###  2.7. Examples of vectorial Lie superalgebras

1) General vectorial Lie superalgebras. Let
$x=(u_{1},\dots,u_{n},\theta_{1},\dots,\theta_{m})$. The the general vectorial
Lie superalgebra is defined to be
${\mathfrak{vect}}(n|m):={\mathfrak{der}}~{}\;{\mathbb{C}}[x]$.

2) Special Lie superalgebras. The divergence of the field
$D=\mathop{\sum}_{i}f_{i}{\frac{\partial}{\partial{u_{i}}}}+\mathop{\sum}_{j}g_{j}{\frac{\partial}{\partial{\theta_{j}}}}$
is the function

(48) $\mathop{\text{\rm div}}\nolimits
D:=\mathop{\sum}\limits_{i}{\frac{\partial{f_{i}}}{\partial{u_{i}}}}+\mathop{\sum}\limits_{j}(-1)^{p(g_{j})}{\frac{\partial{g_{i}}}{\partial{\theta_{j}}}}.$

$\bullet$ The Lie superalgebra
${\mathfrak{svect}}(n|m):=\\{D\in{\mathfrak{vect}}(n|m)\mid\mathop{\text{\rm
div}}\nolimits D=0\\}$ is called the special (or divergence-free) vectorial
superalgebra.

It is clear that one can also describe ${\mathfrak{svect}}(n|m)$ as
$\\{D\in{\mathfrak{vect}}(n|m)\mid
L_{D}\mathop{\mathrm{{vol\/}}}\nolimits_{x}=0\\}$.

$\bullet$ The Lie superalgebra
${\mathfrak{svect}}^{\prime}(1|n):=[{\mathfrak{svect}}(1|n),{\mathfrak{svect}}(1|n)]$
is called the traceless special vectorial superalgebra.

$\bullet$ The deformation of ${\mathfrak{svect}}(0|m)$ is the Lie superalgebra

${\mathfrak{svect}}_{\lambda}(0|m)=\\{D\in{\mathfrak{vect}}(0|m)\mid\mathop{\text{\rm
div}}\nolimits(1+\lambda\theta_{1}\cdot\ldots\cdot\theta_{m})D=0\\},$

where $p(\lambda)\equiv m\pmod{2}$ and if $p(\lambda)={\bar{1}}$, then
${\mathfrak{svect}}_{\lambda}(0|m)$ is considered over
${\mathbb{C}}[\lambda]$. It is called the deformed special (or divergence-
free) vectorial superalgebra. Clearly,
${\mathfrak{svect}}_{\lambda}(0|m)\simeq{\mathfrak{svect}}_{\mu}(0|m)$ if
$\lambda\neq 0$ and $\mu\neq 0$. Accordingly, we briefly denote these
deformations by $\widetilde{{\mathfrak{svect}}}(0|m)$, see Subsection 1.10.4.
Proposition.

3) Lie superalgebras that preserve Pfaff equations or, equivalently, non-
integrable distributions, contact and symplectic forms.

$\bullet$ Recall that a symplectic form is a non-degenerate closed
differential 2-form $\omega$. In agreement with the normal shapes of the
bilinear forms (35) and (37), the descriptions of normal shapes of the
symplectic forms (Darboux’s theorem, when on manifolds) were formulated in
[L1]; this claim was proved in [Shan]. The obstruction to reducing the non-
degenerate differential 2-form $\omega$ to the normal shape is $d\omega$, see
[BGLS]. If $\omega$ is odd, it is called, as A. Weil advised, periplectic.

$\bullet$ The contact form (resp., pericontact form) is a nowhere vanishing
differential 1-form $\alpha$ on a (super)manifold $M$ that singles out in the
tangent bundle $TM$ a maximally non-integrable distribution ${\mathcal{D}}$ of
codimension 1 (resp., $\varepsilon=0|1$), i.e., at every point $m\in M$, there
are given

(49) $\begin{array}[]{l}\text{a~{}subspace ${\mathcal{D}}_{m}$ of codimension
1 (resp., $\varepsilon=0|1$) in the tangent space $T_{m}M$, and}\\\
\text{a~{}differential 1-form $\alpha$ such that
$d\alpha|_{{\mathcal{D}}_{m}}$ is non-degenerate.}\end{array}$

Note that in Differential Geometry on odd-dimensional manifolds (over the
ground field ${\mathbb{K}}$), the contact form is often defined as a nowhere
vanishing differential 1-form $\alpha$ such that

(50) $\alpha\wedge(d\alpha)^{\wedge
n}=\lambda\mathop{\mathrm{{vol\/}}}\nolimits_{x}$ for some
$\lambda\in{\mathbb{K}}$.

Obviously, this definition cannot be superized since on supermanifolds there
are no differential forms of highest degree. Condition (50) is a property of
the contact form on a manifold, whereas (49) is a reasonable definition.

$\bullet$ Let $u=(t,p_{1},\dots,p_{n},q_{1},\dots,q_{n})$ be even
indeterminates and $\Theta=(\theta_{1},\dots,\theta_{m})$ be odd
indeterminates. The form

(51) $\widetilde{\alpha}_{1}=dt+\mathop{\sum}\limits_{1\leq i\leq
n}(p_{i}dq_{i}-q_{i}dp_{i})\ +\mathop{\sum}\limits_{1\leq j\leq
m}\theta_{j}d\theta_{j}\quad\text{and}\quad\widetilde{\omega}_{0}:=d\widetilde{\alpha}_{1}$

is called contact, and the form $\widetilde{\omega}_{0}$ is called symplectic.
Sometimes it is more convenient to redenote the $\theta_{j}$ and set
$\Theta=\begin{cases}(\xi,\eta),&\text{ if }\ m=2k\\\
(\xi,\eta,\theta),&\text{ if }\ m=2k+1,\end{cases}$ where

(52)
$\xi_{j}=\frac{1}{\sqrt{2}}(\theta_{j}-\sqrt{-1}\theta_{k+j}),\quad\eta_{j}=\frac{1}{\sqrt{2}}(\theta_{j}+\sqrt{-1}\theta_{k+j})\;\text{~{}
for~{}}\;j\leq
k=\lfloor\mathchoice{\raisebox{0.25pt}{$\dfrac{m}{2}$}}{\raisebox{0.25pt}{$\dfrac{m}{2}$}}{\raisebox{-0.5pt}{$\tfrac{m}{2}$}}{\raisebox{-0.5pt}{$\tfrac{m}{2}$}}\rfloor,\;\;\theta=\theta_{2k+1},$

and in place of $\widetilde{\omega}_{0}$ or $\widetilde{\alpha}_{1}$ take
$\alpha_{1}$ and $\omega_{0}=d\alpha_{1}$, respectively, where (we mostly
consider the cases where $\ell=0$ or $1$)

(53) $\alpha_{1}=dt+\mathop{\sum}\limits_{1\leq i\leq
n}(p_{i}dq_{i}-q_{i}dp_{i})+\mathop{\sum}\limits_{1\leq j\leq
k}(\xi_{j}d\eta_{j}+\eta_{j}d\xi_{j})+\begin{cases}0,&\text{ if }\ m=2k\\\
\mathop{\sum}\limits_{1\leq s\leq\ell}\theta_{s}d\theta_{s},&\text{ if }\
m=2k+\ell.\end{cases}$

The Lie superalgebra that preserves the distribution given by the Pfaff
equation

(54) $\tilde{\alpha}_{1}(X)=0$ for $X\in{\mathfrak{vect}}(2n+1|m)$

(or by an equivalent equation $\alpha_{1}(X)=0$), i.e., the superalgebra

(55) ${\mathfrak{k}}(2n+1|m):=\\{D\in{\mathfrak{vect}}(2n+1|m)\mid
L_{D}\alpha_{1}=f_{D}\alpha_{1}\text{ for some
}f_{D}\in{\mathbb{C}}[t,p,q,\theta]\\},$

is called the contact superalgebra. The divergence-free subalgebra of
${\mathfrak{k}}(2n+1|m)$ is the Lie superalgebra

(56) ${\mathfrak{po}}(2n|m)=\\{D\in{\mathfrak{k}}(2n+1|m)\mid
L_{D}\alpha_{1}=0\\}.$

It is called the Poisson superalgebra. (The Poisson superalgebra can be
interpreted as the Lie superalgebra that, over a symplectic supermanifold with
the symplectic form $d\alpha_{1}$, preserves the connection with form
$\alpha_{1}$ and curvature proportional to $d\alpha_{1}$ in a line bundle.)

