\mathcal{W}_{N}\otimes\mathbb{C}[D]$ are called super-
differential operators. For a pseudo-differential operator
$A(D)=\sum_{m\in\mathbb{Z}}a_{m}D^{m}$, its adjoint
$A^{*}(D)\in\mathcal{W}_{N}(\\!(D^{-1})\\!)$ is defined by

(3.3)
$\Big{(}\sum_{m\in\mathbb{Z}_{+}}aD^{m}\Big{)}^{*}:=\sum_{m\in\mathbb{Z}_{+}}(-1)^{mp(a)+\frac{m(m+1)}{2}}D^{m}a.$

Thus $D^{*}=-D$, $(D^{2})^{*}=-D^{2}$, $(D^{-1})^{*}=D^{-1}$ and $a^{*}=a$ for
any $a\in\mathcal{W}_{N}$. We have the following lemma.

######  Lemma 3.1.

Let $A$ and $B$ be pseudo-differential operators on $\mathcal{W}_{N}$. Then

(3.4) $(AB)^{*}=(-1)^{p(A)p(B)}B^{*}A^{*}.$

###### Proof.

First, one can easily check (3.4) is true when $A=D^{m}$ and $B=D^{n}$ for any
$m,n\in\mathbb{Z}$ and when $A\in\mathcal{W}_{N}.$ Now suppose that $A=D$ and
$B=bD^{n}$ for some $b\in\mathcal{W}_{N}$ and $n\in\mathbb{Z}.$ Then

(3.5) $\displaystyle(AB)^{*}=(D\cdot
bD^{n})^{*}=(b^{\prime}D^{n}+(-1)^{p(b)}bD^{n+1})^{*}$
$\displaystyle=(-1)^{n(p(b)+1)+\frac{n(n+1)}{2}}D^{n}b^{\prime}+(-1)^{np(b)+\frac{(n+1)(n+2)}{2}}D^{n+1}b$
$\displaystyle=(-1)^{p(b)+n}\big{(}(-1)^{np(b)+\frac{n(n+1)}{2}+1}D^{n}bD\big{)}=(-1)^{p(A)p(B)}B^{*}A^{*}.$

Here, we used $bD=(-1)^{p(b)+1}b^{\prime}+(-1)^{p(b)}Db.$ Inductively, we can
see that (3.4) holds when $A\in\mathcal{W}_{N}[D].$ Moreover, we have

(3.6)
$(D^{-1}B)^{*}D=(D^{-1}B)^{*}(-D)^{*}=(-1)^{p(B)+1}(-DD^{-1}B)^{*}=(-1)^{p(B)}B^{*}$

and hence $(D^{-1}B)^{*}=(-1)^{p(B)}B^{*}D^{-1}=(-1)^{p(B)}B^{*}(D^{-1})^{*}$.
This implies (3.4) holds for any pseudo-differential operator $A$. ∎

For a pseudo-differential operator $A\in\mathcal{W}_{N}(\\!(D^{-1})\\!)$, the
differential and integral part of $A(D)$ are denoted by
$A(D)_{+}:=\sum_{m\in\mathbb{Z_{+}}}a_{m}D^{m}$ and
$A(D)_{-}:=\sum_{m\in\mathbb{Z}_{+}}a_{-m-1}D^{-m-1}$ respectively, and the
residue $\text{ Res }(A(D))$ of $A(D)$ is the $(-1)$-th coefficient $a_{-1}.$
The properties in the following lemma are useful throughout this paper.

######  Lemma 3.2.

Let $A$ and $B$ be two pseudo-differential operators on $\mathcal{W}_{N}$ and
$a\in\mathcal{W}_{N}$. Then the following properties hold.

  1. (1)

$\text{Res }(A)=\text{Res }(A^{*}),$

  2. (2)

$\text{Res }(Aa)=(-1)^{p(a)}\text{Res }(A)a,$

  3. (3)

$\smallint\text{Res }(AB)=(-1)^{p(A)p(B)}\smallint\text{Res }(BA).$

###### Proof.

Let $A=\sum_{i\in\mathbb{Z}}a_{i}D^{i}$. Then $\text{Res }(A)=\text{ Res
}(A^{*})=a_{-1}$ and $\text{ Res }(Aa)=(-1)^{p(a)}\text{ Res
}(A)a=(-1)^{p(a)}a_{-1}a.$ Hence we get (1) and (2). Let us show (3). For
$m,n\in\mathbb{Z},$ one can observe that $\text{ Res
}(aD^{2m}bD^{2n})=0=\text{ Res }(bD^{2n}aD^{2m})$. Also, we have

(3.7) $\displaystyle\text{ Res
}(aD^{2m+1}bD^{2n})=(-1)^{p(b)}{\binom{m}{n+m+1}}ab^{(2m+2n+2)},$
$\displaystyle\text{ Res
}(aD^{2m+1}bD^{2n-1})={\binom{m}{n+m}}ab^{(2m+2n+1)},$

and

(3.8) $\displaystyle\text{ Res
}(bD^{2n}aD^{2m+1})={\binom{n}{n+m+1}}ba^{(2n+2n+2)},$ $\displaystyle\text{
Res }(bD^{2n-1}aD^{2m+1})={\binom{n-1}{n+m}}ba^{(2m+2n+1)},$

Considering the fact that ${\binom{m}{n+m+}}=(-1)^{n+m}{\binom{n-1}{n+m}}$ and
${\binom{m}{n+m+1}}=(-1)^{n+m+1}{\binom{n}{n+m+1}}$ and comparing (3.7) and
(3.8) conclude the proof. ∎

###  3.2. SUSY quadratic Gelfand-Dickey bracket

Consider the monic super-differential operator

(3.9) $L_{N}(D)=D^{N}+u_{1}D^{N-1}+...+u_{N}.$

Following [HN91] and [FFR92] we define the SUSY quadratic Gelfand-Dickey
bracket on $\smallint\mathcal{W}_{N}$ by

(3.10) $\\{\smallint a,\smallint b\\}:=(-1)^{p(a)+N}\text{ Res
}\int\,\Big{(}(L\frac{\delta a}{\delta L})_{+}L-L(\frac{\delta a}{\delta
L}L)_{+}\Big{)}\,\frac{\delta b}{\delta L},$

where for all $\smallint a\in\smallint\mathcal{W}_{N}$, the variational
derivative of $\smallint a$ with respect to $L$ is the pseudo-differential
operator

(3.11) $\frac{\delta a}{\delta
L}:=(-1)^{p(a)}\sum_{k=1}^{N}D^{k-N-1}\sum_{l\in\mathbb{Z}_{+}}(-1)^{k+kl+\frac{l(l+1)}{2}}D^{l}(\frac{\partial
a}{\partial u_{k}^{(l)}})\in\mathcal{W}_{N}(\\!(D^{-1})\\!).$

Hence the parity of $\frac{\delta a}{\delta L}$ is $p(a)+N+1$ (mod $2$). We
remark that, at this moment, it is not clear if the bracket (3.10) is a degree
$1$ Lie superalgebra bracket or not. We will prove this by realizing it as a
reduction of a affine SUSY PVA bracket. Let us consider the odd linear matrix
differential operator $\mathcal{L}^{can}$ in
$\mathfrak{gl}(m|n)\otimes\mathcal{W}_{N}[D]$, where $m=n+1,\,n$ and $N=m+n$:

(3.12) $\displaystyle\mathcal{L}^{can}$
$\displaystyle=\sum_{i=1}^{N}e_{ii}\otimes
D+\sum_{i=1}^{N}e_{iN}\otimes(-1)^{N+1}u_{N+1-i}-\sum_{i=1}^{N-1}e_{i+1,i}\otimes
1.$

Here we use the basis $\\{e_{ij}|i,j=1,\cdots,N\\}$ of $\mathfrak{gl}(m|n)$
introduced in Example 2.4 and 2.5 so that $p(e_{ij})\equiv i+j$ (mod $2$). In
the space of matrix operators $\mathfrak{gl}(m|n)\otimes\mathcal{W}_{N}[D],$
we consider the associative multiplication and Lie bracket defined by

(3.13) $\displaystyle(e_{ij}\otimes a)\cdot(e_{kl}\otimes
b)=(-1)^{(k+l)p(a)}\delta_{jk}e_{il}\otimes ab,$ $\displaystyle[e_{ij}\otimes
a,e_{kl}\otimes b]=(e_{ij}\otimes a)\cdot(e_{kl}\otimes
b)-(-1)^{(i+j+p(a))(k+l+p(b))}(e_{kl}\otimes b)\cdot(e_{ij}\otimes a).$

We can visualize $\mathcal{L}^{can}$ in its matrix form

(3.14) $\mathcal{L}^{can}={\begin{pmatrix}D&0&\dots&\dots&0&(-1)^{N}u_{N}\\\
-1&D&\dots&\dots&0&(-1)^{N-1}u_{N-1}\\\ \vdots&\vdots&\ddots&&\vdots&\vdots\\\
\vdots&\vdots&&\ddots&\vdots&\vdots\\\ 0&0&\dots&\dots&D&u_{2}\\\
0&0&\dots&\dots&-1&D-u_{1}\\\
\end{pmatrix}=\begin{pmatrix}D&0&\dots&\dots&0&s_{N}(b_{N}-Db_{N-1})\\\
-1&D&\dots&\dots&0&s_{N-1}(b_{N-1}-Db_{N-2})\\\
\vdots&\vdots&\ddots&&\vdots&\vdots\\\ \vdots&\vdots&&\ddots&\vdots&\vdots\\\
0&0&\dots&\dots&D&s_{2}(b_{2}-Db_{1})\\\ 0&0&\dots&\dots&-1&s_{1}b_{1}\\\
\end{pmatrix},}$

where $i=0,...,N$, $s_{i}=(-1)^{\frac{i(i-1)}{2}+\frac{N(N+1)}{2}}$ and

(3.15) $b_{i}(D)=(D^{i-N}L^{*}(D))_{+}\in\mathcal{W}_{N}[D].$

In particular, we have $b_{0}=(-1)^{\frac{N(N+1)}{2}}$ and hence
$s_{1}b_{1}=D-s_{1}(b_{1}-Db_{0})$ and $b_{N}=L^{*}(D).$

We associate to any $a\in\mathcal{W}_{N}$ the $N\times N$ matrix pseudo-
differential operator $\nabla_{a}$ whose $ij$-th entry for $i,j=1,2,\cdots,N$
is

(3.16)
$(\nabla_{a})_{ij}=(-1)^{(j+1)p(a)+(j+1)(i+1)+\frac{j(j-1)}{2}+\frac{i(i-1)}{2}}b_{N-i}(\frac{\delta
a}{\delta L})^{*}D^{j-1}.$

Hence the $4\times 4$ left-top corner of $\nabla_{a}$ is

(3.17) $\begin{pmatrix}b_{N-1}(\frac{\delta a}{\delta
L})^{*}&(-1)^{p(a)+1}b_{N-1}(\frac{\delta a}{\delta
L})^{*}D&-b_{N-1}(\frac{\delta a}{\delta
L})^{*}D^{2}&(-1)^{p(a)}b_{N-1}(\frac{\delta a}{\delta L})^{*}D^{3}\\\
-b_{N-2}(\frac{\delta a}{\delta L})^{*}&(-1)^{p(a)+1}b_{N-2}(\frac{\delta
a}{\delta L})^{*}D&b_{N-2}(\frac{\delta a}{\delta
L})^{*}D^{2}&(-1)^{p(a)}b_{N-2}(\frac{\delta a}{\delta L})^{*}D^{3}\\\
-b_{N-3}(\frac{\delta a}{\delta L})^{*}&(-1)^{p(a)}b_{N-3}(\frac{\delta
a}{\delta L})^{*}D&b_{N-3}(\frac{\delta a}{\delta
L})^{*}D^{2}&(-1)^{p(a)+1}b_{N-3}(\frac{\delta a}{\delta L})^{*}D^{3}\\\
b_{N-4}(\frac{\delta a}{\delta L})^{*}&(-1)^{p(a)}b_{N-4}(\frac{\delta
a}{\delta L})^{*}D&-b_{N-4}(\frac{\delta a}{\delta
L})^{*}D^{2}&(-1)^{p(a)+1}b_{N-4}(\frac{\delta a}{\delta
L})^{*}D^{3}\end{pmatrix}.$

######  Lemma 3.3.

Let $a\in\mathcal{W}_{N}$. For $\mathcal{L}^{can}$ in (3.12), we have the
following equalities in
$\mathfrak{gl}(m|n)\otimes\mathcal{W}_{N}(\\!(D^{-1})\\!):$

(3.18)
$\mathcal{L}^{can}\nabla_{a}=\sum_{i=1}^{N}e_{1i}\otimes(-1)^{\frac{i(i-1)}{2}+(p(a)+1)(i+1)}L^{*}(\frac{\delta
a}{\delta L})^{*}$

and

(3.19)
$\nabla_{a}\mathcal{L}^{can}=\left\\{\begin{array}[]{ll}\displaystyle\sum_{i=1}^{N}e_{iN}\otimes(-1)^{p(a)+1+\frac{i(i+1)}{2}}b_{i}(\frac{\delta
a}{\delta L})^{*}L^{*}&\text{ if }N\text{ is even , }\\\
\displaystyle\sum_{i=1}^{N}e_{iN}\otimes(-1)^{\frac{i(i-1)}{2}+1}b_{i}(\frac{\delta
a}{\delta L})^{*}L^{*}&\text{ if }N\text{ is odd . }\end{array}\right.$

###### Proof.

It directly follows from computations. ∎

Simply speaking, (3.18) implies that the only nonzero row of
$\mathcal{L}^{can}\nabla_{a}$ is the first row and the first $4$ entries in
that row are

(3.20) $\begin{pmatrix}L^{*}(\frac{\delta a}{\delta
L})^{*}&(-1)^{p(a)}L^{*}(\frac{\delta a}{\delta L})^{*}D&-L^{*}(\frac{\delta
a}{\delta L})^{*}D^{2}&(-1)^{p(a)+1}L^{*}(\frac{\delta a}{\delta
L})^{*}D^{3}\end{pmatrix}.$