The above “symmetric” expression of $\alpha_{1}$ is popular among algebraists.
However, in analytical mechanics and differential geometry, the following
expression of the form $\alpha_{1}$ (without odd coordinates, of course)
naturally appears:

(57) $\alpha_{1(2)}=dt-\mathop{\sum}\limits_{1\leq i\leq
n}p_{i}dq_{i}+\mathop{\sum}\limits_{1\leq j\leq
k}\xi_{j}d\eta_{j}+\begin{cases}0,&\text{ if }\ m=2k\\\ \theta d\theta,&\text{
if }\ m=2k+1.\end{cases}$

The form $\alpha_{1(2)}$ is the only reasonable expression of the contact form
over fields of characteristic 2, whereas the symmetric expression of the
contact forms is inapplicable: $d\alpha_{1}=2\omega=0$.

$\bullet$ Similarly, let $u=q=(q_{1},\dots,q_{n})$ and
$\theta=(\xi_{1},\dots,\xi_{n};\tau)$ be even and odd indeterminates,
respectively. Set

(58)
$\begin{array}[]{c}\alpha_{0}=d\tau+\mathop{\sum}\limits_{i}(\xi_{i}dq_{i}+q_{i}d\xi_{i}),\qquad\qquad\omega_{1}:=d\alpha_{0}\end{array}$

and call the form $\omega_{1}$, as A. Weil advised, periplectic; accordingly,
we call $\alpha_{0}$ pericontact.

The Lie superalgebra that preserves the distribution given by the Pfaff
equation

(59) $\alpha_{0}(X)=0$ for $X\in{\mathfrak{vect}}(n|n+1)$,

i.e., the superalgebra

(60) ${\mathfrak{m}}(n)=\\{D\in{\mathfrak{vect}}(n|n+1)\mid
L_{D}\alpha_{0}=f_{D}\cdot\alpha_{0}\text{ for some
}\;f_{D}\in{\mathbb{C}}[q,\xi,\tau]\\},$

is called the pericontact superalgebra.151515To call ${\mathfrak{m}}$ “odd
contact superalgebra” is misleading, since the even part of ${\mathfrak{m}}$
is non-zero. To say “the “odd” contact superalgebra” is OK, though a bit too
long, and the pericontact form is even.

####  2.7.1. On normal shapes of contact and symplectic/periplectic forms.

The restriction of $\omega_{0}$ to $\ker\alpha_{1}$ is the orthosymplectic
form $B_{ev}(m|2n)$; the restriction of $\widetilde{\omega}_{0}$ to
$\ker\widetilde{\alpha}_{1}$ is $B^{\prime}_{ev}(m|2n)$. Similarly, the
restriction of $\omega_{1}$ to $\ker\alpha_{0}$ is $B_{odd}(n|n)$. Clearly,
$\ker\alpha_{1}=\ker\widetilde{\alpha}_{1}$.

There is just one class of contact forms (both even and odd), whereas the
obstruction to reducibility of the non-degenerate bilinear form $\omega$ (even
or odd) to the normal shape is $d\omega$, see [BGLS].

####  2.7.2. Generating functions: contact series and their subalgebras

A laconic way to describe ${\mathfrak{k}}$, ${\mathfrak{m}}$ and their
subalgebras is via generating functions. We express the contact fields of
series ${\mathfrak{k}}$ (resp., ${\mathfrak{m}}$) in natural bases $K_{f}$
(resp., $M_{f}$).

$\bullet$ Odd form $\alpha_{1}$. For any $f\in{\mathbb{C}}[t,p,q,\theta]$, set

(61)
$K_{f}=(2-E)(f){\frac{\partial}{\partial{t}}}-H_{f}+{\frac{\partial{f}}{\partial{t}}}E,$

where $E=\mathop{\sum}\limits_{i}y_{i}{\frac{\partial}{\partial{y_{i}}}}$
(here the $y_{i}$ are all the coordinates except $t$) is the Euler operator,
and $H_{f}$ is the Hamiltonian field with Hamiltonian $f$ that preserves
$d\widetilde{\alpha}_{1}$:

(62) $H_{f}=\mathop{\sum}\limits_{i\leq
n}\left({\frac{\partial{f}}{\partial{p_{i}}}}{\frac{\partial}{\partial{q_{i}}}}-{\frac{\partial{f}}{\partial{q_{i}}}}{\frac{\partial}{\partial{p_{i}}}}\right)-(-1)^{p(f)}\left(\mathop{\sum}\limits_{j\leq
m}{\frac{\partial{f}}{\partial{\theta_{j}}}}{\frac{\partial}{\partial{\theta_{j}}}}\right).$

The choice of the form $\alpha_{1}$ instead of $\widetilde{\alpha}_{1}$
affects only the shape of $H_{f}$ which we give for $m=2k+\ell$:

(63) $H_{f}=\mathop{\sum}\limits_{i\leq
n}\left({\frac{\partial{f}}{\partial{p_{i}}}}{\frac{\partial}{\partial{q_{i}}}}-{\frac{\partial{f}}{\partial{q_{i}}}}{\frac{\partial}{\partial{p_{i}}}}\right)-(-1)^{p(f)}\left(\mathop{\sum}\limits_{j\leq
k}\left({\frac{\partial{f}}{\partial{\xi_{j}}}}{\frac{\partial}{\partial{\eta_{j}}}}+{\frac{\partial{f}}{\partial{\eta_{j}}}}{\frac{\partial}{\partial{\xi_{j}}}}\right)+\mathop{\sum}\limits_{j\leq\ell}{\frac{\partial{f}}{\partial{\theta_{j}}}}{\frac{\partial}{\partial{\theta_{j}}}}\right).$

The expression of the contact field corresponding to the form $\alpha_{1}$ or
$\widetilde{\alpha}_{1}$ is

(64)
$K_{f}=(2-E)(f){\frac{\partial}{\partial{t}}}-H_{f}+{\frac{\partial{f}}{\partial{t}}}E,$

where
$E=\mathop{\sum}\limits_{i}p_{i}{\frac{\partial}{\partial{p_{i}}}}+\mathop{\sum}\limits_{j}\xi_{j}{\frac{\partial}{\partial{\xi_{j}}}}$,
and $H_{f}$ is the Hamiltonian field with Hamiltonian $f$ that preserves
$d\widetilde{\alpha}_{1}$, see (62).

$\bullet$ Even form $\alpha_{0}$. For any $f\in{\mathbb{C}}[q,\xi,\tau]$, set

(65)
$M_{f}=(2-E)(f){\frac{\partial}{\partial{\tau}}}-Le_{f}-(-1)^{p(f)}\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial f}{\partial\tau}$}}E,$

where $E=\mathop{\sum}\limits_{i}y_{i}{\frac{\partial}{\partial{y_{i}}}}$
(here the $y_{i}$ are all the coordinates except $\tau$), and

(66) $Le_{f}=\mathop{\sum}\limits_{i\leq
n}\left({\frac{\partial{f}}{\partial{q_{i}}}}\
{\frac{\partial}{\partial{\xi_{i}}}}+(-1)^{p(f)}{\frac{\partial{f}}{\partial{\xi_{i}}}}\
{\frac{\partial}{\partial{q_{i}}}}\right).$

Since

(67)
$\begin{array}[]{l}L_{K_{f}}(\alpha_{1})=2{\frac{\partial{f}}{\partial{t}}}\alpha_{1}=K_{1}(f)\alpha_{1},\\\
L_{M_{f}}(\alpha_{0})=-(-1)^{p(f)}2{\frac{\partial{f}}{\partial{\tau}}}\alpha_{0}=-(-1)^{p(f)}M_{1}(f)\alpha_{0},\end{array}$

it follows that $K_{f}\in{\mathfrak{k}}(2n+1|m)$ and
$M_{f}\in{\mathfrak{m}}(n)$. Note that

$p(Le_{f})=p(M_{f})=p(f)+{\bar{1}}.$

$\bullet$ To the (super)commutators $[K_{f},K_{g}]$ and $[M_{f},M_{g}]$ there
correspond contact brackets of the generating functions:

$\begin{array}[]{l}{}[K_{f},K_{g}]=K_{\\{f,\;g\\}_{k.b.}}\text{~{}~{}and~{}~{}}{}[M_{f},M_{g}]=M_{\\{f,\;g\\}_{m.b.}}.\end{array}$

To give the explicit formulas for the contact brackets, we first define the
brackets on functions that do not depend on $t$ (resp., $\tau$).

The Poisson bracket $\\{-,-\\}_{P.b.}$ in the realization with the form
$\widetilde{\omega}_{0}$ is given by the formula

(68) $\begin{array}[]{ll}\\{f,g\\}_{P.b.}&=\mathop{\sum}_{i\leq n}\
\bigg{(}{\frac{\partial{f}}{\partial{p_{i}}}}\
{\frac{\partial{g}}{\partial{q_{i}}}}-{\frac{\partial{f}}{\partial{q_{i}}}}{\frac{\partial{g}}{\partial{p_{i}}}}\bigg{)}-\\\
&(-1)^{p(f)}\mathop{\sum}_{j\leq m}\
{\frac{\partial{f}}{\partial{\theta_{j}}}}\
{\frac{\partial{g}}{\partial{\theta_{j}}}}\text{ for any
}f,g\in{\mathbb{C}}[p,q,\theta]\end{array}$

and in the realization with the form $\omega_{0}$ for $m=2k+1$ it is given by
the formula

(69) $\begin{array}[]{ll}\\{f,g\\}_{P.b.}&=\mathop{\sum}\limits_{i\leq n}\
\bigg{(}{\frac{\partial{f}}{\partial{p_{i}}}}\
{\frac{\partial{g}}{\partial{q_{i}}}}-\ {\frac{\partial{f}}{\partial{q_{i}}}}\
{\frac{\partial{g}}{\partial{p_{i}}}}\bigg{)}-\\\
&(-1)^{p(f)}\bigg{(}\mathop{\sum}\limits_{j\leq
m}\left({\frac{\partial{f}}{\partial{\xi_{j}}}}\
{\frac{\partial{g}}{\partial{\eta_{j}}}}+{\frac{\partial{f}}{\partial{\eta_{j}}}}\
{\frac{\partial{g}}{\partial{\xi_{j}}}}\right)+{\frac{\partial{f}}{\partial{\theta}}}\
{\frac{\partial{g}}{\partial{\theta}}}\bigg{)}\text{ for any
}f,g\in{\mathbb{C}}[p,q,\xi,\eta,\theta].\end{array}$

The Buttin bracket $\\{-,-\\}_{B.b.}$ is given by the formula

(70) $\\{f,g\\}_{B.b.}=\mathop{\sum}\limits_{i\leq n}\
\bigg{(}{\frac{\partial{f}}{\partial{q_{i}}}}\
{\frac{\partial{g}}{\partial{\xi_{i}}}}+(-1)^{p(f)}\
{\frac{\partial{f}}{\partial{\xi_{i}}}}\
{\frac{\partial{g}}{\partial{q_{i}}}}\bigg{)}\text{ for any
}f,g\in{\mathbb{C}}[q,\xi].$

In terms of the Poisson and Buttin brackets, respectively, the (peri)contact
brackets are

(71)
$\\{f,g\\}_{k.b.}=(2-E)(f){\frac{\partial{g}}{\partial{t}}}-{\frac{\partial{f}}{\partial{t}}}(2-E)(g)-\\{f,g\\}_{P.b.}$

and

(72)
$\\{f,g\\}_{m.b.}=(2-E)(f){\frac{\partial{g}}{\partial{\tau}}}+(-1)^{p(f)}\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}(2-E)(g)-\\{f,g\\}_{B.b.}.$

####  2.7.3. Divergence-free subalgebras

Since, as is easy to calculate,

(73) $\mathop{\text{\rm div}}\nolimits K_{f}=(2n+2-m)K_{1}(f),$

it follows that the divergence-free subalgebra of the contact Lie superalgebra
either coincides with the whole algebra (for $m=2n+2$) or is the Poisson
superalgebra., cf. (56)

(74) ${\mathfrak{po}}(2n|m):=\\{K_{f}\in{\mathfrak{k}}(2n+1|m)\mid
K_{1}(f)=0\\}.$

Since for $(n|m)\neq(0|2)$ the codimension of
$({\mathfrak{k}}(2n+1|m)_{0})^{\prime}$ in ${\mathfrak{k}}(2n+1|m)_{0}$ is
equal to 1, there is exactly one (up to a non-zero factor) trace on
${\mathfrak{k}}(2n+1|m)_{0}$. Hence, there should be exactly one (up to a non-
zero factor) intrinsic divergence on ${\mathfrak{k}}(2n+1|m)$, see [BGLLS].
Setting $\mathop{\mathrm{tr}}\nolimits(K_{t})=1$, the divergence is given by
$\partial_{t}$ for any $n$ and $m$. For $m\neq 2n+2$, this intrinsic
divergence coincides with the restriction to ${\mathfrak{k}}$ of the
divergence on ${\mathfrak{vect}}$.

##### The case of ${\mathfrak{k}}(1|2)$

Although ${\mathfrak{k}}(1|2)$ is not W-graded, let us consider it for
completeness. This case is exceptional: the algebra ${\mathfrak{k}}(1|2)_{0}$
of degree-0 elements is commutative and $\dim{\mathfrak{k}}(1|2)_{0}=2$, so
there are 2 traces on ${\mathfrak{k}}(1|2)_{0}$, and hence 2 divergences on
${\mathfrak{k}}(1|2)$. Let $\alpha_{1}:=dt+\xi d\eta+\eta d\xi$. To the
operators

$\tilde{K}_{\xi}(f)=(-1)^{p(f)}(\partial_{\eta}-\xi\partial_{t})(f),\
\tilde{K}_{\eta}(f)=(-1)^{p(f)}(\partial_{\xi}-\eta\partial_{t})(f)\text{~{}~{}and~{}~{}}\tilde{K}_{1}=\partial_{t}$

there corresponds the second divergence

(75) $\mathop{\text{\rm
div}}\nolimits_{2}:=\tilde{K}_{\eta}\tilde{K}_{\xi}-\tilde{K}_{1}.$

Let $\beta$ be the symbol of the class of a pseudo-differential form161616If
the reader wishes to “visualize” this form, one can consider
$\beta:=[d\xi(d\eta)^{-1}]$. Actually, the expression of $\beta$ in terms of
$d\xi$ and $d\eta$ is irrelevant, we need to know only how the Lie derivative
acts on it. on which the Lie derivative acts as follows

(76)
$\begin{array}[]{l}L_{K_{f}}(\alpha_{1}^{a}\beta^{b})=(2a\partial_{t}(f)+(-1)^{p(f)}b\mathop{\text{\rm
div}}\nolimits_{2}(K_{f}))(\alpha_{1}^{a}\beta^{b})\\\
=((2a-(-1)^{p(f)}b)\partial_{t}(f)+(-1)^{p(f)}b\tilde{K}_{\eta}\tilde{K}_{\xi}(f))(\alpha_{1}^{a}\beta^{b}).\end{array}$

Let the space ${\mathcal{F}}_{a,b}$ of weighted densities over
${\mathfrak{k}}(1|2)$ be defined as the rank-1 ${\mathcal{F}}$-module
generated by $\alpha^{a/2}_{1}\beta^{b}$. (For the modified formula (76) when
$\mathop{\text{\rm char}}\nolimits{\mathbb{K}}=2$, see [BGLLS].)

##### The pericontact series

Here, the situation is more interesting: the divergence-free subalgebra is
simple.

Since the restriction of the divergence from ${\mathfrak{vect}}$ to
${\mathfrak{m}}$ is equal to

(77) $\mathop{\text{\rm div}}\nolimits
M_{f}=(-1)^{p(f)}2\left((1-E)\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}-\mathop{\sum}\limits_{i\leq
n}\frac{\partial^{2}f}{\partial q_{i}\partial\xi_{i}}\right),$

it follows that the divergence-free subalgebra of the pericontact superalgebra
is

(78)
${\mathfrak{sm}}(n)=\mathop{\mathrm{{Span}}}\nolimits\left(M_{f}\in{\mathfrak{m}}(n)\mid(1-E)\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}=\mathop{\sum}\limits_{i\leq
n}\frac{\partial^{2}f}{\partial q_{i}\partial\xi_{i}}\right).$

In particular,

(79) $\mathop{\text{\rm div}}\nolimits
Le_{f}=(-1)^{p(f)}2\mathop{\sum}\limits_{i\leq n}\frac{\partial^{2}f}{\partial
q_{i}\partial\xi_{i}}.$

The odd analog of the Laplacian, namely, the operator

(80) $\Delta:=\mathop{\sum}\limits_{i\leq n}\frac{\partial^{2}}{\partial
q_{i}\partial\xi_{i}}$

on a periplectic supermanifold appeared in physics under the name of BRST
operator, cf. [GPS], or Batalin-Vilkovysky operator. Observe that $\Delta$ is
the Fourier transform (with respect to the “ghost indeterminates” $\check{x}$
(the odd ones, if considered on manifolds) of the exterior differential $d$.

The divergence-free vector fields from ${\mathfrak{sle}}(n)$ are generated by
harmonic functions, i.e., such that $\Delta(f)=0$.

The Lie superalgebra

(81) $\begin{array}[]{c}{\mathfrak{b}}(n)=\\{D\in{\mathfrak{m}}(n)\mid
L_{D}\alpha_{0}=0\\}\end{array}$

is called the Buttin superalgebra in honor of C. Buttin, see Section 2.7.4. (A
geometric interpretation: over a periplectic supermanifold, i.e., over a
supermanifold with the periplectic form $d\alpha_{0}$, the Buttin superalgebra
is the Lie superalgebra that preserves the connection with form $\alpha_{0}$
and curvature proportional to $d\alpha_{0}$ in a line bundle of superrank
$\varepsilon=(0|1)$.)