In a similar way, (3.19) says that the only nontrivial column of
$\nabla_{a}\mathcal{L}^{can}$ is the last column and its first 4 entries are
as follows:

(3.21) $(-1)^{p(a)}\begin{pmatrix}b_{1}(\frac{\delta a}{\delta L})^{*}L^{*}\\\
b_{2}(\frac{\delta a}{\delta L})^{*}L^{*}\\\ -b_{3}(\frac{\delta a}{\delta
L})^{*}L^{*}\\\ -b_{4}(\frac{\delta a}{\delta L})^{*}L^{*}\\\
\end{pmatrix}\text{ when $N$ is even},\quad\begin{pmatrix}-b_{1}(\frac{\delta
a}{\delta L})^{*}L^{*}\\\ b_{2}(\frac{\delta a}{\delta L})^{*}L^{*}\\\
b_{3}(\frac{\delta a}{\delta L})^{*}L^{*}\\\ -b_{4}(\frac{\delta a}{\delta
L})^{*}L^{*}\\\ \end{pmatrix}\text{ when $N$ is odd}.$

######  Lemma 3.4.

Let $a,b$ be two elements in $\mathcal{W}_{N}$. We have

(3.22) $\int\text{ Res }((L\frac{\delta a}{\delta L})_{+}L\frac{\delta
b}{\delta L})=(-1)^{(p(a)+p(b))N}\int\sum_{i=1}^{N}\text{ Res }((\frac{\delta
a}{\delta L})^{*}D^{i-1})\text{ Res }(b_{N-i}(\frac{\delta a}{\delta
L})^{*}L^{*})\,.$

###### Proof.

For $j\in\mathbb{Z}$ and $v\in\mathcal{W}_{N}$, the pseudo-differential
operator $Z=D^{-j}v$ satisfies $\text{ Res
}(D^{i-1}Z)=\delta_{ij}(-1)^{p(v)}v$. Hence for any pseudo-differential
operator $Z$, we have

(3.23) $\sum_{i=1}^{N}(-1)^{i}(LD^{-i})_{+}\text{ Res
}(D^{i-1}Z)=(-1)^{p(Z)}(LZ_{-})_{+}.$

In particular, if we substitute $Z=Z_{-}=\frac{\delta a}{\delta L}$ then

(3.24) $\sum_{i=1}^{N}(-1)^{i}(LD^{-i})_{+}\text{ Res }(D^{i-1}\frac{\delta
a}{\delta L})=(-1)^{N+p(a)+1}(L\frac{\delta a}{\delta L})_{+}.$

After multiplying on the left both sides of (3.24) by $L\frac{\delta b}{\delta
L}$ and taking residues, we get

$\sum_{i=1}^{N}\text{ Res }(L\frac{\delta b}{\delta L}(LD^{-i})_{+})\text{ Res
}(D^{i-1}\frac{\delta a}{\delta L})=\text{ Res }(L\frac{\delta b}{\delta
L}(L\frac{\delta a}{\delta L})_{+})$

using (2) of Lemma 3.2. Finally, by (1) and (3) of Lemma 3.2, we have

$\int\sum_{i=1}^{N}\text{ Res }((\frac{\delta a}{\delta L})^{*}D^{i-1})\text{
Res }((D^{-i}L^{*})_{+}(\frac{\delta b}{\delta
L})^{*}L^{*})=(-1)^{(p(a)+p(b))N}\int\text{ Res }((L\frac{\delta a}{\delta
L})_{+}L\frac{\delta b}{\delta L})$

for any pseudo-differential operators $A$ and $B$, which is the same equality
as (3.22). Here we used $(D^{i-1})^{*}=(-1)^{\frac{(i-1)i}{2}}(D^{i-1})$,
$(D^{-i})^{*}=(-1)^{\frac{(-i)(-i+1)}{2}}D^{-i}$ and
$\frac{(i-1)i}{2}\equiv\frac{(-i)(-i+1)}{2}$ (mod 2). ∎

######  Lemma 3.5.

Let $a,b$ be two elements in $\mathcal{W}_{N}$. We have

(3.25) $\int\text{ Res }(L(\frac{\delta a}{\delta L}L)_{+}\frac{\delta
b}{\delta L})=(-1)^{p(b)+N+1}\int\sum_{i=1}^{N}(-1)^{i(p(a)+p(b))}\text{ Res
}(b_{N-i}(\frac{\delta a}{\delta L})^{*})\text{ Res }(L^{*}(\frac{\delta
b}{\delta L})^{*}D^{i-1}).$

###### Proof.

For pseudo-differential operators $A$ and $B$, we know $\text{ Res
}(A_{+}B_{+})=\text{ Res }(A_{-}B_{-})=0$. Hence

(3.26) $\text{ Res }(AB_{+})=\text{ Res }(A_{-}B_{+})=\text{ Res }(A_{-}B).$

Also, recall that the differential part of $\frac{\delta a}{\delta L}$ is
trivial. Hence

(3.27) $\begin{split}\int\text{ Res }(L(\frac{\delta a}{\delta
L}L)_{+}\frac{\delta b}{\delta L})&=\int(-1)^{(p(a)+N+1)(p(b)+N+1)}\text{ Res
}(\frac{\delta b}{\delta L}L(\frac{\delta a}{\delta L}L)_{+})\\\
&=\int(-1)^{N(p(a)+p(b)+1)}\text{ Res }(\frac{\delta a}{\delta
L}L(\frac{\delta b}{\delta L}L)_{-})\\\ &=\int(-1)^{N(p(a)+p(b)+1)}\text{ Res
}(\frac{\delta a}{\delta L}(L(\frac{\delta b}{\delta
L}L)_{-})_{+}).\end{split}$

Applying (3.23) in the preceding proof for $Z=\frac{\delta b}{\delta L}L$, we
have

(3.28) $(-1)^{p(b)+1}(L(\frac{\delta b}{\delta
L}L)_{-})_{+}=\sum_{i=1}^{N}(-1)^{i}(LD^{-i})_{+}\text{ Res
}(D^{i-1}\frac{\delta b}{\delta L}L).$

Multiplying on the left both sides of (3.28) by $\frac{\delta a}{\delta L}$,
taking the residue and applying (3.27), one gets

$\begin{split}\int\text{ Res }(L(\frac{\delta a}{\delta L}L)_{+}\frac{\delta
b}{\delta L})&=\sum_{i=1}^{N}(-1)^{N(p(a)+p(b)+1)+i+p(b)+1}\int\text{ Res
}\big{(}\frac{\delta a}{\delta L}(LD^{-i})_{+}\text{ Res }(D^{i-1}\frac{\delta
b}{\delta L}L)\big{)}\\\ &=\sum_{i=1}^{N}(-1)^{N(p(a)+p(b)+1)}\int\text{ Res
}(\frac{\delta a}{\delta L}(LD^{-i})_{+})\text{ Res }(D^{i-1}\frac{\delta
b}{\delta L}L).\end{split}$

Here, for the last equality, we used (2) of Lemma 3.2. Finally, the result
follows from Lemma 3.2 (1) and the fact that $(A_{+})^{*}=(A^{*})_{+}$ for any
pseudo-differential operator $A$. ∎

######  Theorem 3.6.

Recall that the integers $m,n$ are uniquely defined by $N=m+n$ and $n\leq
m\leq n+1$. Let
$(\quad|\quad):\mathfrak{gl}(m|n)\otimes\mathcal{W}_{N}\times\mathfrak{gl}(m|n)\otimes\mathcal{W}_{N}\to\mathcal{W}_{N}$
be the bilinear map defined by

(3.29) $(e_{ij}\otimes a|e_{kl}\otimes
b)=(-1)^{p(a)(k+l)}(-1)^{i+1}\delta_{il}\delta_{jk}\,ab$

for $i,j,k,l\in\\{1,2,\cdots,N=m+n\\}$ and $a,b\in\mathcal{W}_{N}.$ Then the
Gelfand-Dickey bracket can be written in terms of the matrix operators
$\mathcal{L}^{can}$ and $\nabla_{a}$. More precisely, we have

(3.30) $\int(\text{ Res }(\nabla_{a})|[\mathcal{L}^{can},\text{ Res
}(\nabla_{b})])=(-1)^{N+p(a)}\int\text{ Res }((L\frac{\delta a}{\delta
L})_{+}L\frac{\delta b}{\delta L}-L(\frac{\delta a}{\delta
L}L)_{+}\frac{\delta b}{\delta L}),$

where the linear map $\text{ Res
}:\mathfrak{gl}(m|n)\otimes\mathcal{W}_{N}(\\!(D^{-1})\\!)\to\mathcal{W}_{N}$
is given by $\text{ Res }(e\otimes A):=e\otimes\text{ Res }(A).$

###### Proof.

Recall (3.16) and Lemma 3.3. By direct computations, we get

(3.31) $\begin{split}(\text{ Res }\nabla_{a}|\text{ Res
}(\nabla_{b}\mathcal{L}^{can}))&=(-1)^{(p(a)+p(b))(N+1)+N+1}\sum_{i=1}^{N}\text{
Res }((\frac{\delta a}{\delta L})^{*}D^{i-1})\text{ Res }(b_{N-i}(\frac{\delta
a}{\delta L})^{*}L^{*}),\\\ (\text{ Res }\nabla_{a}|\text{ Res
}(\mathcal{L}^{can}\nabla_{b}))&=(-1)^{p(a)+p(b)}\sum_{i=1}^{N}(-1)^{i(p(a)+p(b))}\text{
Res }(b_{N-i}(\frac{\delta a}{\delta L})^{*})\text{ Res }(L^{*}(\frac{\delta
b}{\delta L})^{*}D^{i-1}).\end{split}$

Hence by Lemma 3.4 and 3.5, we have

(3.32) $\begin{split}\int\text{ Res }((L\frac{\delta a}{\delta
L})_{+}L\frac{\delta b}{\delta L})&=(-1)^{p(a)+p(b)+N+1}\int(\text{ Res
}\nabla_{a}|\text{ Res }(\nabla_{b}\mathcal{L}^{can})),\\\ \int\text{ Res
}(L(\frac{\delta a}{\delta L}L)_{+}\frac{\delta b}{\delta
L})&=(-1)^{p(a)+N+1}\int(\text{ Res }\nabla_{a}|\text{ Res
}(\mathcal{L}^{can}\nabla_{b})).\end{split}$

Moreover, if we denote by $(\nabla_{b})_{-2}$ the coefficient of $D^{-2}$ in
$\nabla_{b}$, it is clear that

(3.33) $\begin{split}\text{ Res }(\nabla_{b}\mathcal{L}^{can})&=-\text{ Res
}(\nabla_{b})\mathcal{L}^{can}+\text{ Res }(\nabla_{b})D+(\nabla_{b})_{-2},\\\
\text{ Res }(\mathcal{L}^{can}\nabla_{b})&=\mathcal{L}^{can}\text{ Res
}(\nabla_{b})+(-1)^{p(b)}\text{ Res
}(\nabla_{b})D+(-1)^{p(b)}(\nabla_{b})_{-2}.\end{split}$

It follows from equations (3.32) and (3.33) that

(3.34) $\begin{split}\int&(\text{ Res }(\nabla_{a})|[\mathcal{L}^{can},\text{
Res }(\nabla_{b})])=\int(\text{ Res }(\nabla_{a})|\mathcal{L}^{can}\text{ Res
}(\nabla_{b})+(-1)^{p(b)}\text{ Res }(\nabla_{b})\mathcal{L}^{can})\\\ &\hskip
28.45274pt=\int(\text{ Res }(\nabla_{a})|\text{ Res
}(\mathcal{L}^{can}\nabla_{b})+(-1)^{p(b)+1}\text{ Res
}(\nabla_{b}\mathcal{L}^{can}))\\\ &\hskip 28.45274pt=(-1)^{p(b)+1}\int(\text{
Res }(\nabla_{a})|\text{ Res }(\nabla_{b}\mathcal{L}^{can}))+\int(\text{ Res
}(\nabla_{a})|\text{ Res }(\mathcal{L}^{can}\nabla_{b}))\\\ &\hskip
28.45274pt=(-1)^{N+p(a)}\int\text{ Res }((L\frac{\delta a}{\delta
L})_{+}L\frac{\delta b}{\delta L}-L(\frac{\delta a}{\delta
L}L)_{+}\frac{\delta b}{\delta L}).\end{split}$

∎

###  3.3. SUSY W-algebras and SUSY Gelfand-Dickey brackets

In this section, we consider the classical SUSY W-algebras introduced in
Examples 2.4 and 2.5. The Lie superalgebra $\mathfrak{g}$ is
$\mathfrak{gl}(m|n)$ for $m=n+1$ or $m=n$ and is spanned by the elements
$e_{ij}$, $i,j\in\\{1,2,\cdots,N=m+n\\}$ whose parity is $i+j$ (mod $2$). The
odd element $f=\sum_{i=1}^{N-1}e_{i+1,i}$ and the subspace
$V=\bigoplus_{i=1}^{N}e_{iN}$ satisfy the decomposition

(3.35) $\mathfrak{b}_{+}=[\mathfrak{n},f]\oplus V,$

where $\mathfrak{n}=\bigoplus_{i<j}e_{ij}$ and
$\mathfrak{b}_{+}=\bigoplus_{i\leq j}e_{ij}$. Recall that for any choice of
dual bases $q:=\\{(q^{i},q_{i})|i\in S\\}$ of $\mathfrak{b}_{+}$ and
$\mathfrak{b}$ we associate the universal Lax operator

$\mathcal{L}^{u}=D+\sum_{i\in S}q^{i}\otimes\bar{q_{i}}+f\otimes 1\,\,.\,$

Proposition 2.6 states that there exists a unique operator $\mathcal{L}^{c}$
gauge equivalent to $\mathcal{L}^{u}$ of the form

(3.36) $\mathcal{L}^{c}=D+{\begin{pmatrix}0&0&\dots&\dots&0&w_{N}\\\
-1&0&\dots&\dots&0&w_{N-1}\\\ \vdots&\vdots&\ddots&&\vdots&\vdots\\\
\vdots&\vdots&&\ddots&\vdots&\vdots\\\ 0&0&\dots&\dots&0&w_{2}\\\
0&0&\dots&\dots&-1&w_{1}\\\ \end{pmatrix}}$

Moreover, the set $\\{w_{1},w_{2},\cdots,w_{N}\\}$ generates
$\mathcal{W}(\bar{\mathfrak{g}},f)$. We construct a differential algebra
isomorphism from $\mathcal{W}_{N}$ to $\mathcal{W}(\bar{\mathfrak{g}},f)$ by
letting

(3.37) $\phi(u_{i}):=(-1)^{i}w_{i}\ ,\ i=1,2,\cdots,N.$

Note that under this isomorpshim, we can identify $\mathcal{L}^{c}$ to the
matrix $\mathcal{L}^{can}$ in (3.14). We will denote them both by
$\mathcal{L}^{c}$ and omit $\phi$.

Recall that in Proposition 2.7, we used the fact that every element in
$a\in\mathcal{V}(\bar{\mathfrak{b}})$ can be considered as a function from the
odd space
$\mathcal{F}:=(\mathfrak{b}_{+}\otimes\mathcal{V}(\bar{\mathfrak{b}}))_{\bar{1}}$
to $\mathcal{V}(\bar{\mathfrak{b}})$. Throughout this section, we let

(3.38) $Q:=\sum_{i\in S}q^{i}\otimes\bar{q}_{i}.$

In addition, for $a\in\mathcal{V}(\bar{\mathfrak{b}})$ and a pair
$q:=\\{(q^{i},q_{i})|i\in S\\}$ of bases of $\mathfrak{b}_{+}$ and
$\mathfrak{b}$, the variational derivative $\frac{\delta a}{\delta q}$ is
defined by (2.21). The variational derivative can be interpreted as a
differential algebra analogue of the gradient in the following sense.

######  Lemma 3.7.

Let $a\in\mathcal{V}(\bar{\mathfrak{b}})$, $h\in\mathcal{F}$. For a constant
even variable $\epsilon$, we have

(3.39)
$\int\Big{(}h\,|\,\frac{\delta{a}}{\delta{q}}\Big{)}=\int\frac{d}{d\epsilon}a(Q+\epsilon
h)|_{\epsilon=0}.$

###### Proof.

Let us denote $h:=\sum_{i\in S}q^{i}\otimes h_{i}$. Since $h$ is an odd
element, the parity $p(h_{i})=1-\tilde{\imath}$ for
$\tilde{\imath}:=p(q^{i}).$ By (2.20), we have $a(Q)=a$ and hence

(3.40) $\int\frac{d}{d\epsilon}a(Q+\epsilon h)|_{\epsilon=0}=\sum_{i\in
S,\,n\in\mathbb{Z}_{+}}\int h_{i}^{(n)}\frac{\partial
a}{\partial\bar{q}_{i}^{(n)}}=\sum_{i\in
S,\,n\in\mathbb{Z}_{+}}\int(-1)^{n(\tilde{\imath}+1)+\frac{n(n+1)}{2}}h_{i}\Big{(}\frac{\partial
a}{\partial\bar{q}_{i}^{(n)}}\Big{)},$

for $\tilde{\imath}=p(q_{i}).$ On the other hand, we have

(3.41) $\Big{(}h\,|\,\frac{\delta a}{\delta
q}\Big{)}=(-1)^{n(\tilde{\imath}+1)+\frac{n(n+1)}{2}}h_{i}\bigg{(}\frac{\partial
a}{\partial{\bar{q}}_{i}^{(n)}}\bigg{)}.$

This proves the lemma. ∎

######  Lemma 3.8.

Let $a$ be an element of $\mathcal{W}_{N}$ and $Q$, $h$ and $\epsilon$ be the
same as in Lemma 3.7. If we consider the function
$u[h](\epsilon):=u(Q+\epsilon h)$ where $u=(u_{i})_{i=1}^{N}$ is the sequence
of elements in (3.37), then we have

(3.42) $\int\Big{(}h|\frac{\delta a}{\delta
q}\Big{)}=\int\sum_{i=1}^{N}u_{i}[h]^{\prime}(0)\frac{\delta a}{\delta
u_{i}}.$

###### Proof.

By Lemma 3.7, we get

(3.43) $\begin{split}\int\Big{(}h|\frac{\delta a}{\delta
q}\Big{)}&=\int\frac{d}{d\epsilon}a(Q+\epsilon
h)|_{\epsilon=0}=\int\frac{d}{d\epsilon}a(u(Q+\epsilon h))|_{\epsilon=0}\\\
&=\int\frac{d}{d\epsilon}a(u+\epsilon
u[h]^{\prime}(0)+O(\epsilon^{2}))|_{\epsilon=0}=\int\sum_{i=1}^{N}{u_{i}[h]^{\prime}(0)\frac{\delta
a}{\delta u_{i}}}.\end{split}$

Here in $a(Q+\epsilon h)$, $a$ is regarded as a function on $\mathcal{F}$. On
the other hand, since $\mathcal{W}_{N}$ is an algebra of differential
polynomials generated by $N$ elements $u_{1},\cdots,u_{N}$, when we write
$w(u(Q+\epsilon h))$, $a$ is regarded as a differential polynomial on $N$
variables. ∎

For a pseudo-differential operator
$M=\sum_{i\in\mathbb{Z}}M_{i}D^{i}\in\mathcal{V}(\bar{\mathfrak{b}}_{-})(\\!(D^{-1})\\!)$,
we denote

(3.44) $M(h):=M_{i}(h)D^{i}$

for $h\in\mathcal{F}.$ Then the following lemma holds.

######  Lemma 3.9.

Let $a$ be in $\mathcal{W}_{N}$. Then

(3.45) $\text{ Res }\,(\frac{\delta a}{\delta L})^{*}\,\,L^{*}(Q+\epsilon
h)=(-1)^{p(a)N}\sum_{i=1}^{N}u_{i}(Q+\epsilon h)\frac{\delta a}{\delta
u_{i}}.$

###### Proof.

This result follows from the fact that $\text{ Res }\,(\frac{\delta a}{\delta
L})^{*}\,L^{*}=(-1)^{N(p(a)+N+1)}\text{ Res }\,L\,\frac{\delta a}{\delta L}$
and the definition of $\frac{\delta a}{\delta L}$. ∎

For a given $h\in\mathcal{F}$, we consider the following deformation of
$\nabla_{a}$:

(3.46)
$(\nabla_{a}[h](\epsilon))_{ij}=(-1)^{(j+1)p(a)+(j+1)(i+1)+\frac{j(j-1)}{2}+\frac{i(i-1)}{2}}b_{N-i}(Q)(\frac{\delta
a}{\delta L})^{*}(Q+\epsilon h),$

where $Q=\sum_{i\in S}q^{i}\otimes\bar{q}_{i}\in\mathcal{F}.$

######  Lemma 3.10.

Let $a$ be in $\mathcal{W}_{N}$. Then

(3.47) $\text{ Res }\,(\nabla_{a}[h](\epsilon)|\mathcal{L}^{c}(Q+\epsilon
h))=(-1)^{p(a)}\sum_{i=1}^{N}u_{i}(Q+\epsilon h)\frac{\delta a}{\delta
u_{i}}.$

###### Proof.

It follows from the fact that for all $w$ in $\mathcal{W}_{N}$ we have

(3.48) $(\nabla_{a}[h](\epsilon)|\mathcal{L}^{c}(Q+\epsilon
h))=(-1)^{p(a)(N+1)}(\frac{\delta a}{\delta L})^{*}(Q)L^{*}(Q+\epsilon h).$

∎

Let $\mathcal{M}$ be a matrix superdifferential operator entries
$\mathcal{M}_{ij}\in\mathcal{V}(\bar{\mathfrak{b}})(\\!(D^{-1})\\!)$ for
$i,j=1,\cdots,N=m+n$. For $h\in\mathcal{F}$, we denote by $\mathcal{M}(h)$ the
matrix differential operator with entries $\mathcal{M}_{ij}(h)$. Here
$\mathcal{M}_{ij}(h)$ is defined in (3.44). Recall that
$\mathfrak{n}\subset\mathfrak{gl}(m|n)$ is the set of positive degree elements
and we have defined
$N_{c}\in\mathfrak{n}\otimes\mathcal{V}(\bar{\mathfrak{b}})$ in Proposition
2.6. $N_{c}$ is fixing the gauge as follows $e^{ad\,N_{c}}(D+Q-f\otimes
1)=\mathcal{L}^{c}$.

######  Lemma 3.11.

For all $a\in\mathcal{W}_{N},$

(3.49) $e^{-ad\,N_{c}}\text{ Res }\nabla_{a}-\frac{\delta a}{\delta
q}\in\mathfrak{n}\otimes\mathcal{V}(\bar{\mathfrak{b}}).$

###### Proof.

In order to prove the lemma, it is enough to show that

(3.50) $\int(e^{-ad\,N_{c}}\text{ Res }\nabla_{a}|h)=\int\Big{(}\frac{\delta
a}{\delta q}\,|\,h\Big{)}$

for any $h\in\mathcal{F}.$ Using Lemma 3.7 and 3.10, the RHS of (3.50) can be
written as follows:

(3.51) $\begin{split}\int\Big{(}\frac{\delta a}{\delta
q}\,|\,h\Big{)}&=(-1)^{p(a)+1}\int\Big{(}h|\frac{\delta a}{\delta q}\Big{)}\\\
&=(-1)^{p(a)+1}\int\sum_{i=1}^{N}{u_{i}^{\prime}(Q+\epsilon h)(0)\frac{\delta
f}{\delta u_{i}}}\\\ &=-\frac{d}{d\epsilon}\int\text{ Res
}\big{(}\nabla_{a}(Q+\epsilon h)|\mathcal{L}^{c}(Q+\epsilon
h)\big{)}|_{\epsilon=0}.\\\ \end{split}$

Now let us show the last term in (3.51) also equals to (3.50). There is a
$\mathfrak{n}$ valued function $M$ such that $N_{c}(Q+\epsilon
h)=N_{c}(Q)+\epsilon M(Q)+O(\epsilon^{2})$. For such $M$, we have :

(3.52) $\begin{split}&\mathcal{L}^{c}(Q+\epsilon h)=e^{adN_{c}(Q+\epsilon
h)}(D+Q+\epsilon h-f\otimes 1)=e^{adN_{c}+\epsilon
adM+O(\epsilon^{2})}(\mathcal{L}(Q)+\epsilon h)\\\
&=\mathcal{L}^{c}(Q)+\epsilon\,e^{adN_{c}}(h)+\epsilon\,[M+\frac{1}{2}[N_{c},M]+\frac{1}{6}[N_{c},[N_{c},M]]+\cdots\,,\mathcal{L}^{c}(Q)]+O(\epsilon^{2})\\\
&=\mathcal{L}^{c}(Q)+\epsilon\,e^{adN_{c}}(h)+\epsilon\,[\phi_{h},\mathcal{L}^{c}(Q)]+O(\epsilon^{2})\end{split}$

where $\phi_{h}=M+\frac{1}{2}[N_{c},M]+\frac{1}{6}[N_{c},[N_{c},M]]+\cdots.$
Note that since both $\mathcal{L}^{c}(Q)\nabla_{a}(Q)$ and
$\nabla_{a}(Q)\mathcal{L}^{c}(Q)$ are upper triangular,

(3.53) $\int\text{ Res }(\nabla_{a}(Q)|[\mathcal{M},\mathcal{L}^{c}(Q)])=0.$

when $\mathcal{M}$ is an $\mathfrak{n}$-valued function. Moreover, since the
last row of $\nabla_{a}$ does not vary with $\epsilon$ but only the last
column of $\mathcal{L}^{c}$ is nonzero, for all $\epsilon$, we see that

(3.54)
$(\nabla_{a}[h](\epsilon)\,|\,\mathcal{L}^{c}(Q))=str(\nabla_{a}(Q)\mathcal{L}^{c}(Q)),$

and hence

(3.55) $(\nabla_{a}[h](0)^{\prime}|\mathcal{L}^{c}(Q))=0.$

By (3.53) and (3.55), we have

(3.56) $\begin{split}\frac{d}{d\epsilon}&\int\text{ Res
}(\nabla_{a}(Q+\epsilon h)|\mathcal{L}^{c}(Q+\epsilon h))|_{\epsilon=0}\\\
&=\int\text{ Res }(\nabla_{a}[h](0)^{\prime}|\mathcal{L}^{c}(Q))+\int\text{
Res }\,(e^{-adN_{c}}\nabla_{a}(Q)|h)+\int\text{ Res
}(\nabla_{a}(Q)|[\phi_{h},\mathcal{L}^{c}(Q)])\\\ &=\int\text{ Res
}\,(e^{-adN_{c}}\nabla_{a}(Q)|h)=-\int\,(e^{-adN_{c}}\text{ Res
}\nabla_{a}|h).\end{split}$

Hence we proved that

(3.57) $\int\Big{(}\frac{\delta a}{\delta
q}\,|\,h\Big{)}=\smallint(e^{-adN_{c}}\text{ Res }(\nabla_{a}(Q))|h)$

for any $h\in\mathcal{F}$. ∎

######  Theorem 3.12.

Let $a,b\in\mathcal{W}_{N}$ and $\mathcal{L}^{u}:=D+Q-f\otimes 1$ for
$Q=\sum_{i\in I}q^{i}\otimes\bar{q}_{i}$. Then

(3.58) $\int\Big{(}\frac{\delta a}{\delta
q}\big{|}\big{[}\mathcal{L}^{u},\frac{\delta b}{\delta
q}\big{]}\Big{)}=(-1)^{N+p(a)}\int\text{ Res }((L\frac{\delta a}{\delta
L})_{+}L\frac{\delta b}{\delta L}-L(\frac{\delta a}{\delta
L}L)_{+}\frac{\delta b}{\delta L}).$

###### Proof.

Since we have Theorem 3.6, the following equality directly implies the
theorem:

(3.59) $\int\Big{(}\frac{\delta a}{\delta
q}\big{|}\big{[}\mathcal{L}^{u},\frac{\delta b}{\delta
q}\big{]}\Big{)}=\int(\text{ Res }(\nabla_{a})|[\mathcal{L}^{c},\text{ Res
}(\nabla_{b})])\,.$