The Lie superalgebras

(82)
$\begin{array}[]{c}{\mathfrak{sm}}(n)=\\{D\in{\mathfrak{m}}(n)\mid\mathop{\text{\rm
div}}\nolimits\ D=0\\},\\\
{\mathfrak{s}}{\mathfrak{b}}(n)=\\{D\in{\mathfrak{b}}(n)\mid\mathop{\text{\rm
div}}\nolimits\ D=0\\}\end{array}$

are called the divergence-free (or special) pericontact and special Buttin
superalgebras, respectively.

Lie superalgebras ${\mathfrak{sle}}(n)$, ${\mathfrak{s}}{\mathfrak{b}}(n)$ and
${\mathfrak{svect}}(1|n)$ have traceless ideals
${\mathfrak{sle}}^{\prime}(n)$, ${\mathfrak{s}}{\mathfrak{b}}^{\prime}(n)$ and
${\mathfrak{svect}}^{\prime}(1|n)$ of codimension 1 (i.e., $\varepsilon^{n}$,
$\varepsilon^{n+1}$ and $n$, respectively) defined from the exact sequences

(83)
$\begin{array}[]{c}0\longrightarrow{\mathfrak{sle}}^{\prime}(n)\longrightarrow{\mathfrak{sle}}(n)\longrightarrow{\mathbb{C}}\cdot\mathop{\mathrm{Le}}\nolimits_{\xi_{1}\dots\xi_{n}}\longrightarrow
0,\\\
0\longrightarrow{\mathfrak{s}}{\mathfrak{b}}^{\prime}(n)\longrightarrow{\mathfrak{s}}{\mathfrak{b}}(n)\longrightarrow{\mathbb{C}}\cdot
M_{\xi_{1}\dots\xi_{n}}\longrightarrow 0,\\\
0\longrightarrow{\mathfrak{svect}}^{\prime}(1|n)\longrightarrow{\mathfrak{svect}}(1|n)\longrightarrow{\mathbb{C}}\cdot\xi_{1}\dots\xi_{n}\partial_{t}\longrightarrow
0.\end{array}$

The Lie superalgebra of Hamiltonian fields (also called Hamiltonian
superalgebra) and its traceless subalgebra (defined only if $n=0$) are

(84)
$\begin{array}[]{l}{\mathfrak{h}}(2n|m):=\mathop{\mathrm{{Span}}}\nolimits(D\in{\mathfrak{vect}}(2n|m)\mid
L_{D}\omega_{0}=0),\\\
{\mathfrak{h}}^{\prime}(0|m):=\mathop{\mathrm{{Span}}}\nolimits(H_{f}\in{\mathfrak{h}}(0|m)\mid\int
f\mathop{\mathrm{{vol\/}}}\nolimits_{\theta}=0).\end{array}$

The “odd” analogues of the Lie superalgebra of Hamiltonian fields are the Lie
superalgebra of vector fields $\mathop{\mathrm{Le}}\nolimits_{f}$ introduced
in [L1], and its special (divergence-free) subalgebra:

(85)
$\begin{array}[]{l}{\mathfrak{le}}(n):=\mathop{\mathrm{{Span}}}\nolimits(D\in{\mathfrak{vect}}(n|n)\mid
L_{D}\omega_{1}=0),\\\
{\mathfrak{sle}}(n):=\mathop{\mathrm{{Span}}}\nolimits(D\in{\mathfrak{le}}(n)\mid\mathop{\text{\rm
div}}\nolimits D=0).\end{array}$

It is not difficult to prove the following isomorphisms (as superspaces):

(86)
$\begin{array}[]{rcl}{\mathfrak{k}}(2n+1|m)&\simeq&\mathop{\mathrm{{Span}}}\nolimits(K_{f}\mid
f\in{\mathbb{C}}[t,p,q,\xi]),\\\
{\mathfrak{le}}(n)&\simeq&\mathop{\mathrm{{Span}}}\nolimits(Le_{f}\mid
f\in{\mathbb{C}}[q,\xi]),\\\
{\mathfrak{m}}(n)&\simeq&\mathop{\mathrm{{Span}}}\nolimits(M_{f}\mid
f\in{\mathbb{C}}[\tau,q,\xi]),\\\
{\mathfrak{h}}(2n|m)&\simeq&\mathop{\mathrm{{Span}}}\nolimits(H_{f}\mid
f\in{\mathbb{C}}[p,q,\xi]).\end{array}$

We see that

$\begin{array}[]{l}{\mathfrak{po}}(0|m)\simeq\mathop{\mathrm{{Span}}}\nolimits(K_{f}\in{\mathfrak{k}}(1|m)\mid
K_{1}(f)=0),\\\
{\mathfrak{po}}^{\prime}(0|m)\simeq\mathop{\mathrm{{Span}}}\nolimits(K_{f}\in{\mathfrak{po}}(0|m)\mid\int
f\mathop{\mathrm{{vol\/}}}\nolimits_{\theta}=0),\\\
{\mathfrak{h}}^{\prime}(0|m)\simeq{\mathfrak{po}}^{\prime}(0|m)/{\mathbb{C}}\cdot
K_{1}.\end{array}$

####  2.7.4. On the history of the antibracket.

What we call here the “Buttin bracket” was discovered in pre-super era by
Schouten; Buttin was the first to prove that this bracket satisfies the super
anti-commutativity and Jacobi identities, see [Bu]. The interpretations of the
Buttin superalgebra similar to that of the Poisson algebra and of the elements
of ${\mathfrak{le}}$ as analogs of Hamiltonian vector fields was given in
[L1]. The Buttin bracket and “odd mechanics” introduced in [L1] were
rediscovered by Batalin with Vilkovisky (and, even earlier, by Zinn-Justin,
but his papers went even more unnoticed, see [FLSf], than [L1]). The Schouten
bracket gained a great deal of currency under the name antibracket, cf. [GPS].
The Schouten bracket$=$Antibracket$=$Buttin bracket was originally defined on
the superspace of multi-vector fields on a manifold, i.e., on the superspace
of sections of the exterior algebra (over the algebra ${\mathcal{F}}$ of
functions) of the tangent bundle, $\Gamma(\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(T(M)))\simeq\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}_{\mathcal{F}}(Vect(M))$. The explicit formula for the
Schouten bracket (hereafter, the hatted slot should be ignored, as usual) is
(mind the Sign Rule!)

(87) $\begin{array}[]{c}[X_{1}\wedge\dots\wedge\dots\wedge
X_{k},Y_{1}\wedge\dots\wedge Y_{l}]=\\\
\mathop{\sum}\limits_{i,j}(-1)^{k-j+i-1}[X_{i},Y_{j}]\wedge
X_{1}\wedge\dots\wedge\widehat{X}_{i}\wedge\dots\wedge X_{k}\wedge
Y_{1}\wedge\dots\wedge\widehat{Y}_{j}\wedge\dots\wedge Y_{l}.\end{array}$

With the help of the Sign Rule we easily superize formula (87), i.e., replace
any manifold $M$ by a supermanifold ${\mathcal{M}}$. Let $x$ and $\xi$ be the
even and odd coordinates on ${\mathcal{M}}$. Setting

(88)
$\theta_{i}=\Pi\left({\frac{\partial}{\partial{x_{i}}}}\right)=\check{x}_{i},\quad
q_{j}=\Pi\left({\frac{\partial}{\partial{\xi_{j}}}}\right)=\check{\xi}_{j}$

we get an identification of the Schouten bracket of multi-vector fields on
${\mathcal{M}}$ with the Buttin bracket of functions on the supermanifold
$\check{\mathcal{M}}$ with coordinates $x,\xi$ and $\check{x}$, $\check{\xi}$,
and the transformation rule of the checked variables induced by that of
unchecked ones via (88). Grozman discovered a deformation of the Schouten
bracket, see [G], but his result was not understood at that time.