Let us show (3.59). For $N_{c}$ in Lemma 3.11, we have

(3.60) $\int(\text{ Res }\nabla_{a}|[\mathcal{L}^{c},\text{ Res
}\nabla_{b}])=\int(e^{-ad\,N_{c}}\text{ Res
}\nabla_{a}|e^{-ad\,N_{c}}[\mathcal{L}^{c},\text{ Res }\nabla_{b}]).$

Since $[\mathcal{L}^{c},\text{ Res }\nabla_{b}]$ is an upper triangular matrix
by Lemma 3.3, we know that

(3.61) $\eqref{eq:2.3thorem_proof1}=\int\Big{(}\frac{\delta a}{\delta
q}\,\big{|}\,\big{[}e^{-ad\,N_{c}}\mathcal{L}^{c},e^{-ad\,N_{c}}\nabla_{b}\big{]}\Big{)}=\int\Big{(}\Big{[}\frac{\delta
a}{\delta q},\,\mathcal{L}^{u}\Big{]}\big{|}e^{-ad\,N_{c}}\nabla_{b}\Big{)}.$

By Lemma 3.11, there is a $\mathfrak{n}$-valued function $M$ such that
$e^{-ad\,N_{c}}\text{ Res }\nabla_{a}+M=\frac{\delta a}{\delta q}$. Hence

(3.62) $\Big{[}\frac{\delta a}{\delta
q},\mathcal{L}^{u}\Big{]}=[e^{-ad\,N_{c}}\text{ Res
}\nabla_{a}+M,\mathcal{L}^{u}]=e^{-ad\,N_{c}}[\text{ Res
}\nabla_{a},\mathcal{L}^{c}]+[M,\mathcal{L}^{u}]$

is also an upper triangular matrix. Therefore using Lemma 3.11 again, we get

(3.63) $\eqref{eq:2.3thorem_proof2}=\int\Big{(}\frac{\delta a}{\delta
q}\big{|}\big{[}\mathcal{L}^{u},\frac{\delta b}{\delta q}\big{]}\Big{)}.$

and conclude (3.59) holds. ∎

######  Lemma 3.13.

The $\chi$-bracket between the generators $u_{i}$’s of $\mathcal{W}_{N}$ can
be equivalently defined by the relation

(3.64)
$\begin{split}\\{L(z)\,{}_{\chi}\,L(w)\\}&=\langle\big{(}L(D+\chi+w)(D+\chi+w-z)(D^{2}-\chi^{2}+w^{2}-z^{2})^{-1}\big{)}_{+}L(D+w)\\\
&-L(D+\chi+w)\big{(}(D+\chi+w-z)(D^{2}-\chi^{2}+w^{2}-z^{2})^{-1}L(D+w)\big{)}_{+}\rangle\end{split}$

where for any pseudo-differential operator $A=\sum_{n\leq M}a_{n}D^{n}$, we
denote $a_{0}$ by $\langle\,A\,\rangle.$ Moreover, (3.64) determines the
entire $\chi$-bracket on $\mathcal{W}_{N}$ by (2.7).

###### Proof.

By equation (2.11) we have for all $a\in\mathcal{W}_{N}$ and all $j=1,...,N$

(3.65) $\\{\smallint a,\,\smallint
u_{j}\\}=\sum_{i=1}^{N}(-1)^{(a+i)j}\int\\{u_{i}{}_{D}u_{j}\\}\frac{\delta
a}{\delta u_{i}}\,.$

This implies that for any $a\in\mathcal{W}_{N}$,

$(-1)^{ja}\\{u_{i}{}_{D}u_{j}\\}(a)=(-1)^{i+N}\Big{(}[LD^{i-N-1}a]_{+}L-L[D^{i-N-1}aL]_{+}\Big{)}_{[N-j]}$

where the subscript $[N-j]$ means the coefficient of $D^{N-j}$. For any
$a,b\in\mathcal{W}_{N}$, we have
$\\{a\,{}_{\chi}\,b\\}=\\{a\,{}_{\chi+D}\,b\\}(1)$. Hence

$(-1)^{i+N}\\{u_{i}\,{}_{\chi}\,u_{j}\\}=\Big{(}[L(\chi+D)(\chi+D)^{i-N-1}]_{+}L(D)-L(\chi+D)[(\chi+D)^{i-N-1}L(D)]_{+}\Big{)}_{[N-j]}.$

Consider two independent odd indeterminate constants $z$ and $w$ and extend
the domain of $\chi$-bracket to $\mathcal{W}_{N}[z,w]$ by the Leibniz rule.
Thanks to the identities

$(D+\chi-z)^{-1}=\sum_{i=0}^{\infty}(-1)^{\frac{i(i+1)}{2}}z^{i}(D+\chi)^{-i-1}$

and

$\\{az^{m}{}_{\chi}bw^{n}\\}:=(-1)^{(p(a)+1)m}z^{m}\\{a{}_{\chi}b\\}w^{n}$

it follows that

$\displaystyle\sum_{i,j=1}^{N}(-1)^{\frac{(N-i)(N-i+3)}{2}}\\{u_{i}z^{N-i}\,{}_{\chi}\,u_{j}w^{N-j}\\}$
$\displaystyle=\sum_{i=1}^{N}(-1)^{\frac{(N-i)(N-i+1)}{2}}\Big{(}(L(D+\chi)z^{N-i}(D+\chi)^{i-N-1})_{+}L(D)-L(D+\chi)(z^{N-i}(D+\chi)^{i-N-1}L(D))_{+}\Big{)}(w)$
$\displaystyle=\Big{(}\big{(}L(D+\chi)(D+\chi-z)^{-1}\big{)}_{+}L(D)-L(D+\chi)\big{(}(D+\chi-z)^{-1}L(D)\big{)}_{+}\Big{)}(w).$

The Lemma follows from the identities

$L(z)=\frac{1-\mathbf{i}}{2}\sum_{j=1}^{N}u_{j}(-1)^{(N-j)(N-j+3)/2}(\mathbf{i}z)^{N-j}+\frac{1+\mathbf{i}}{2}\sum_{j=1}^{N}u_{j}(-1)^{(N-j)(N-j+3)/2}(-\mathbf{i}z)^{N-j}\,$

and

$(D+\chi+\omega+\mathbf{i}z)^{2}=(D+\chi+\omega-\mathbf{i}z)^{2}\,$

where $\mathbf{i}$ is the imaginary number in $\mathbf{i}^{2}=-1.$

∎

######  Definition 3.14.

We denote the odd SUSY PVA bracket (3.64) on $\mathcal{W}_{N}$ by
$\\{\,{}_{\chi}\,\\}^{q}$ since it induces the quadratic SUSY Gelfand-Dickey
bracket on the space of functionals $\smallint\mathcal{W}_{N}$.

###  3.4. Integrable hierarchy on $\mathcal{W}_{2n}$

In this section we consider the case where the super-differential operator

$L=D^{2n}+u_{1}D^{2n-1}+...+u_{2n}$

has even degree $2n$ for $n>2$. The quadratic bracket (3.64) can be deformed
after replacing $L$ by $L+\epsilon$ where $\epsilon$ is a even constant.
Explicitly, the following formula defines an odd SUSY PVA bracket on
$\mathcal{W}_{2n}$ compatible with the quadratic one, which we will call the
linear odd SUSY Gelfand-Dickey bracket

(3.66)
$\begin{split}\\{L(z)\,{}_{\chi}\,L(w)\\}^{o}&=\langle\,L(D+\chi+w)(D+\chi+w-z)(D^{2}-\chi^{2}+w^{2}-z^{2})^{-1}\,\rangle\\\
&-\langle\,(D+\chi+w-z)(D^{2}-\chi^{2}+w^{2}-z^{2})^{-1}L(D+w)\,\rangle.\end{split}$

Moreover, Theorem 3.12 implies that its reduced form to the quotient
$\smallint\mathcal{W}_{2n}$ is

(3.67) $\smallint\\{\,a\,{}_{\chi}\,b\,\\}^{o}|_{\chi=0}=(-1)^{p(a)}\int\text{
Res }\,(L\frac{\delta a}{\delta L}-\frac{\delta a}{\delta L}L)\frac{\delta
b}{\delta L}.$

Recall that given an odd SUSY PVA bracket on a differential algebra
$\mathcal{P}$ and a functional $\smallint a\in\smallint\mathcal{P}$, the map
$\\{\,\smallint a\,{}_{\chi}\,\cdot\,\\}|_{\chi=0}$ defines a derivation of
$\mathcal{P}$ of parity $p(a)+1$. This derivation is evolutionary, which means
that it supercommutes with $D$. Moreover, if $\mathcal{P}$ is an algebra of
differential polynomials generated by elements $\\{u_{i}\\}_{i\in I}$ and $d$
is an even or odd evolutionary derivation, we have for all $b\in\mathcal{V}$

$\smallint d(b)=\sum_{i\in I}\int\,d(u_{i})\frac{\delta b}{\delta u_{i}}\,.$

Hence we deduce from (3.67) and (3.58) that for all $\smallint
a\in\smallint\mathcal{W}_{2n}$

(3.68)
$\begin{split}&\\{\smallint\,a\,{}_{\chi}\,L\,\\}^{q}|_{\chi=0}=(-1)^{p(a)}((L\frac{\delta
a}{\delta L})_{+}L-L(\frac{\delta a}{\delta L}L)_{+})\,,\\\
&{\\{\smallint\,a\,{}_{\chi}\,L\,\\}^{o}}|_{\chi=0}=(-1)^{p(a)}[\,L\,,\,\frac{\delta
a}{\delta L}\,]_{+}\,,\end{split}$

where
$\\{\,\smallint\,a\,{}_{\chi}\,L\,\\}:=\sum_{i=1}^{2n}\\{\,\smallint\,a\,{}_{\chi}\,u_{i}\,\\}D^{2n-i}$
and $[\,A\,,\,B\,]$ denotes the commutator of two operators $A$ and $B$.

We now relate both the quadratic and odd linear Gelfand-Dickey brackets to a
hierarchy of evolutionary derivations on the differential algebra
$\mathcal{W}_{2n}$. There exists a unique $n$-th root of $L$ denoted by
$L^{1/n}$ with leading term $D^{2}$. Note that the $2n$-th root of $L$ does
not exist in $\mathcal{W}_{2n}((D^{-1}))$. Using $L^{1/n}$ we define the even
derivations $(d/dt_{k})_{k\geq 1}$ of $\mathcal{W}_{2n}$ in the classical way

(3.69)
$\frac{dL}{dt_{k}}=[\,L^{k/n}_{+}\,,\,L\,]\,,\,\,\,\,\,\,\,\,\,[\,d/dt_{k}\,,\,D\,]=0\,.$

Due to the fact that $L$ is even, the nonSUSY picture translates directly to
our setting. Even though these arguments are very standard [Dic91], we rewrite
them here for the unfamiliar reader.

######  Lemma 3.15.

Let $k$ and $l$ be positive integers.

  1. $(1)$

The derivations $d/dt_{k}$ and $d/dt_{l}$ commute,

  2. $(2)$

The residue of $L^{l/n}$ is conserved for $d/dt_{k}$, i.e.
$\int\frac{d}{dt_{k}}\,\textit{ Res }\,L^{l/n}=0.$

###### Proof.

Let $X=\frac{d}{dt_{k}}(L^{1/n})-[\,L^{k/n}_{+}\,,\,L^{1/n}\,]$. We have

$\sum_{j=0}^{n-1}L^{j/n}XL^{(n-j-1)/n}=\frac{dL}{dt_{k}}-[\,L^{k/n}_{+}\,,\,L\,]=0\,,$

hence $X=0$ by considering the leading term of the LHS. It follows by the
Leibniz rule that

$\frac{d}{dt_{k}}(L^{l/n})=[\,L^{k/n}_{+}\,,\,L^{l/n}\,]\,\,\,\text{ for all
}\,l\geq 1\,.$

From that equation and the fact that the residue of a commutator is always a
total derivative, we get $(2)$. As for the first statement,

(3.70)
$\begin{split}\frac{d^{2}L}{dt_{k}dt_{l}}-\frac{d^{2}L}{dt_{l}dt_{k}}&=[\,[\,L^{k/n}_{+}\,,\,L^{l/n}\,]_{+},\,L\,]+[\,L^{l/n}_{+}\,[\,L^{k/n}_{+}\,,\,L\,]\,]\\\
&-[\,[\,L^{l/n}_{+}\,,\,L^{k/n}\,]_{+},\,L\,]-[\,L^{k/n}_{+}\,[\,L^{l/n}_{+}\,,\,L\,]\,]\\\
&=[\,[\,L^{k/n}_{+}\,,\,L^{l/n}\,]_{+}-[\,L^{k/n}_{+}\,,\,L^{l/n}_{+}\,]\,-[\,L^{l/n}_{+}\,,\,L^{k/n}\,]_{+}\,,\,L\,]\\\
&=[\,[\,L^{k/n}_{+}\,,\,L^{l/n}_{-}\,]_{+}+[\,L^{k/n}_{+}\,,\,L^{l/n}_{+}\,]_{+}\,+[\,L^{k/n}_{-}\,,\,L^{l/n}_{+}\,]_{+}\,,\,L\,]\\\
&=[\,[\,L^{k/n}\,,\,L^{l/n}\,]_{+}-[\,L^{k/n}_{-}\,,\,L^{l/n}_{-}\,]_{+}\,,\,L\,]=0\,.\end{split}$

∎

Now we compute the variational derivatives of the residues of fractional
powers of $L$. For any integer $k\geq 1$ we define the odd functionals

(3.71) $\int\,h_{k}=-\frac{n}{k}\,\int\,\text{res}\,L^{k/n}\,.$

######  Lemma 3.16.

For any integer $k\geq 1$, we have

(3.72) $\frac{\delta h_{k}}{\delta L}=-(L^{k/n-1})_{-}\,\,.$

###### Proof.

Recall that for any $\smallint a\in\smallint\mathcal{W}_{2n}$, the variational
derivative $\frac{\delta a}{\delta L}$ is uniquely defined by the universal
property

(3.73) $\int\,a(u+\epsilon h)=\int\,a(u)\,+\epsilon\int\,\text{ Res
}(\sum_{i=1}^{2n}h_{i}D^{2n-i})\frac{\delta a}{\delta
L}+\text{o}(\epsilon)\,\,,$

where $(h_{i})_{i=1,...,2n}\in(\mathcal{W}_{2n})^{2n}$ is any $2n$-tuple
satisfying $p(h_{i})=p(u_{i})$ and $\epsilon$ is a even constant parameter. In
the formula above, differential polynomials are interpretated as functions of
the generators $u$. Let us fix such a $2n$-tuple $(h_{i})_{i=1,...,2n}$ and
write $\delta L=\sum_{i=1}^{2n}h_{i}D^{2n-i}$. We denote $L^{k/n}(u)$ by $X$.
There exists a unique pseudo-differential operator $\delta X$ such that

$L^{k/m}(u+\epsilon h)=X+\epsilon\delta X+o(\epsilon)\,.$

Taking the differential of both sides in the equality $(L^{k/n})^{n}=L^{k}$ we
get

$\delta XX^{n-1}+...+X^{n-1}\delta X=\delta LL^{k-1}+...+L^{k-1}\delta L,$

hence

$\delta X+...+X^{n-1}\delta XX^{1-n}=\delta LL^{k-1}X^{1-n}+...+L^{k-1}\delta
LX^{1-n}.$

Using Lemma 3.2 we deduce that

$n\int\text{ Res }\delta X=k\text{ Res }\int\delta LL^{k/n-1}$

which ends the proof since we have

$\int\text{ Res }L^{k/n}(u+\epsilon h)-\int\text{ Res
}L^{k/n}(u)\,=\epsilon\int\text{ Res }\delta X+o(\epsilon)$

and one can take $a(u)=\text{ Res }L^{k/n}(u)$ in equation (3.73).

∎

It follows directly from the previous Lemma and equation (3.68) that for all
$k\geq 1$

(3.74)
$\\{\smallint\,h_{k}\,{}_{\chi}\,L\,\\}^{q}|_{\chi=0}=[\,L^{k/n}_{+}\,,\,L\,]\,\
=\,{\\{\smallint\,h_{k+n}\,{}_{\chi}\,L\,\\}^{o}}|_{\chi=0}\,.$

Hence we have shown that

######  Theorem 3.17.

The hierarchy of evolutionary derivations (3.69) is biHamiltonian for the two
compatible odd SUSY PVA brackets $\\{\,\,{}_{\chi}\,\,\\}^{q}$ and
$\\{\,\,{}_{\chi}\,\,\\}^{o}$. Its hamiltonians are the functionals
$(\smallint\,h_{k})_{k\geq 1}$ which are conserved for each derivation in the
hierarchy. They are in involution for the induced odd linear and quadratic
SUSY Gelfand-Dickey brackets.

##  4\. Even SUSY PVAs and $\mathcal{W}_{2n+1}$

We now consider the case where our algebra of differential polynomials is
generated by an odd monic differential operator

$L(D)=D^{2n+1}+u_{1}D^{2n}+...+u_{2n+1}.$

There exists a unique $(2n+1)$-th monic root of $L$ which enables us to
construct the hierarchy of evolutionary even derivations of
$\mathcal{W}_{2n+1}$

(4.1)
$\frac{dL}{dt_{k}}=[(L^{\frac{2k}{2n+1}})_{+},L]\,,\,\,\,\,[\frac{d}{dt_{k}}\,,\,D]=0\,,\,\,\,k\geq
1.$

These derivations pairwise commute using the same argument as in section 3.4.
They have common conserved densities given by the residues of fractional
powers of $L$. However, those corresponding to even operators are trivial
since

(4.2) $\int\text{ Res }L^{\frac{2q}{2n+1}}=\int\text{ Res
}[\,L^{\frac{q}{2n+1}}\,,\,L^{\frac{q}{2n+1}}\,]=0\,.$

Hence, the non trivial conserved densities of the hierarchy are even
functionals. To find an underlying Hamiltonian structure one must adapt our
language since the usual (odd) SUSY PVA brackets map even functionals to odd
derivations.

###  4.1. Even SUSY brackets

Let $\mathcal{R}$ be a $\mathbb{C}[D]$-module. A bilinear map

(4.3) $[\ {}_{\chi}\
]:\mathcal{R}\times\mathcal{R}\to\mathbb{C}[\chi]\otimes\mathcal{R}$

is called an even $\chi$-bracket if it satisfies

  1. (even parity) $p([a{}_{\chi}b])=p(a)+p(b),$

  2. (sesquilinearity) $[Da{}_{\chi}b]=-\chi[a{}_{\chi}b],$ $[a{}_{\chi}Db]=(-1)^{p(a)}(D+\chi)[a{}_{\chi}b]$

for $a,b\in\mathcal{R}.$

######  Definition 4.1.

A $\mathbb{C}[D]$-module $\mathcal{R}$ is called an even SUSY LCA if it is
endowed with an even $\chi$-bracket satisfying the following properties:

  1. (skewsymmetry) $[b{}_{\chi}a]=(-1)^{p(a)p(b)+1}[a{}_{-D-\chi}b]_{\leftarrow},$

  2. (Jacobi identity) $[a{}_{\chi}[b{}_{\gamma}c]]-(-1)^{p(a)p(b)}[b{}_{\gamma}[a{}_{\chi}c]]=[[a{}_{\chi}b]_{\chi+\gamma}c]$.

In the Jacobi identity, we assume $\chi$ and $\gamma$ commute with the
brackets. For example, $[\chi^{n}A{}_{\gamma}B]:=\chi^{n}[A{}_{\gamma}B]$.

For an even SUSY LCA $\mathcal{R}$, the superspace
$\int\mathcal{R}:=\mathcal{R}/\partial\mathcal{R}$ is a Lie superalgebra for
the induced bracket

(4.4) $[\smallint a\,\,\smallint b]:=\smallint[a{}_{\chi}b]|_{\chi=0}\,.$

######  Definition 4.2.

Let $\mathcal{P}$ be a unital SUSY differential algebra with an odd derivation
$D$. An even SUSY LCA $(\mathcal{P},D,\\{\ {}_{\chi}\ \\})$ is called an even
SUSY PVA if it satisfies the (right) Leibniz rule:

(4.5)
$\\{a{}_{\chi}bc\\}=\\{a{}_{\chi}b\\}c+(-1)^{p(a)p(b)}b\\{a{}_{\chi}c\\}\text{
for }a,b,c\in\mathcal{P}.$

######  Example 4.3.

If $\mathcal{P}$ is the algebra of polynomials generated by an odd variable
$u$ and an odd derivation $D$, then for any constant polynomial
$Q\in\mathbb{C}[\chi]$ the following defines a even SUSY PVA structure on
$\mathcal{P}$

$\\{u\,{}_{\chi}\,u\\}=Q(\chi^{4})\ \,.$

This follows from Theorem $4.12$. We were not able to find another example on
this specific differential algebra.

Let us define the following super-differential operator

(4.6)
$\\{a\,{}_{D}\,b\\}:=\sum_{m\in\mathbb{Z}_{+}}(-1)^{m(p(a)+p(b))+\frac{m(m+1)}{2}}a_{[m]}b\,D^{m}$

where $a_{[m]}b$ is the $m$-th coefficient in the $\chi$-bracket
$\\{a{}_{\chi}b\\}:=\sum_{m\in\mathbb{Z}_{+}}\chi^{m}a_{[m]}b$. Conversely,
one retrieves the even $\chi$-bracket from this super-differential operator
via the formula

(4.7) $\\{a{}_{\chi}b\\}=\\{a{}_{\chi+D}b\\}(1).$

######  Lemma 4.4.

Let $a,b,c$ be elements in an even SUSY PVA $\mathcal{P}$. Then

(4.8)
$\\{a{}_{\chi+D}b\\}(c)=(-1)^{p(a)p(b)+1}\\{b{}_{-\chi-D}a\\}_{\leftarrow}\,c.$

###### Proof.

The statement can be written as an identity between super-differential
operators

$(-1)^{p(a)p(b)+1}\sum_{n\in\mathbb{Z}_{+}}(-\chi-D)^{m}b_{[m]}a=\sum_{m\in\mathbb{Z}_{+}}(-1)^{m(a+b)+\frac{m(m+1)}{2}}a_{[m]}b(\chi+D)^{m}$

which is equivalent to

$(-1)^{p(a)p(b)+1}\sum_{m\in\mathbb{Z}_{+}}(-D)^{m}b_{[m]}a=\sum_{m\in\mathbb{Z}_{+}}(-1)^{m(a+b)+\frac{m(m+1)}{2}}a_{[m]}b\,\,D^{m}\,.$

Taking the adjoints of both sides yields

(4.9)
$(-1)^{p(a)p(b)+1}\sum_{m\in\mathbb{Z}_{+}}(-1)^{m(a+b)+\frac{m(m+1)}{2}}b_{[m]}a\,\,D^{n}=\sum_{m\in\mathbb{Z}_{+}}(-D)^{m}a_{[m]}b\,.$

Note that two differential operators are equal if and only if their symbols
are equal. Hence in order to show (4.9) it is enough to see

$\begin{split}\sum_{m\in\mathbb{Z}_{+}}(-D-\chi)^{m}(a_{[m]}b)&=(-1)^{p(a)p(b)+1}\sum_{m\in\mathbb{Z}_{+}}(-1)^{m(a+b)+\frac{m(m+1)}{2}}b_{[m]}a\,\,(D+\chi)^{m}(1)\,\\\
&=(-1)^{p(a)p(b)+1}\sum_{n\in\mathbb{Z}_{+}}(-1)^{m(a+b)+\frac{m(m+1)}{2}}b_{[m]}a\,\,(-1)^{m+\frac{n(n+1)}{2}}\chi^{m}\,\\\
&=(-1)^{p(a)p(b)+1}\sum_{m\in\mathbb{Z}_{+}}\chi^{m}\,b_{[m]}a\end{split}$

which holds by the skewsymmetry axiom. ∎

######  Proposition 4.5 (Left Leibniz rule).

Let $a,b,c$ be elements in an even SUSY PVA $\mathcal{P}.$ Then

(4.10)
$\\{ab{}_{\chi}c\\}=(-1)^{p(b)p(c)}\\{a{}_{\chi+D}c\\}(b)+(-1)^{p(a)(p(b)+p(c))}\\{b{}_{\chi+D}c\\}(a).$

###### Proof.

By the Leibniz rule,

(4.11)
$\\{c{}_{\chi}ab\\}=(-1)^{p(a)p(b)}\\{c{}_{\chi}b\\}a+\\{c{}_{\chi}a\\}b.$

By the skewsymmetry and Lemma 4.4, we get the proposition. ∎

######  Lemma 4.6.

Suppose that $\mathcal{P}$ is an even SUSY PVA. Then we have

(4.12)
$\\{a^{(m)}{}_{\chi+D}\,b^{(n)}\\}c=(-1)^{m(p(a)+p(b)+n)+\frac{m(m+1)}{2}+np(a)}(\chi+D)^{n}\\{a{}_{\chi+D}b\\}(\chi+D)^{m}c$

for $a,b,c\in\mathcal{P}.$

###### Proof.

We use (4.6) and the sesquilinearity. More precisely,

(4.13) $\displaystyle\\{a^{(m)}{}_{\chi+D}b^{(n)}\\}c$
$\displaystyle=(-1)^{m(p(a)+p(b)+n)+\frac{m(m+1)}{2}}\\{a{}_{\chi+D}b^{(n)}\\}(\chi+D)^{m}c$
$\displaystyle=(-1)^{m(p(a)+p(b)+n)+\frac{m(m+1)}{2}+np(a)}(\chi+D)^{n}\\{a{}_{\chi+D}b\\}(\chi+D)^{m}c.$

∎

Consider the freely generated differential algebra

(4.14) $\mathcal{P}:=\mathbb{C}\\{u_{i}^{(m)}|i\in I,\,m\in\mathbb{Z}\\},$

where $I$ is a finite index set and $u_{i}^{(m)}:=D^{m}(u_{i}).$

######  Theorem 4.7 (Master Formula).

Let $\mathcal{P}$ be an even SUSY PVA in (4.14) and $a,b\in\mathcal{P}.$ Then

(4.15) $\displaystyle\\{a{}_{\chi}b\\}=\sum_{i,j\in I,\,m,n\in\mathbb{Z}_{+}}$
$\displaystyle(-1)^{p(a)p(b)+(m+n)\tilde{\imath}+(p(b)+\tilde{\imath}+1)(\tilde{\jmath}+n)+\frac{m(m+1)}{2}}$
$\displaystyle\frac{\partial b}{\partial
u_{j}^{(n)}}(\chi+D)^{n}\\{u_{i}{}_{\chi+D}u_{j}\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}.$

###### Proof.

By the right and left Leibniz rules,

(4.16) $\displaystyle\\{a{}_{\chi}b\\}$
$\displaystyle=\sum_{\begin{subarray}{c}i,j\in I\\\
m,n\in\mathbb{Z}_{+}\end{subarray}}(-1)^{(b+\tilde{\jmath}+n)(a+\tilde{\jmath}+n)+(a+\tilde{\imath}+m)(\tilde{\jmath}+n)}\frac{\partial
b}{\partial u_{j}^{(n)}}\\{u_{i}^{(m)}{}_{\chi+D}u_{j}^{(n)}\\}\frac{\partial
a}{\partial u_{i}^{(m)}}$ $\displaystyle=\sum_{\begin{subarray}{c}i,j\in I\\\
m,n\in\mathbb{Z}_{+}\end{subarray}}(-1)^{(b+\tilde{\jmath}+n)(a+\tilde{\jmath}+n)+(a+\tilde{\imath}+m)(\tilde{\jmath}+n)+m(\tilde{\imath}+\tilde{\jmath}+n)+\frac{m(m+1)}{2}+n\tilde{\imath}}$
$\displaystyle\hskip 56.9055pt\frac{\partial b}{\partial
u_{j}^{(n)}}(\chi+D)^{n}\\{u_{i}\,{}_{\chi+D}\,u_{j}\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}.$