####  2.7.5. The explicit form of the pericontact bracket

Set $F_{x}:={\frac{\partial}{\partial{x}}}F$. Let $M_{F}\alpha$ be a short for
$L_{M_{F}}\alpha$. Since $M_{F}\alpha_{0}=(-1)^{p(F)+1}2F_{\tau}\alpha_{0}$,
it follows that

(89)
$\begin{array}[]{l}(M_{F}M_{G}-(-1)^{(p(F)+1)(p(G)+1)}M_{G}M_{F})\alpha_{0}=\\\
\\{(-1)^{p(G)+1}2[(\xi
F_{\tau}+F_{q})G_{\xi\tau}+(-1)^{p(F)}(F_{\xi}-qF_{\tau})G_{q\tau}+(-1)^{p(G)(p(F)+1)}2G_{\tau}F_{\tau}]\\\
-(-1)^{(p(F)+1)p(G)}2[(\xi
G_{\tau}+G_{q})F_{\xi\tau}+(-1)^{p(G)}(G_{\xi}-qG_{\tau})F_{q\tau}+(-1)^{p(F)(p(G)+1)}2F_{\tau}G_{\tau}]\\}\alpha_{0}.\end{array}$

On the other hand, we have

(90) $((2-E)(F)G_{\tau})_{\tau}=2F_{\tau}G_{\tau}+\xi
F_{\tau\xi}G_{\tau}-qF_{\tau q}G_{\tau};\ $ (91)
$(F_{\tau}(2-E)(G))_{\tau}=(-1)^{p(F)+1}(2F_{\tau}G_{\tau}+F_{\tau}\xi
G_{\tau\xi}-F_{\tau}qG_{\tau q});\ $ (92)
$(F_{q}G_{\xi}+(-1)^{p(F)}F_{\xi}G_{q})_{\tau}=F_{\tau
q}G_{q}+(-1)^{p(F)}F_{q}G_{\tau\xi}-F_{\xi}G_{\tau q}.$

Clearly,

$(-1)^{p(F)p(G)}~{}\eqref{2.3.11}+(-1)^{p(G)}~{}\eqref{2.3.12}+(-1)^{p(F)+p(G)}~{}\eqref{2.3.13}=\\{\cdots\\},$

where $\\{\cdots\\}$ is the coefficient of $\alpha_{0}$ in the right-hand side
of (89). Therefore,

$\\{F,G\\}_{m.b.}=(-1)^{p(F)+p(G)}\left((2-E)(F)G_{\tau}+(-1)^{p(F)}F_{\tau}(2-E)(G)+\\{F,G\\}_{\text{B.b.}}\right).$

The arguments for the contact bracket are similar (with $\alpha_{0}$ replaced
with $\alpha_{1}$ or $\widetilde{\alpha}_{1}$, and the field $M_{f}$ with
$K_{f}$, respectively).

###  2.8. Cartan prolongations

Recall that the graded Lie superalgebra ${\mathfrak{b}}=\oplus_{k\geqslant-d}\
{\mathfrak{b}}_{k}$ is said to be transitive if for all $k\geq 0$ we have

$\\{x\in{\mathfrak{b}}_{k}\mid[x,{\mathfrak{b}}_{-}]=0\\}=0,\text{~{}~{}where
${\mathfrak{b}}_{-}:=\oplus_{k<0}\,{\mathfrak{b}}_{k}$}.$

Let ${\mathfrak{g}}_{-}=\mathop{\oplus}_{-d\leq i\leq-1}\ {\mathfrak{g}}_{i}$
be a nilpotent ${\mathbb{Z}}$-graded Lie (super)algebra and let
${\mathfrak{g}}_{0}$ be a Lie sub(super)algebra of the Lie (super)algebra
$\mathfrak{der}_{0}({\mathfrak{g}}_{-})$ of degree-0 derivations of
${\mathfrak{g}}_{-}$. The maximal transitive ${\mathbb{Z}}$-graded Lie
(super)algebra ${\mathfrak{g}}=\oplus_{k\geqslant-d}\ {\mathfrak{g}}_{k}$
whose non-positive part is the given
${\mathfrak{g}}_{-}\oplus{\mathfrak{g}}_{0}$ is called the Cartan prolong —
the result of Cartan prolongation — of the pair
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})$ and is denoted by
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*}$.

We can realize ${\mathfrak{g}}_{-}$ by elements of negative degree of
${\mathfrak{vect}}(n|m;\vec{r})$ and ${\mathfrak{g}}_{0}$ by degree-0 elements
in ${\mathfrak{vect}}(n|m;\vec{r})$ in a non-standard (see Subsection 2.9)
grading $\vec{r}$ of ${\mathfrak{vect}}(n|m)$, where $n|m={\rm
sdim}{\mathfrak{g}}_{-}$. Then, the degree-$k$ components of the Cartan
prolongation
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*}:=\mathop{\oplus}_{k\geq-d}\
{\mathfrak{g}}_{k}$ of the pair $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})$ are
obtained for any $k>0$ by setting

${\mathfrak{g}}_{k}:=\\{D\in{\mathfrak{vect}}(n|m;\vec{r})_{k}\mid[D,{\mathfrak{g}}_{i}]\subset{\mathfrak{g}}_{k+i}\text{~{}~{}for
any $i<0$}\\}.$

In what follows ${\mathbin{\hbox{\raise 1.72218pt\hbox{\bf.}}}}$ in
superscript denotes, as is now customary, the collection of all degrees, while
$*$ is reserved for dualization; in the subscripts we retain the old-fashioned
$*$ instead of ${\mathbin{\hbox{\raise 1.72218pt\hbox{\bf.}}}}$ to avoid
confusion with the punctuation marks.

Suppose that the ${\mathfrak{g}}_{0}$-module ${\mathfrak{g}}_{-1}$ is
faithful. Then, clearly,

(93)
$\begin{array}[]{ll}({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}&\subset{\mathfrak{vect}}(n|m)={\mathfrak{der}}~{}{\mathbb{C}}[x_{1},\dots,x_{n},\theta_{1},\dots\theta_{m}],\;\text{
where }\;n|m=\mathop{\mathrm{sdim}}\nolimits~{}{\mathfrak{g}}_{-1}\;\text{ and
}\\\ &{\mathfrak{g}}_{i}=\\{D\in{\mathfrak{vect}}(n|m)\mid\deg
D=i,[D,X]\in{\mathfrak{g}}_{i-1}\text{ for any
}X\in{\mathfrak{g}}_{-1}\\}.\end{array}$

It is subject to an easy verification that the Lie algebra structure on
${\mathfrak{vect}}(n|m)$ induces a Lie algebra structure on
$({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}$; actually,
$({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}$ has a natural Lie superalgebra
structure even if the ${\mathfrak{g}}_{0}$-module ${\mathfrak{g}}_{-1}$ is not
faithful.

####  2.8.1. Partial Cartan prolongation: involving positive components

Let ${\mathfrak{h}}_{1}\subset{\mathfrak{g}}_{1}$ be a proper
${\mathfrak{g}}_{0}$-submodule such that
$[{\mathfrak{g}}_{-1},{\mathfrak{h}}_{1}]={\mathfrak{g}}_{0}$. If such
${\mathfrak{h}}_{1}$ exists (usually,
$[{\mathfrak{g}}_{-1},{\mathfrak{h}}_{1}]\subsetneq{\mathfrak{g}}_{0}$),
define the 2nd prolongation of $(\mathop{\oplus}_{i\leq
0}{\mathfrak{g}}_{i})\oplus{\mathfrak{h}}_{1}$ to be

${\mathfrak{h}}_{2}:=\\{D\in{\mathfrak{g}}_{2}\mid[D,{\mathfrak{g}}_{-1}]\subset{\mathfrak{h}}_{1}\\}.$

The terms ${\mathfrak{h}}_{i}$, where $i>2$, are similarly defined by
induction. Set ${\mathfrak{h}}_{i}:={\mathfrak{g}}_{i}$ for $i\leq 0$ and call
${\mathfrak{h}}_{*}:=\oplus{\mathfrak{h}}_{i}$ the partial Cartan prolong
involving positive components.

The Lie superalgebra ${\mathfrak{vect}}(1|n;n)$ is a subalgebra of
${\mathfrak{k}}(1|2n;n)$. The former is obtained as the Cartan prolong of the
same nonpositive part as ${\mathfrak{k}}(1|2n;n)$ and a submodule of
${\mathfrak{k}}(1|2n;n)_{1}$. The simple exceptional superalgebra
${\mathfrak{k}}{\mathfrak{as}}$ introduced in [Sh5, Sh14], see also line
$N=12$ in Table (131), is another example.

####  2.8.2. Lie superalgebras of vector fields as Cartan prolongs

The superization of the constructions from Subsection 2.8 are straightforward,
via the Sign Rule. We thus get Lie superalgebras

(94)
$\begin{array}[]{l}{\mathfrak{vect}}(m|n)=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{gl}}(m|n))_{*},\\\
{\mathfrak{svect}}(m|n)=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{sl}}(m|n))_{*},\\\
{\mathfrak{h}}(2m|n)=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{osp}}^{a}(m|2n))_{*},\\\
{\mathfrak{le}}(n)=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}^{a}(n))_{*},\\\
{\mathfrak{s}}{\mathfrak{le}}(n)=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{spe}}^{a}(n))_{*}.\\\
\end{array}$

For an algorithm for constructing prolongs, see [Shch]; it is implemented in
the code package SuperLie, see [Gr].

Note that $(\mathop{\mathrm{id}}\nolimits,{\mathfrak{osp}}^{sy}(m|2n))_{*}$
and $(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}^{sy}(n))_{*}$ are finite-
dimensional, unlike

$(\Pi(\mathop{\mathrm{id}}\nolimits),{\mathfrak{osp}}^{sy}(m|2n))_{*}=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{osp}}^{a}(m|2n))_{*}$
and
$(\Pi(\mathop{\mathrm{id}}\nolimits),{\mathfrak{pe}}^{sy}(n))_{*}=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}^{a}(n))_{*}$.