Simplifying the sign, we get the theorem. ∎

######  Corollary 4.8.

Let $a,b\in\mathcal{P}$. Then

(4.17) $\\{\smallint a\,,\,\smallint b\\}|_{\chi=0}=\sum_{i,j\in
I}\int(-1)^{p(a)p(b)+\tilde{\jmath}(p(b)+1)+\tilde{\imath}\tilde{\jmath}}\frac{\delta
b}{\delta u_{j}}\\{u_{i}{}_{D}u_{j}\\}\frac{\delta a}{\delta u_{i}}.$

###### Proof.

Recall $\frac{\delta a}{\delta
u_{i}}=\sum_{m\in\mathbb{Z}_{+}}(-1)^{m\tilde{\imath}+\frac{m(m+1)}{2}}$ and

(4.18) $\int\frac{\partial b}{\partial
u_{j}^{(n)}}D^{n}A=\int(-1)^{n(p(b)+\tilde{\jmath})+\frac{n(n-1)}{2}}D^{n}\frac{\partial
b}{\partial u_{j}^{(n)}}A$

for any $A,b\in\mathcal{P}.$ By the master formula, we get the corollary. ∎

######  Lemma 4.9.

For $f(\chi):=\sum_{i\in\mathbb{Z}_{+}}\chi^{i}\otimes
a_{i}\in\mathbb{C}[\chi]\otimes\mathcal{P}$, let us denote
$f(-\chi-D):=\sum_{i\in\mathbb{Z}_{+}}(-\chi-D)^{i}\otimes a_{i}$. Suppose
$a,b$ are elements in $\mathcal{P}$ and
$g(\chi):=a(\chi+D)^{m}b\in\mathbb{C}[\chi]\otimes\mathcal{P}.$ Then

(4.19)
$g(-\chi-D)=(-1)^{p(a)p(b)+m(p(a)+p(b))+\frac{m(m+1)}{2}}b(\chi+D)^{n}a.$

###### Proof.

Since $D$ is a derivation, we have

(4.20)
$g(-\chi-D)=(-1)^{mp(a)+\frac{m(m+1)}{2}}(\chi+D)^{m}(a)b=(-1)^{p(a)p(b)+m(p(a)+p(b))+\frac{m(m+1)}{2}}b(\chi+D)^{m}a.$

Here, the sign $\frac{m(m+1)}{2}$ comes from the fact that

$\displaystyle
a(\chi+D)b\xrightarrow{\quad\chi\to(-\chi-D)\quad}(-1)^{p(a)}(-\chi-D)(a)b$
$\displaystyle a(\chi+D)^{2}b=a(-\chi^{2}+\partial)b$ $\displaystyle\hskip
28.45274pt\xrightarrow{\quad\chi^{2}\to(-\chi-D)^{2}\quad}-(-\chi-D)^{2}(a)b+aD^{2}(b)=-(\chi+D)^{2}(a)b$

Inductively, we get

$a(\chi+D)^{m}b\xrightarrow{\quad\chi\to(-\chi-D)\quad}(-1)^{mp(a)+\frac{m(m+1)}{2}}(\chi+D)^{m}(a)b.$

∎

######  Proposition 4.10.

Let $\mathcal{P}$ be the freely generated differential algebra in (4.14)
endowed with the even $\chi$-bracket $\\{\cdot\,{}_{\chi}\cdot\,\\}$. Suppose
the $\chi$-bracket is defined by the master formula (4.15). If the
skewsymmetry axiom holds on the generators, i.e. if

(4.21)
$\\{u_{i}{}_{\chi}u_{j}\\}=(-1)^{\tilde{\imath}\tilde{\jmath}+1}\\{u_{j}{}_{-\chi-D}u_{i}\\}_{\leftarrow}$

for any $i,j\in I$, then it holds for any $a,b\in\mathcal{P}$.

###### Proof.

Let $a,b\in\mathcal{P}.$ By the master formula and Lemma 4.9, we have

(4.22) $\displaystyle\\{a{}_{-\chi-D}b\\}_{\leftarrow}=\sum_{i,j\in
I,\,m,n\in\mathbb{Z}_{+}}$
$\displaystyle(-1)^{p(a)p(b)+(m+n)\tilde{\imath}+(p(b)+\tilde{\imath}+1)(\tilde{\jmath}+n)+\frac{m(m+1)}{2}}(-1)^{S_{1}+S_{2}+S_{3}+S_{4}+S_{5}}$
$\displaystyle\frac{\partial a}{\partial
u_{i}^{(m)}}(\chi+D)^{m}\\{u_{i}{}_{-\chi-D}u_{j}\\}(\chi+D)^{n}\frac{\partial
b}{\partial u_{j}^{(n)}},$

where

(4.23) $\displaystyle
S_{1}:=(p(a)+\tilde{\imath}+m)(m+\tilde{\imath}+\tilde{\jmath}+p(b)+\tilde{\jmath}),$
$\displaystyle
S_{2}:=m(\tilde{\imath}+\tilde{\jmath}+p(b)+\tilde{\jmath})+\frac{m(m+1)}{2},$
$\displaystyle S_{3}:=(\tilde{\imath}+\tilde{\jmath})(p(b)+\tilde{\jmath}),$
$\displaystyle S_{4}:=n(p(b)+\tilde{\jmath}+n)+\frac{n(n+1)}{2}.$

Here $S_{1},$ $S_{2},$ $S_{3}$ and $S_{4}$ appear when we change the positions
of $\frac{\partial a}{\partial u_{i}^{(m)}},$ $(\chi+D)^{m}$, $(\chi+D)^{n}$
and $\frac{\partial b}{\partial u_{j}^{(n)}}$ using Lemma 4.9. Now, by Lemma
4.4, we have

(4.24)
$\\{u_{i}{}_{-\chi-D}u_{j}\\}_{\leftarrow}c=(-1)^{\tilde{\imath}\tilde{\jmath}+1}\\{u_{j}{}_{\chi+D}u_{i}\\}c.$

Applying (4.23) and (4.24) to (4.23), we get

(4.25) $\displaystyle\\{a{}_{-\chi-D}b\\}_{\leftarrow}$
$\displaystyle=\sum_{i,j\in
I,\,m,n\in\mathbb{Z}_{+}}(-1)^{(m+n)\tilde{\jmath}+(p(a)+\tilde{\jmath}+1)(\tilde{\imath}+m)+\frac{n(n+1)}{2}+1}$
$\displaystyle\hskip 56.9055pt\frac{\partial a}{\partial
u_{i}^{(m)}}(\chi+D)^{m}\\{u_{j}{}_{\chi+D}u_{i}\\}(\chi+D)^{n}\frac{\partial
b}{\partial u_{j}^{(n)}}$ $\displaystyle=(-1)^{p(a)p(b)+1}\\{b{}_{\chi}a\\}.$

∎

######  Proposition 4.11.

Let $\mathcal{P}$ be the freely generated differential algebra in (4.14)
endowed with a even $\chi$-bracket $\\{\,{}_{\chi}\,\\}$. Suppose that this
$\chi$-bracket satisfies the Leibniz rule. If the $\chi$-bracket is skew-
symmetric and the Jacobi identity holds for any triple of generators, that is
for $i,j,k\in I$,

(4.26)
$\\{u_{i}{}_{\chi}\\{u_{j}{}_{\gamma}u_{k}\\}\\}-(-1)^{\tilde{\imath}\tilde{\jmath}}\\{u_{j}{}_{\gamma}\\{u_{i}{}_{\chi}u_{k}\\}\\}=\\{\\{u_{i}{}_{\chi}u_{j}\\}_{\chi+\gamma}u_{k}\\},$

then the Jacobi identity holds for any elements in $\mathcal{P}$.

###### Proof.

We aim to show that

(4.27)
$\\{a{}_{\chi}\\{b{}_{\gamma}c\\}\\}-(-1)^{p(a)p(b)}\\{b{}_{\gamma}\\{a{}_{\chi}c\\}\\}=\\{\\{a{}_{\chi}b\\}_{\chi+\gamma}c\\}.$

for $a,b,c\in\mathcal{P}.$ The first term in (4.27) is

$\displaystyle\\{a{}_{\chi}\\{b{}_{\gamma}c\\}\\}=\sum_{i,j\in
I,\,m,n\in\mathbb{Z}_{+}}(-1)^{S_{1}}\Big{\\{}\,a\,{}_{\chi}\,\frac{\partial
c}{\partial
u_{j}^{(n)}}(\chi+D)^{n}\\{u_{i}{}_{\chi+D}u_{j}\\}(\chi+D)^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}\,\Big{\\}}$ (4.28) $\displaystyle=\sum_{i,j\in
I,\,m,n\in\mathbb{Z}_{+}}(-1)^{S_{1}}\bigg{[}\Big{\\{}a{}_{\chi}\frac{\partial
c}{\partial
u_{j}^{(n)}}\Big{\\}}(\gamma+D)^{n}\\{u_{i}{}_{\gamma+D}u_{j}\\}(\gamma+D)^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}$ (4.29) $\displaystyle\hskip
71.13188pt+(-1)^{S_{2}}\frac{\partial c}{\partial
u_{j}^{(n)}}(\chi+\gamma+D)^{n}\\{u_{i}{}_{\chi+\gamma+D}u_{j}\\}(\chi+\gamma+D)^{m}\Big{\\{}a{}_{\chi}\frac{\partial
b}{\partial u_{i}^{(m)}}\Big{\\}}$ (4.30) $\displaystyle\hskip
71.13188pt+(-1)^{S_{3}}\frac{\partial c}{\partial
u_{j}^{(n)}}(\chi+\gamma+D)^{n}\\{a{}_{\chi+D}\\{u_{i}{}_{\gamma+D}u_{j}\\}\\}(\gamma+D)^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}\bigg{]}.$

where
$S_{1}=p(b)p(c)+(m+n)\tilde{\imath}+(p(c)+\tilde{\imath}+1)(\tilde{\jmath}+n)+\frac{m(m+1)}{2}$,
$S_{2}=p(a)(p(c)+m+\tilde{\imath})$ and $S_{3}=p(a)(p(c)+\tilde{\jmath}).$ We
expand (4.28), (4.29) and (4.30) using the formula

(4.31) $\displaystyle\Big{\\{}a{}_{\chi}\frac{\partial c}{\partial
u_{j}^{(n)}}\Big{\\}}=\sum_{k,r\in
I,\,l,q\in\mathbb{Z}_{+}}(-1)^{S_{4}}\frac{\partial^{2}c}{\partial
u_{r}^{(q)}\partial
u_{j}^{(n)}}(\chi+D)^{q}\\{u_{k}{}_{\chi+D}u_{r}\\}(\chi+D)^{l}\frac{\partial
a}{\partial u_{k}^{(l)}},$ (4.32)
$\displaystyle\Big{\\{}a{}_{\chi}\frac{\partial b}{\partial
u_{i}^{(m)}}\Big{\\}}=\sum_{k,r\in
I,\,l,q\in\mathbb{Z}_{+}}(-1)^{S_{5}}\frac{\partial^{2}b}{\partial
u_{r}^{(q)}\partial
u_{i}^{(m)}}(\chi+D)^{q}\\{u_{k}{}_{\chi+D}u_{r}\\}(\chi+D)^{l}\frac{\partial
a}{\partial u_{k}^{(l)}},$ (4.33)
$\displaystyle\\{a{}_{\chi+D}\\{u_{i}{}_{\gamma+D}u_{j}\\}\\}=\sum_{k\in
I,l\in\mathbb{Z}_{+}}(-1)^{S_{6}}\\{u_{k}{}_{\chi+D}\\{u_{i}{}_{\gamma+D}u_{j}\\}\\}(\chi+D)^{l}\frac{\partial
a}{\partial u_{k}^{(l)}},$

where
$S_{4}=p(a)(p(c)+\tilde{\jmath}+n)+(q+l)\tilde{k}+((p(c)+\tilde{\jmath}+n)+\tilde{k}+1)(\tilde{r}+q)+\frac{l(l+1)}{2}$,
$S_{4}=p(a)(p(b)+\tilde{\imath}+m)+(q+l)\tilde{k}+((p(b)+\tilde{\imath}+m)+\tilde{k}+1)(\tilde{r}+q)+\frac{l(l+1)}{2}$
and
$S_{6}=(\tilde{\imath}+\tilde{\jmath})(p(a)+\tilde{k}+l)+l(\tilde{k}+\tilde{\imath}+\tilde{\jmath})+\frac{l(l+1)}{2}$.
Then (4.28) is

(4.34) $\displaystyle\sum_{\begin{subarray}{c}i,j,k,r\in I\\\
m,n,l,q\in\mathbb{Z}_{+}\end{subarray}}(-1)^{S_{7}}\frac{\partial^{2}c}{\partial
u_{r}^{(q)}\partial u_{j}^{(n)}}$
$\displaystyle\Big{[}(\chi+D_{1})^{q}\\{u_{k}{}_{\chi+D_{1}}u_{r}\\}(\chi+D_{1})^{l}\frac{\partial
a}{\partial u_{k}^{(l)}}\Big{]}_{1}$ $\displaystyle\hskip
28.45274pt\Big{[}(\gamma+D_{2})^{n}\\{u_{i}{}_{\gamma+D_{2}}u_{j}\\}(\gamma+D_{2})^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}\Big{]}_{2}$

where $D_{1}$ and $D_{2}$ are the derivation $D$ which act in the first and
second parentheses $[\ ]_{1}$ and $[\ ]_{2}$, respectively and
$S_{7}=(p(a)+p(b))p(c)+p(c)(\tilde{\jmath}+n+\tilde{r}+q)+\tilde{\imath}(\tilde{\jmath}+m)+\tilde{\jmath}(\tilde{r}+q+1)+\tilde{k}(l+\tilde{r})+(\tilde{r}+q)(n+1)+n+\frac{m(m+1)}{2}+\frac{l(l+1)}{2}.$
Similarly, (4.29) is

(4.35) $\displaystyle\sum_{\begin{subarray}{c}i,j,k,r\in I\\\
m,n,l,q\in\mathbb{Z}_{+}\end{subarray}}(-1)^{S_{8}}\frac{\partial c}{\partial
u_{j}^{(n)}}$
$\displaystyle(\chi+\gamma+D)^{n}\\{u_{i}{}_{\chi+\gamma+D}u_{j}\\}(\chi+\gamma+D)^{m}\frac{\partial^{2}b}{\partial
u_{r}^{(q)}\partial u_{i}^{(m)}}$ $\displaystyle\hskip
28.45274pt(\chi+D)^{q}\\{u_{k}{}_{\chi+D}u_{r}\\}(\chi+D)^{l}\frac{\partial
a}{\partial u_{k}^{(l)}}$

where $D_{1}$ and $D_{2}$ are the derivation $D$ which only act in the first
and second parentheses $[\ ]_{1}$ and $[\ ]_{2}$, respectively and
$S_{8}=p(a)p(c)+p(a)p(b)+p(b)p(c)+p(c)(\tilde{\jmath}+n)+p(b)(\tilde{r}+q)+\tilde{\imath}(\tilde{\jmath}+m+\tilde{r}+q)+\tilde{\jmath}+\tilde{k}(\tilde{r}+l)+(\tilde{r}+q)(m+1)+n+\frac{m(m+1)}{2}+\frac{l(l+1)}{2}.$
Finally, (4.30) is

(4.36) $\displaystyle\sum_{\begin{subarray}{c}i,j,k\in I\\\
m,n,l\in\mathbb{Z}_{+}\end{subarray}}(-1)^{S_{9}}\frac{\partial c}{\partial
u_{j}^{(n)}}\\{u_{k}{}_{\chi+D}\\{u_{i}{}_{\gamma+D}u_{j}\\}\\}\Big{[}(\chi+D_{1})^{l}\frac{\partial
a}{\partial u_{k}^{(l)}}\Big{]}_{1}\Big{[}(\gamma+D_{2})^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}\Big{]}_{2},$

where $D_{1}$ and $D_{2}$ is the derivation $D$ which only act in the
parentheses $[\ ]_{1}$ and $[\ ]_{2}$ and
$S_{9}=p(a)p(c)+p(b)p(c)+p(a)\tilde{\imath}+p(c)(\tilde{\jmath}+n)+\tilde{\imath}\tilde{\jmath}+\tilde{\imath}m+\tilde{k}l+\tilde{\imath}\tilde{k}+\tilde{\jmath}\tilde{k}+\tilde{\jmath}+n+\frac{m(m+1)}{2}+\frac{l(l+1)}{2}.$

The second term in (4.27) is

$\displaystyle\\{b{}_{\gamma}\\{a{}_{\chi}c\\}\\}$ (4.37)
$\displaystyle=\sum_{\begin{subarray}{c}i,j,k,r\in I\\\
m,n,l,q\in\mathbb{Z}_{+}\end{subarray}}(-1)^{T_{1}}\frac{\partial^{2}c}{\partial
u_{j}^{(n)}\partial
u_{r}^{(q)}}\Big{[}(\gamma+D_{1})^{n}\\{u_{i}{}_{\gamma+D_{1}}u_{j}\\}(\gamma+D_{1})^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}\Big{]}_{1}$ $\displaystyle\hskip
170.71652pt\Big{[}(\chi+D_{2})^{q}\\{u_{k}{}_{\chi+D_{2}}u_{r}\\}(\chi+D_{2})^{l}\frac{\partial
a}{\partial u_{k}^{(l)}}\Big{]}_{2}$ (4.38)
$\displaystyle+\sum_{\begin{subarray}{c}i,j,k,r\in I\\\
m,n,l,q\in\mathbb{Z}_{+}\end{subarray}}(-1)^{T_{2}}\frac{\partial c}{\partial
u_{j}^{(n)}}(\chi+\gamma+D)^{n}\\{u_{i}{}_{\chi+\gamma+D}u_{j}\\}(\chi+\gamma+D)^{m}$
$\displaystyle\hskip 142.26378pt\frac{\partial^{2}a}{\partial
u_{r}^{(q)}\partial
u_{i}^{(m)}}(\gamma+D)^{q}\\{u_{k}{}_{\gamma+D}u_{r}\\}(\gamma+D)^{l}\frac{\partial
b}{\partial u_{k}^{(l)}}$ (4.39)
$\displaystyle+\sum_{\begin{subarray}{c}i,j,k\in I\\\
m,n,l\in\mathbb{Z}_{+}\end{subarray}}(-1)^{T_{3}}\frac{\partial c}{\partial
u_{j}^{(n)}}\\{u_{i}{}_{\gamma+D}\\{u_{k}{}_{\chi+D}u_{j}\\}\\}\Big{[}(\gamma+D_{1})^{m}\frac{\partial
b}{\partial u_{i}^{(m)}}\Big{]}_{1}\Big{[}(\chi+D_{2})^{l}\frac{\partial
a}{\partial u_{k}^{(l)}}\Big{]}_{2},$

where
$T_{1}=p(a)p(c)+p(b)p(c)+p(b)(\tilde{r}+q)+p(c)(\tilde{r}+q+\tilde{\jmath}+n)+\tilde{r}(\tilde{k}+\tilde{\jmath}+n+1)+\tilde{\jmath}(\tilde{\imath}+q)+nq+q+n+\tilde{\jmath}+\tilde{\imath}m+\tilde{k}l+\frac{m(m+1)}{2}+\frac{n(n+1)}{2},$
$T_{2}=p(a)p(b)+p(b)p(c)+p(a)p(c)+p(c)(\tilde{\jmath}+n)+p(a)(\tilde{r}+q)+\tilde{\imath}(\tilde{\jmath}+m\tilde{r}+q)+\tilde{\jmath}+\tilde{k}(\tilde{r}+q)+\tilde{p}m+mq+n+\tilde{r}+q+\frac{m(m+1)}{2}+\frac{l(l+1)}{2}$
and
$T_{3}=p(b)p(c)+p(a)p(c)+p(b)\tilde{k}+c\tilde{\jmath}+p(c)n+\tilde{k}\tilde{\jmath}+\tilde{k}l+\tilde{\jmath}+\tilde{\imath}\tilde{k}+\tilde{\imath}\tilde{\jmath}+\tilde{\imath}m+n+\frac{m(m+1)}{2}+\frac{l(l+1)}{2}$.
The last term in (4.27) can be expanded as follows:

$\displaystyle\\{\\{a{}_{\chi}b\\}_{\chi+\gamma}c\\}$
$\displaystyle=\sum_{\begin{subarray}{c}i,j\in I\\\
m,n\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{1}}\Big{\\{}\frac{\partial
b}{\partial
u_{j}^{(n)}}(\chi+D)^{n}\\{u_{i}{}_{\chi+D}u_{j}\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}\,{}_{\chi+\gamma}\,c\Big{\\}}$ (4.40)
$\displaystyle=\sum_{\begin{subarray}{c}i,j\in I\\\
m,n\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{1}+\alpha_{2}}\Big{\\{}\frac{\partial
b}{\partial
u_{j}^{(n)}}{}_{\chi+\gamma+D}c\Big{\\}}(\chi+D)^{n}\\{u_{i}{}_{\chi+D}u_{j}\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}$ (4.41)
$\displaystyle+\sum_{\begin{subarray}{c}i,j\in I\\\
m,n\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{1}+\alpha_{3}}\Big{\\{}(\chi+D)^{m}\frac{\partial
a}{\partial
u_{i}^{(m)}}\,{}_{\chi+\gamma+D}\,c\Big{\\}}\\{u_{i}{}_{-\chi-D}u_{j}\\}_{\leftarrow}(\gamma+D)^{n}\frac{\partial
b}{\partial u_{j}^{(n)}}$ (4.42)
$\displaystyle+\sum_{\begin{subarray}{c}i,j\in I\\\
m,n\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{1}+\alpha_{4}}\\{\\{u_{i}{}_{\chi+\gamma+D}u_{j}\\}_{\chi+\gamma+D}c\\}\Big{[}(\chi+D_{1})^{m}\frac{\partial
a}{\partial u_{i}^{(m)}}\Big{]}_{1}\Big{[}(\gamma+D_{2})^{n}\frac{\partial
a}{\partial u_{j}^{(n)}}\Big{]}_{2},$

where
$\alpha_{1}=p(a)p(b)+(m+n)\tilde{\imath}+(p(b)+\tilde{\imath}+1)(\tilde{\jmath}+n)+\frac{m(m+1)}{2}$,
$\alpha_{2}=p(c)(p(a)+n+\tilde{\jmath}),$
$\alpha_{3}=n(p(a)+p(c)+\tilde{\jmath})+\frac{n(n+1)}{2}+(\tilde{\imath}+\tilde{\jmath})(p(a)+p(c)+\tilde{\imath})$
and
$\alpha_{4}=n(p(a)+p(c)+\tilde{\jmath})+\frac{n(n+1)}{2}+(p(a)+\tilde{\imath})p(c)$.

Using the master formula once more, (4.40) is

(4.43) $\displaystyle\sum_{\begin{subarray}{c}i,j,k,r\in I\\\
m,n,l,q\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{5}}\frac{\partial
c}{\partial u_{k}^{(l)}}(\chi+\gamma+D)^{l}$
$\displaystyle\\{u_{r}{}_{\chi+\gamma+D}u_{k}\\}(\chi+\gamma+D)^{q}$
$\displaystyle\frac{\partial^{2}b}{\partial u_{j}^{(n)}\partial u_{r}^{(q)}}$
$\displaystyle(\chi+D)^{n}\\{u_{i}{}_{\chi+D}u_{j}\\}(\chi+D)^{m}\frac{\partial
a}{\partial u_{i}^{(m)}},$

where
$\alpha_{5}=p(a)p(b)+p(a)p(c)+p(b)p(c)+p(c)(\tilde{k}+l)+p(b)(n+\tilde{\jmath})+\tilde{r}(n+\tilde{\jmath}+\tilde{k}+q)+\tilde{\imath}(m+\tilde{\jmath})+q(\tilde{\jmath}+n)$.
The next term (4.41) is

(4.44) $\displaystyle\sum_{\begin{subarray}{c}i,j,k,r\in I\\\
m,n,l,q\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{6}}\frac{\partial
c}{\partial u_{k}^{(l)}}(\chi+\gamma+D)^{l}$
$\displaystyle\\{u_{r}{}_{\chi+\gamma+D}u_{k}\\}(\chi+\gamma+D)^{q}$
$\displaystyle\frac{\partial^{2}a}{\partial u_{i}^{(m)}u_{r}^{(q)}}$
$\displaystyle(\gamma+D)^{m}\\{u_{j}{}_{\gamma+D}u_{i}\\}(\gamma+D)^{n}\frac{\partial
b}{\partial u_{j}^{(n)}},$

where
$\alpha_{6}=p(a)p(c)+p(b)p(c)+\tilde{r}(\tilde{\imath}+m+\tilde{k}+q)+\tilde{\jmath}(\tilde{\imath}+n)+q(\tilde{\imath}+m)+p(a)(\tilde{\imath}+m)+p(c)(\tilde{k}+l)+\tilde{\imath}+\tilde{k}+m+l+\frac{q(q+1)}{2}+\frac{n(n+1)}{2}+1$.
Here, we used

(4.45) $\frac{\partial}{\partial u_{r}^{(q)}}\frac{\partial}{\partial
u_{i}^{(m)}}=(-1)^{(\tilde{r}+q)(\tilde{\imath}+m)}\frac{\partial}{\partial
u_{i}^{(m)}}\frac{\partial}{\partial u_{r}^{(q)}}$

The last term (4.42) is

(4.46) $\displaystyle\sum_{\begin{subarray}{c}i,j,k\in I\\\
m,n,l\in\mathbb{Z}_{+}\end{subarray}}(-1)^{\alpha_{7}}$
$\displaystyle\frac{\partial c}{\partial
u_{k}^{(l)}}(\chi+\gamma+D)^{l}\\{\\{u_{i}{}_{\chi+D}u_{j}\\}{}_{\chi+\gamma+D}u_{k}\\}$
$\displaystyle\Big{[}(\chi+D_{1})^{m}\frac{\partial a}{\partial
u_{i}^{(m)}}\Big{]}_{1}\Big{[}(\gamma+D_{2})^{n}\frac{\partial b}{\partial
u_{j}^{(n)}}\Big{]}_{2},$

where
$\alpha_{7}=p(a)p(c)+p(b)p(c)+p(c)(\tilde{k}+l)+\tilde{\imath}m+\tilde{\imath}\tilde{k}+\tilde{\imath}\tilde{\jmath}+\tilde{\jmath}\tilde{k}+\tilde{\jmath}n+\tilde{k}+l+\frac{m(m+1)}{2}+\frac{n(n+1)}{2}$.
By tracking all the signs, we get

(4.47)
$\begin{array}[]{ccccc}&\eqref{eq:proof_Jacobi-1}&-&(-1)^{p(a)p(b)}\eqref{eq:proof_Jacobi-4}&=0\\\
&\eqref{eq:proof_Jacobi-2}&+&0&=\eqref{eq:proof_Jacobi-7}\\\
&0&-&(-1)^{p(a)p(b)}\eqref{eq:proof_Jacobi-2}&=\eqref{eq:proof_Jacobi-8}\\\
&\eqref{eq:proof_Jacobi-3}&-&(-1)^{p(a)p(b)}\eqref{eq:proof_Jacobi-6}&=\eqref{eq:proof_Jacobi-9},\end{array}$

which implies the Jacobi identity. ∎

By Proposition 4.10 and 4.11, we conclude the following Theorem.

######  Theorem 4.12.

Let $\mathcal{P}$ be the freely generated differential algebra in (4.14)
endowed with an even $\chi$-bracket $\\{\,{}_{\chi}\,\\}$ given by the master
formula. If the skewsymmetry axiom and the Jacobi identity hold for the
generators then they hold for any elements in $\mathcal{P}$.