There are two superizations of the contact series: ${\mathfrak{k}}$ and
${\mathfrak{m}}$. Let us describe them again.

$\bullet$ Let the space of the Heisenberg Lie superalgebra
${\mathfrak{hei}}(2n|m)$ be the direct sum of a $(2n,m)$-dimensional
superspace $W$ endowed with a non-degenerate anti-symmetric even bilinear form
$B$ and a $(1,0)$-dimensional space spanned by $z$. Define the bracket in
${\mathfrak{hei}}$ by the formula

(95) $z$ is in the center and $[v,w]=B(v,w)\cdot z$ for any $v,w\in W$.

Clearly,

(96)
${\mathfrak{k}}(2n+1|m)=({\mathfrak{hei}}(2n|m),{\mathfrak{c}}({\mathfrak{osp}}^{a}(m|2n)))_{*}\,.$

More generally, given ${\mathfrak{hei}}(2n|m)$ and a subalgebra
${\mathfrak{g}}$ of ${\mathfrak{c}}({\mathfrak{osp}}^{a}(m|2n))$, we call
$({\mathfrak{hei}}(2n|m),{\mathfrak{g}})_{*}$ the $k$-prolong of
$(W,{\mathfrak{g}})$, where $W$ is the tautological
${\mathfrak{osp}}^{a}(m|2n)={\mathfrak{osp}}(W)$-module.

$\bullet$ The “odd” analog of ${\mathfrak{k}}$ is associated with the
following “odd” analog of ${\mathfrak{hei}}(2n|m)$. Denote by
${\mathfrak{ba}}(n)$ the antibracket Lie superalgebra (${\mathfrak{ba}}$
stands for Anti-Bracket read backwards) whose space is
$W\oplus{\mathbb{C}}\cdot z$, where $W$ is an $n|n$-dimensional superspace
endowed with a non-degenerate anti-symmetric odd bilinear form $B$ and $z$ is
odd; the bracket in ${\mathfrak{ba}}(n)$ is given by the relations

(97) $z$ lies in the center and $[v,w]=B(v,w)\cdot z$ for any $v,w\in W$.

Clearly,

(98)
${\mathfrak{m}}(n)=({\mathfrak{ba}}(n),{\mathfrak{c}}({\mathfrak{pe}}^{a}(n)))_{*}\,.$

More generally, given ${\mathfrak{ba}}(n)$ and a subalgebra ${\mathfrak{g}}$
of ${\mathfrak{c}}{\mathfrak{pe}}^{a}(n)$, we call
$({\mathfrak{ba}}(n),{\mathfrak{g}})_{*}$ the $m$-prolong of
$(W,{\mathfrak{g}})$, where $W$ is the tautological
${\mathfrak{pe}}^{a}(n)={\mathfrak{pe}}(W)$-module.

Generally, given a non-degenerate ${\mathfrak{g}}$-invariant form $B$ on a
superspace $W$ and a Lie subsuperalgebra
${\mathfrak{g}}\subset{\mathfrak{aut}}(B)$, we call any of the above
generalized prolongations (96) and (98) the $mk$-prolongation of the pair
$(W,{\mathfrak{g}})$ and denote it by $(W,{\mathfrak{g}})_{*}^{mk}$.

##### On the history of Cartan prolongations

The above-described procedure is sometimes called a generalized prolongation
because the initial Cartan prolongation was defined for $d=1$ only. Or rather
it was customary to write so without reading works by Cartan and Killing who
did consider the above-described generalized Cartan prolongation, see [Cart,
Ki, Co].

In the non-super setting, the “generalized” Cartan prolongation was introduced
by Cartan (see [Cart]) or, perhaps, Killing (see [Ki]). The Cartan
prolongation was rediscovered in two steps by Tanaka [T], see [Y]. The
emphasis of these papers is different from ours, this delayed recognition of
the similarity of constructions. The partial prolongation seems to be first
described in [ALSh, Sh14]. It also appears in the classification of simple Lie
(super)algebras in characteristic $p>0$, see [BGLLS], where the correct
definition of Cartan prolongation in characteristic $p>0$ was first given,
after several approximations to the truth in earlier papers.

###  2.9. Nonstandard realizations

The Step 7 of our proof (see Subsection 1.8) implies that the nonstandard
gradings listed in table (99) exhaust all the W-gradings of all the simple
vectorial Lie superalgebras. In particular, the gradings in the series
${\mathfrak{vect}}$ induce the gradings in the series ${\mathfrak{svect}}$,
${\mathfrak{svect}}^{\prime}$ and the exceptional families
${\mathfrak{v}}{\mathfrak{le}}(4|3)$ and ${\mathfrak{v}}{\mathfrak{as}}(4|4)$;
the gradings in ${\mathfrak{m}}$ induce the gradings in
${\mathfrak{b}}_{\lambda}$, ${\mathfrak{le}}$, ${\mathfrak{sle}}$,
${\mathfrak{sle}}^{\prime}$, ${\mathfrak{b}}$, ${\mathfrak{s}}{\mathfrak{b}}$,
${\mathfrak{s}}{\mathfrak{b}}^{\prime}$ and the exceptional family
${\mathfrak{m}}{\mathfrak{b}}$; the gradings in ${\mathfrak{k}}$ induce the
gradings in ${\mathfrak{po}}$, ${\mathfrak{h}}$, ${\mathfrak{h}}^{\prime}$ and
the exceptional families ${\mathfrak{k}}{\mathfrak{as}}$ and
${\mathfrak{k}}{\mathfrak{le}}$.

The ${\mathbb{Z}}$-gradings of vectorial Lie superalgebras are defined by the
vector $\vec{r}$ of degrees of the indeterminates. For W-gradings, this vector
can be shorthanded to a symbol or the number $r$ of degree-0 indeterminates;
we do not indicate $r$ if $r=0$. Let the indeterminates $t$ and $u_{i}$ be
even, let $\tau$ and the $\theta_{j}$ be odd. In Table (99), we consider
${\mathfrak{k}}(2n+1|m)$ preserving the Pfaff equation with the 1-form
$\alpha_{1})$; here $(t;p,q)$ are the even indeterminates, the odd
indeterminates being $\tau;\theta,\xi,\eta$, see eq. (53). For the series
${\mathfrak{m}}$, the indeterminates are denoted as in eq. (58).

The standard realizations correspond to $r=0$, they are marked by an $(*)$.
Note that the codimension of ${\mathcal{L}}_{0}$ attains its minimum in the
standard realization.

(99) Lie superalgebra |  its ${\mathbb{Z}}$-grading
---|---
${\mathfrak{vect}}(n|m;r)$, |  $\deg u_{i}=\deg\xi_{j}=1$ for any $i,j$ $(*)$
$0\leq r\leq m$ |  $\deg\xi_{j}=0$ for $1\leq j\leq r;$
|  $\deg u_{i}=\deg\xi_{r+s}=1$ for any $i,s$
${\mathfrak{m}}(n;r),$ |  $\deg\tau=2$, $\deg q_{i}=\deg\xi_{i}=1$ for any $i$ $(*)$
$\;0\leq r<n-1$ |  $\deg\tau=\deg q_{i}=2$, $\deg\xi_{i}=0$ for $1\leq i\leq r<n$;
|  $\deg q_{r+j}=\deg\xi_{r+j}=1$ for any $j$
${\mathfrak{m}}(n;n)$ |  $\deg\tau=\deg q_{i}=1$, $\deg\xi_{i}=0$ for $1\leq i\leq n$
${\mathfrak{k}}(2n+1|m;r)$, |  $\deg t=2$, whereas, for any $i,j,s$ $(*)$
see eq. (53); | $\deg p_{i}=\deg q_{i}=\deg\xi_{j}=\deg\eta_{j}=\deg\theta_{s}=1$
$0\leq r\leq[\frac{m}{2}]$ |  $\deg t=\deg\xi_{i}=2$, $\deg\eta_{i}=0$ for $1\leq i\leq r\leq[\frac{m}{2}]$;
$r\neq k-1$ for $m=2k$ and $n=0$ |  $\deg p_{i}=\deg q_{i}=\deg\theta_{j}=1$ for $j\geq 1$ and all $i$
${\mathfrak{k}}(1|2m;m)$ |  $\deg t=\deg\xi_{i}=1$, $\deg\eta_{i}=0$ for $1\leq i\leq m$

In table (100), the five families of exceptional simple vectorial Lie
superalgebras are given in their W-gradings as Cartan prolongs
$({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}$ or generalized Cartan prolongs
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*}^{mk}$. For depth $2$, for
${\mathfrak{g}}_{-}=\mathop{\oplus}_{-2\leq i\leq-1}{\mathfrak{g}}_{i}$, we
write $({\mathfrak{g}}_{-2},{\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}^{mk}$
instead of $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*}^{mk}$. For
${\mathfrak{k}}{\mathfrak{a}}{\mathfrak{s}}^{\xi}$, the meaning of the
notation $r=0$, or $1\xi$, or $3\xi$ is clear: none, or one, or three of the
$\xi$’s have degree 0 (and the corresponding $\eta$’s acquire degree 2). Table
(100) contains exceptional superalgebras whose grading $r$ is described in
table (101), indeterminates and their respective degrees in the regrading
$R(r)$ of the ambient superalgebra and describes these regradings:

(the degrees of the even indeterminates $\mid$ the degrees of the odd indeterminates). (100) hv ${\mathfrak{v}}{\mathfrak{le}}(4|3)$ | $R(0)=\begin{pmatrix}x_{1}&x_{2}&x_{3}&y&|&\xi_{1}&\xi_{2}&\xi_{3}\\\ 1&1&1&1&|&1&1&1\\\ \end{pmatrix}$
---|---
vv ${\mathfrak{v}}{\mathfrak{le}}(5|4)$ | $R(1)=\begin{pmatrix}x_{1}&x_{2}&x_{3}&y&|&\xi_{1}&\xi_{2}&\xi_{3}\\\ 2&1&1&0&|&0&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{v}}{\mathfrak{le}}(3|6)$ | $R(K)=\begin{pmatrix}x_{1}&x_{2}&x_{3}&y&|&\xi_{1}&\xi_{2}&\xi_{3}\\\ 2&2&2&0&|&1&1&1\\\ \end{pmatrix}$
vhv ${\mathfrak{m}}{\mathfrak{b}}(4|5)$ | $R(0)=\begin{pmatrix}x_{0}&x_{1}&x_{2}&x_{3}&|&\xi_{0}&\xi_{1}&\xi_{2}&\xi_{3}&\tau\\\ 1&1&1&1&|&1&1&1&1&2\\\ \end{pmatrix}$
vv ${\mathfrak{m}}{\mathfrak{b}}(5|6)$ | $R(1)=\begin{pmatrix}x_{0}&x_{1}&x_{2}&x_{3}&|&\xi_{0}&\xi_{1}&\xi_{2}&\xi_{3}&\tau\\\ 0&2&1&1&|&2&0&1&1&2\\\ \end{pmatrix}$
vv ${\mathfrak{m}}{\mathfrak{b}}(3|8)$ | $R(K)=\begin{pmatrix}x_{0}&x_{1}&x_{2}&x_{3}&|&\xi_{0}&\xi_{1}&\xi_{2}&\xi_{3}&\tau\\\ 0&2&2&2&|&3&1&1&1&3\\\ \end{pmatrix}$
vhv ${\mathfrak{k}}{\mathfrak{as}}(1|6)$ | $R(0)=\begin{pmatrix}t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 2&|&1&1&1&1&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{k}}{\mathfrak{as}}(5|5)$ | $R(1\xi)=\begin{pmatrix}t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 2&|&0&1&1&2&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{k}}{\mathfrak{as}}(4|4)$ | $R(3\xi)=\begin{pmatrix}t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 1&|&0&0&0&1&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{k}}{\mathfrak{as}}(4|3)$ | $R(3\eta)=\begin{pmatrix}t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 1&|&1&1&1&0&0&0\\\ \end{pmatrix}$
vhv ${\mathfrak{k}}{\mathfrak{le}}(9|6)$ | $R(0)=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}q_{1}&q_{2}&q_{3}&q_{4}&p_{1}&p_{2}&p_{3}&p_{4}&t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 1&1&1&1&1&1&1&1&2&|&1&1&1&1&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{k}}{\mathfrak{le}}(11|9)$ | $R(2)=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}q_{1}&q_{2}&q_{3}&q_{4}&p_{1}&p_{2}&p_{3}&p_{4}&t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 1&1&2&2&1&1&0&0&2&|&0&1&1&2&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{k}}{\mathfrak{le}}(5|10)$ | $R(K)=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}q_{1}&q_{2}&q_{3}&q_{4}&p_{1}&p_{2}&p_{3}&p_{4}&t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 2&2&2&2&1&1&1&1&2&|&1&1&1&1&1&1\\\ \end{pmatrix}$
vv ${\mathfrak{k}}{\mathfrak{le}}(9|11)$ | $R(CK)=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}q_{1}&q_{2}&q_{3}&q_{4}&p_{1}&p_{2}&p_{3}&p_{4}&t&|&\xi_{1}&\xi_{2}&\xi_{3}&\eta_{1}&\eta_{2}&\eta_{3}\\\ 3&2&2&2&0&1&1&1&3&|&2&2&2&1&1&1\\\ \end{pmatrix}$
vh |

In Table (101), indicated is also one of the Lie superalgebras from the list
of series (22) as an ambient which contains the exceptional one as a maximal
subalgebra. The W-graded superalgebras of depth $3$ appear as regradings of
the listed ones at certain values of $\vec{r}$; the corresponding terms
${\mathfrak{g}}_{i}$ for $i\leq 0$ are given in Subsection 2.15. Let the label
$CK$ commemorate an exceptional grading found by Cheng and Kac, see [CK2], see
Theorem 2.14.1. Theorem.

(101) ${\mathfrak{v}}{\mathfrak{le}}(4|3;r)=(\Pi(\Lambda(3))/{\mathbb{C}}\cdot 1,{\mathfrak{c}}{\mathfrak{vect}}(0|3))_{*}$ | $\subset{\mathfrak{vect}}(4|3;R(r))$ | $r=0,1,K$
---|---|---
${\mathfrak{v}}{\mathfrak{as}}(4|4)=(\mathop{\mathrm{spin}}\nolimits,{\mathfrak{as}})_{*}$, see Subsection 2.11 | $\subset{\mathfrak{vect}}(4|4)$ |
${\mathfrak{k}}{\mathfrak{as}}^{\xi}(1|6;r)$ | $\subset{\mathfrak{k}}(1|6;r)$ | $r=0,1\xi,3\xi$
${\mathfrak{k}}{\mathfrak{as}}^{\xi}(1|6;3\eta)=(\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|3),{\mathfrak{c}}({\mathfrak{vect}}(0|3)))_{*}$ | $\subset{\mathfrak{svect}}(4|3)$ | $r=3\eta$
${\mathfrak{m}}{\mathfrak{b}}(4|5;r)=({\mathfrak{ba}}(4),{\mathfrak{c}}{\mathfrak{vect}}(0|3))_{*}^{m}$ | $\subset{\mathfrak{m}}(4|5;R(r))$ | $r=0,1,K$
${\mathfrak{k}}{\mathfrak{le}}(9|6;r)=({\mathfrak{hei}}(8|6),{\mathfrak{svect}}(0|4)_{3,4})_{*}^{k}$ | $\subset{\mathfrak{k}}(9|6;r)$ |  $r=0,2$, CK
${\mathfrak{k}}{\mathfrak{le}}(9|6;K)=(\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(5)},\Lambda^{2}(\mathop{\mathrm{id}}\nolimits^{*}_{{\mathfrak{sl}}(5)}),{\mathfrak{sl}}(5))_{*}^{k}$ | $\subset{\mathfrak{svect}}(5|10;R(K))$ | $r=K$

###  2.10. The modules of tensor fields

To advance further, we have to recall the definition of the modules of tensor
fields over ${\mathfrak{vect}}(m|n)$ and its subalgebras, see [BL4, L2, GLS3].

Let ${\mathfrak{g}}={\mathfrak{vect}}(m|n)$ and
${\mathfrak{g}}_{\geq}=\mathop{\oplus}_{i\geq 0}\ {\mathfrak{g}}_{i}$. For any
other ${\mathbb{Z}}$-graded vectorial Lie superalgebra for whose component
${\mathfrak{g}}_{0}$ the notion of lowest weight can be defined, the
construction is identical.

Let $V$ be the ${\mathfrak{g}}_{0}$-module with the lowest weight
$\lambda=\mathop{\mathrm{lwt}}\nolimits(V)$ and even lowest weight vector.
Make $V$ into a ${\mathfrak{g}}_{\geq}$-module by setting
${\mathfrak{g}}_{+}\cdot V=0$ for ${\mathfrak{g}}_{+}=\mathop{\oplus}_{i>0}\
{\mathfrak{g}}_{i}$. Let us realize
${\mathfrak{g}}={\mathfrak{g}}_{-}\oplus{\mathfrak{g}}_{0}\oplus{\mathfrak{g}}_{+}$
by vector fields on the linear supermanifold ${\mathcal{C}}^{m|n}$
corresponding to the superspace ${\mathfrak{g}}_{-}:=\mathop{\oplus}_{i<0}\
{\mathfrak{g}}_{i}$ with coordinates $x=(u,\xi)$. The superspace
$T(V):=\mathop{\mathrm{Hom}}\nolimits_{U({\mathfrak{g}}_{\geq})}(U({\mathfrak{g}}),V)$
is isomorphic, due to the Poincaré–Birkhoff–Witt theorem, to
${{\mathbb{C}}}[[x]]\otimes V$. Its elements admit a natural interpretation as
formal tensor fields of type $V$. When $\lambda=(a,\dots,a)$ we will simply
write $T(\vec{a})$ instead of $T(\lambda)$. We will usually need
${\mathfrak{g}}$-modules coinduced from irreducible
${\mathfrak{g}}_{0}$-modules.