###  4.2. Linear affine even SUSY PVA

In the rest of this paper, we let $\mathfrak{g}=\mathfrak{gl}(n+1|n)$,
$N=2n+1$. Recall Example 2.4 that $\mathfrak{g}$ is spanned by
$\\{e_{ij}|i,j\in I\\}$, where $p(e_{ij})=i+j.$ Consider the differential
superalgebra

(4.48)
$\mathcal{V}(\bar{\mathfrak{g}})=S(\mathbb{C}[D]\otimes\bar{\mathfrak{g}}).$

######  Definition 4.13.

The linear affine even SUSY PVA bracket on the differential algebra
$\mathcal{V}(\bar{\mathfrak{g}})$ is given by

(4.49) $\\{\bar{e}_{1i}\,{}_{\chi}\,\bar{e}_{i\,2n+1}\\}=(-1)^{i+1}\,$

for $i=1,2,\cdots,2n+1.$ All other brackets between generators are set as
being trivial.

It follows from Theorem 4.12 that this bracket is well-defined. Moreover, from
Corollary 4.8 we have for all $a,b\in\mathcal{V}(\bar{\mathfrak{g}})$,

(4.50)
$\begin{split}\smallint\,\\{a\,{}_{\chi}\,b\\}|_{\chi=0}&=\int\sum_{i=1}^{2n+1}(-1)^{(a+1)i}\Big{(}\frac{\delta
a}{\delta\bar{e}_{i\,2n+1}}\frac{\delta
b}{\delta\bar{e}_{1i}}+(-1)^{i+1}\frac{\delta
a}{\delta\bar{e}_{1i}}\frac{\delta b}{\delta\bar{e}_{i\,2n+1}}\Big{)}\\\
&=(-1)^{p(a)+1}\Big{(}\frac{\delta a}{\delta q}|(e_{1\,2n+1}\otimes
1)\frac{\delta b}{\delta q}+\frac{\delta b}{\delta q}(e_{1\,2n+1}\otimes
1)\Big{)},\end{split}$

where $\frac{\delta a}{\delta q}=\sum_{i,j\in I}e_{ij}\otimes\frac{\delta
a}{\delta\,\bar{e}_{ij}}.$ Finally, by Lemma 3.11, we deduce that for all
$a,b\in\mathcal{W}_{2n+1}$

(4.51) $\int\,\\{a\,{}_{\chi}\,b\\}|_{\chi=0}=(-1)^{p(a)+1}\int\,\big{(}\
\text{ Res }\nabla_{a}\,|\,(e_{1\,2n+1}\otimes 1)\text{ Res }\nabla_{b}+\text{
Res }\nabla_{b}\,(e_{1\,2n+1}\otimes 1)\ \big{)}\,.$

######  Lemma 4.14.

Let $a,b$ be two elements in $\mathcal{W}_{2n+1}$. We have

(4.52) $\int\text{ Res }(L\frac{\delta a}{\delta L}\frac{\delta b}{\delta
L})=(-1)^{p(a)}\int\sum_{i=1}^{2n+1}\text{ Res }((\frac{\delta a}{\delta
L})^{*}D^{i-1})\text{ Res }(b_{N-i}(\frac{\delta b}{\delta L})^{*})$

###### Proof.

For $j\in\mathbb{Z}$ and $v\in\mathcal{W}_{2n+1}$, the pseudo-differential
operator $Z=D^{-j}v$ satisfies $Res(D^{i-1}Z)=\delta_{ij}(-1)^{p(v)}v$. Hence
for any pseudo-differential operator $Z$, we have

(4.53) $\sum_{i=1}^{2n+1}(-1)^{i}(LD^{-i})_{+}\text{ Res
}(D^{i-1}Z)=(-1)^{p(Z)}(LZ_{-})_{+}.$

In particular, if we substitute $Z=Z_{-}=\frac{\delta a}{\delta L}$ then

(4.54) $\sum_{i=1}^{2n+1}(-1)^{i}(LD^{-i})_{+}\text{ Res }(D^{i-1}\frac{\delta
a}{\delta L})=(-1)^{p(a)}(L\frac{\delta a}{\delta L})_{+}.$

After multiplying by $\frac{\delta b}{\delta L}$ on the left to the both sides
of (4.54) and taking residues, we get

$\sum_{i=1}^{2n+1}\text{ Res }(\frac{\delta b}{\delta L}(LD^{-i})_{+})\text{
Res }(D^{i-1}\frac{\delta a}{\delta L})=\text{ Res }(\frac{\delta b}{\delta
L}L\frac{\delta a}{\delta L}),$

using (2) of Lemma 3.2. Finally, by (1) and (3) of Lemma 3.2, we have

$\int\sum_{i=1}^{2n+1}\text{ Res }((\frac{\delta a}{\delta
L})^{*}D^{i-1})\text{ Res }((D^{-i}L^{*})_{+}(\frac{\delta b}{\delta
L})^{*})=(-1)^{p(a)}\int\text{ Res }(L\frac{\delta a}{\delta L}\frac{\delta
b}{\delta L})\,.$

∎

######  Lemma 4.15.

Let $a,b$ be two elements in $\mathcal{W}_{2n+1}$. We have

(4.55) $\int\text{ Res }(\frac{\delta a}{\delta L}L\frac{\delta b}{\delta
L})=(-1)^{p(a)+p(b)}\int\sum_{i=1}^{2n+1}(-1)^{i(p(a)+p(b)+1)}\text{ Res
}(b_{N-i}(\frac{\delta a}{\delta L})^{*})\text{ Res }((\frac{\delta b}{\delta
L})^{*}D^{i-1}).$

###### Proof.

Applying (4.53) in the preceding proof for $Z=\frac{\delta b}{\delta L}$, we
have

(4.56) $(-1)^{p(b)}(L\frac{\delta b}{\delta
L})_{+}=\sum_{i=1}^{2n+1}(-1)^{i}(LD^{-i})_{+}\text{ Res }(D^{i-1}\frac{\delta
b}{\delta L}).$

On (4.56), by multiplying $\frac{\delta a}{\delta L}$ on the left, taking the
residue and applying (3.27), one gets

$\int\text{ Res }(\frac{\delta a}{\delta L}L\frac{\delta b}{\delta
L})=\sum_{i=1}^{2n+1}\int\text{ Res }(\frac{\delta a}{\delta
L}(LD^{-i})_{+})\text{ Res }(D^{i-1}\frac{\delta b}{\delta L}).$

One ends the proof by taking adjoints inside the residues. ∎

######  Lemma 4.16.

For all $a,b\in\mathcal{W}_{2n+1}$,

(4.57) $\int\,\\{a\,{}_{\chi}\,b\\}|_{\chi=0}=\int\text{ Res }(L\frac{\delta
a}{\delta L}\frac{\delta b}{\delta L}+(-1)^{p(a)+1}\frac{\delta a}{\delta
L}L\frac{\delta b}{\delta L}).$

###### Proof.

It can be checked by direct computation that

(4.58) $\begin{split}\int\big{(}\,\text{ Res }(\nabla_{a})|(e_{1\,2n+1}\otimes
1)\,\,\text{ Res
}(\nabla_{b})\,\big{)}&=\int\sum_{i=1}^{2n+1}(-1)^{i(p(a)+p(b)+1)}\text{ Res
}(b_{N-i}(\frac{\delta a}{\delta L})^{*})\text{ Res }((\frac{\delta b}{\delta
L})^{*}D^{i-1})\,,\\\ \int\big{(}\,\text{ Res }(\nabla_{a})|\,\text{ Res
}(\nabla_{b})\,(e_{1\,2n+1}\otimes 1)\,\big{)}&=-\int\sum_{i=1}^{2n+1}\text{
Res }((\frac{\delta a}{\delta L})^{*}D^{i-1})\text{ Res }(b_{N-i}(\frac{\delta
b}{\delta L})^{*})\,.\end{split}$

We conclude the proof using equation (LABEL:eq11) and the previous two Lemmas.
∎

######  Theorem 4.17.

The subalgebra $\mathcal{W}_{2n+1}\subset\mathcal{V}(\bar{\mathfrak{g}})$ is
stabilized by the linear affine even bracket. In particular, the even bracket
on $\mathcal{W}_{2n+1}$ satisfies the Jacobi identity.

###### Proof.

Immediately follows from the previous Lemma and the master formula. ∎

######  Lemma 4.18.

The linear even SUSY PVA bracket reduced to $\mathcal{W}_{2n+1}$ is given
explicitly by the formula

(4.59) $\begin{split}\\{L(z)\,{}_{\chi}\,L(w)\\}&=\langle
L(D+\chi+w)(D+\chi+w-z)(D^{2}-\chi^{2}+w^{2}-z^{2})^{-1}\rangle\\\
&+\langle(D+\chi+w-z)(D^{2}-\chi^{2}+w^{2}-z^{2})^{-1}L(D+w)\rangle\,.\end{split}$

###### Proof.

By the master formula, we have

$\\{u_{i}{}_{D}u_{j}\\}(\frac{\delta a}{\delta
u_{i}})=\Big{(}(-1)^{(j+1)(p(a)+i)}LD^{i-2n-2}\frac{\delta a}{\delta
u_{i}}+(-1)^{i+1+j(p(a)+i)}D^{i-2n-2}\frac{\delta a}{\delta
u_{i}}L\Big{)}_{[2n+1-j]}\,.$

This implies that for any $a\in\mathcal{W}_{N}$

$\\{u_{i}{}_{D}u_{j}\\}(a)=\Big{(}(-1)^{jp(a)+p(a)}LD^{i-2n-2}a+(-1)^{i+1+jp(a)}D^{i-2n-2}aL\Big{)}_{[2n+1-j]}\,.$

In other words,

$\\{u_{i}\,{}_{\chi}\,u_{j}\\}=\Big{(}L(D+\chi)(D+\chi)^{i-2n-2}+(-1)^{i+1}(D+\chi)^{i-2n-2}L(D)\Big{)}_{[2n+1-j]}\,.$

Therefore

$\begin{split}\\{u_{i}\,{}_{\chi}\,L(w)\\}&=\langle
L(D+\chi+w)(D+\chi+w)^{i-2n-2}\rangle\\\
&+(-1)^{i+1}\langle(D+\chi+w)^{i-2n-2}L(D+w)\rangle\,.\end{split}$

Finally

$\begin{split}\sum_{i=1}^{2n+1}\\{u_{i}(-1)^{(2n+1-i)(2n-i+4)/2}z^{2n+1-i}\,{}_{\chi}\,L(w)\\}&=\langle
L(D+\chi+w)(D+\chi+w-z)^{-1}\rangle\\\
&+\langle(D+\chi+w-z)^{-1}L(D+w)\rangle\,.\end{split}$

The result follows from the identity

$\begin{split}L(z)&=\frac{1-\mathbf{i}}{2}\sum_{j=1}^{2n+1}u_{j}(-1)^{(2n+1-j)(2n-j+4)/2}(\mathbf{i}z)^{2n+1-j}\\\
&\hskip
14.22636pt+\frac{1+\mathbf{i}}{2}\sum_{j=1}^{2n+1}u_{j}(-1)^{(2n+1-j)(2n-j+4)/2}(-\mathbf{i}z)^{2n+1-j}\,\end{split}$

where $\mathbf{i}^{2}=-1$. ∎

######  Definition 4.19.

We denote the even SUSY PVA bracket (4.59) on $\mathcal{W}_{2n+1}$ by
$\\{\,{}_{\chi}\,\\}^{e}$ and call it the even linear GD bracket.

###  4.3. Hamiltonian hierarchy on $\mathcal{W}_{2n+1}$

In this subsection we relate the linear even Gelfand-Dickey bracket to the
hierarchy (4.60). Recall that the evolutionary derivations $dt_{k}$ are
defined by the Lax equations

(4.60)
$\frac{dL}{dt_{k}}=[(L^{\frac{2k}{2n+1}})_{+},L],\quad\,[\frac{d}{dt_{k}}\,,\,D]=0\,,\,\,\,k\geq
1\,.$

######  Lemma 4.20.

For all $k,l\geq 1$ we have

  1. $(1)$

The derivations $d/dt_{k}$ and $d/dt_{l}$ commute ,

  2. $(2)$

The residue of $L^{\frac{l}{2n+1}}$ is conserved for $d/dt_{k}$, i.e.

$\int\frac{d}{dt_{k}}\,\textit{ Res }\,L^{\frac{l}{2n+1}}=0\,.$

###### Proof.

Identical to the even case in section 3. ∎

Now we assume $k$ is an odd integer and define the density

(4.61) $\int\,h_{k}=\,\frac{2n+1}{k}\int\text{ Res }L^{\frac{k}{2n+1}}\,.$

######  Lemma 4.21.

For all odd integer $k$, we have

$\frac{\delta{h_{k}}}{\delta L}=(L^{\frac{k}{2n+1}-1})_{-}\,\,.$

###### Proof.

The proof is identical to Lemma 3.16. ∎

######  Theorem 4.22.

The hierarchy of even evolutionary derivations (4.60) is Hamiltonian for the
linear even SUSY PVA bracket $\\{\,\,{}_{\chi}\,\,\\}^{e}$. Its hamiltonians
are the functionals (4.61) which are conserved for each derivation in the
hierarchy. They are in involution for the even linear Gelfand-Dickey bracket.

###### Proof.

Recall that for any evolutionary derivation $d$ of $\mathcal{W}_{2n+1}$ one
has for all $a\in\mathcal{W}_{2n+1}$

$\int\,d(a)=\int\sum_{i=1}^{2n+1}d(u_{i})\frac{\delta a}{\delta u_{i}}\,.$

Hence it follows from Lemma 4.16 that for any homogeneous element $\smallint
a\in\smallint\mathcal{W}_{2n+1}$ the derivation
$\\{\smallint\,a\,{}_{\chi}\,\cdot\\}^{e}|_{\chi=0}$ of $\mathcal{W}_{2n+1}$
is defined by

$\\{a\,{}_{\chi}\,L\\}^{e}|_{\chi=0}=\Big{(}L\frac{\delta a}{\delta
L}+(-1)^{p(a)+1}\frac{\delta a}{\delta L}L\Big{)}_{+}\,.$

In particular, for all positive odd integer $k$,

(4.62)
${\\{\,\smallint\,h_{k}\,{}_{\chi}\,L(w)\,\\}^{e}}|_{\chi=0}=[L^{\frac{k}{2n+1}-1}_{+},L](w).$

∎

## References