####  2.10.1. Examples.

As ${\mathfrak{vect}}(m|n)$\- and ${\mathfrak{svect}}(m|n)$-modules,
${\mathfrak{vect}}(m|n)=T(\mathop{\mathrm{id}}\nolimits)$.

$T(\vec{0})$ is the superspace of functions;

$\mathop{\mathrm{{Vol\/}}}\nolimits(m|n):=T(\underbrace{1,\dots,1}_{m};\underbrace{-1,\dots,-1}_{n})$,
where the semicolon separates the first $m$ (“even”) coordinates of the weight
with respect to the matrix units $E_{ii}$ of ${\mathfrak{gl}}(m|n)$, is the
superspace of volume forms, see [LSoS, Ch.4] or [Del, Ch.1, §1.11].

We denote the generator of $\mathop{\mathrm{{Vol\/}}}\nolimits(m|n)$,
considered as a ${\mathbb{C}}[[x]]$\- or ${\mathbb{C}}[x]$-module,
corresponding to the ordered set of coordinates $x$ by
$\mathop{\mathrm{{vol\/}}}\nolimits(x)$ or
$\mathop{\mathrm{{vol\/}}}\nolimits_{x}$. The space of $\lambda$-densities is

$\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(m|n):=T(\underbrace{\lambda,\dots,\lambda}_{m};\underbrace{-\lambda,\dots,-\lambda}_{n}).$

In particular,
$\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(m|0)=T(\vec{\lambda})$, whereas
$\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|n)=T(\overrightarrow{-\lambda})$.
In dimension $(0|n)$, we set

(102)
$\begin{array}[]{l}\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|n):=\\{v\in\mathop{\mathrm{{Vol\/}}}\nolimits(0|n)\mid\int
v=0\\},\\\ T_{0}(\vec{0}):=\Lambda(n)/{\mathbb{C}}\cdot 1.\end{array}$

As ${\mathfrak{svect}}(m|n)$-modules,
$\mathop{\mathrm{{Vol\/}}}\nolimits\simeq T(\vec{0})$. So, we can define the
${\mathfrak{svect}}(0|n)$-module

(103)
$T^{0}_{0}(\vec{0}):=\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|n)/{\mathbb{C}}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}.$

##### Remark: Volume element as a quotient

To view the volume element $\mathop{\mathrm{{vol\/}}}\nolimits(x)$ as
“$d^{m}ud^{n}\xi$” is wrong, but in some works this wrong notation is used to
this day. Observe that the superdeterminant of the Jacobi matrix can never
appear as a factor under the changes of “variables”, i.e., indeterminates, in
“$d^{m}ud^{n}\xi$”. We can try to use the usual notation of differentials
provided all the differentials anticommute and $\deg du_{i}=1$ while $\deg
d\xi_{j}=-1$ for all $i,j$. Then, at least the linear transformations that do
not intermix the even $u$’s with the odd $\xi$’s multiply the volume element
$\mathop{\mathrm{{vol\/}}}\nolimits_{x}$, viewed as the fraction
$\frac{du_{1}\cdot...\cdot du_{m}}{d\xi_{1}\cdot...\cdot d\xi_{n}}$, by the
correct factor, the Berezinian $\mathop{\mathrm{ber}}\nolimits(A)$, see
[Lber], of the transformation $A$. But how to justify this? Let $x:=(u,\xi)$.
If we consider the exterior differential forms, then the $dX_{i}$’s super
anticommute, hence, the $d\xi_{i}$ commute and so should $(d\xi_{i})^{-1}$;
whereas if we consider the symmetric product of the differentials, as in
metrics, then the $dx_{i}$’s supercommute, hence, the $du_{i}$ commute, so
neither the exterior, nor the symmetric product is what we need: all factors
should anticommute.

Note that the ${\frac{\partial}{\partial{\xi_{i}}}}$ anticommute, and
moreover, from the point of view of linear transformations,
${\frac{\partial}{\partial{\xi_{i}}}}=\frac{1}{d\xi_{i}}$. The expression
$du_{1}\cdot...\cdot
du_{m}\cdot{\frac{\partial}{\partial{\xi_{1}}}}\cdot\ldots\cdot{\frac{\partial}{\partial{\xi_{n}}}}$
is, nevertheless, still wrong: many transformations
$A:(u,\xi)\longmapsto(v,\eta)$ send $du_{1}\cdot....\cdot
du_{m}\cdot{\frac{\partial}{\partial{\xi_{1}}}}\cdot...\cdot{\frac{\partial}{\partial{\xi_{n}}}}$
to the correct element, $\mathop{\mathrm{ber}}\nolimits(A)(dv_{1}\cdot...\cdot
dv_{m}\cdot{\frac{\partial}{\partial{\eta_{1}}}}\cdot...\cdot{\frac{\partial}{\partial{\eta_{n}}}})$
plus extra terms. Indeed, the term $dv_{1}\cdot...\cdot
dv_{m}\cdot{\frac{\partial}{\partial{\eta_{1}}}}\cdot...\cdot{\frac{\partial}{\partial{\eta_{n}}}}$
is the highest weight vector of an indecomposable
${\mathfrak{gl}}(m|n)$-module and
$\mathop{\mathrm{{vol\/}}}\nolimits_{x}:=\mathop{\mathrm{{vol\/}}}\nolimits(u,\xi)$
is the notation of the image of this vector in the 1-dimensional quotient
module by the invariant submodule that consists precisely of all the extra
terms. Deligne, Witten, and their followers denote the volume element — the
generator of the quotient module — by $[dx/\partial_{\xi}]$, see [Del, Ch.1].

###  2.11. A. Sergeev’s central extension of ${\mathfrak{spe}}(4)$

In 1970’s A. Sergeev proved (cited in [FuLe], see also [BGLL1]) that over
${\mathbb{C}}$ there is just one nontrivial central extension of
${\mathfrak{spe}}(n)$ for $n>2$. It exists only for $n=4$ and we denote it by
Sergeev’s initials: ${\mathfrak{as}}$. Let us represent an arbitrary element
$\widehat{X}\in{\mathfrak{as}}$ as a pair $\widehat{X}=X+a\cdot 1_{4|4}$,
where $X\in{\mathfrak{spe}}(4)$ and $a\in{{\mathbb{C}}}$. The bracket in
${\mathfrak{as}}$ is

(104) $\left[\text{\footnotesize$\begin{pmatrix}A&B\\\
C&-A^{t}\end{pmatrix}$}+a\cdot
1_{4|4},\text{\footnotesize$\begin{pmatrix}A^{\prime}&B^{\prime}\cr
C^{\prime}&-A^{\prime}{}^{t}\end{pmatrix}$}+a^{\prime}\cdot
1_{4|4}\right]=\left[\text{\footnotesize$\begin{pmatrix}A&B\cr
C&-A^{t}\end{pmatrix}$},\text{\footnotesize$\begin{pmatrix}A^{\prime}&B^{\prime}\cr
C^{\prime}&-A^{\prime}{}^{t}\end{pmatrix}$}\right]+\mathop{\mathrm{tr}}\nolimits(C\widetilde{C}^{\prime})\cdot
1_{4|4},$

where $\ \widetilde{}\ $ is extended by linearity from matrices
$C_{ij}:=E_{ij}-E_{ji}$, on which $\widetilde{C}_{ij}=C_{kl}$ for any even
permutation $(1234)\longmapsto(ijkl)$.

The Lie superalgebra ${\mathfrak{as}}$ can also be described with the help of
the spinor representation, see [LSh4]. Consider the particular case of
${\mathfrak{o}}(6)=({\mathfrak{po}}(0|6))_{0}$. The superspace of
${\mathfrak{po}}(0|6)$ is the Grassmann superalgebra $\Lambda(\xi,\eta)$
generated by $\xi=(\xi_{1},\xi_{2},\xi_{3})$ and
$\eta=(\eta_{1},\eta_{2},\eta_{3})$ and the Lie superalgebra structure given
by the Poisson bracket (68).

Recall that ${\mathfrak{h}}(0|6)=\mathop{\mathrm{{Span}}}\nolimits(H_{f}\mid
f\in\Lambda(\xi,\eta))$. Now, note that ${\mathfrak{spe}}(4)$ can be embedded
into ${\mathfrak{h}}(0|6)$. Indeed, setting $\deg\xi_{i}=\deg\eta_{i}=1$ for
all $i$ we introduce a ${\mathbb{Z}}$-grading on $\Lambda(\xi,\eta)$ which, in
turn, induces a ${\mathbb{Z}}$-grading on ${\mathfrak{h}}(0|6)$ of the form
${\mathfrak{h}}(0|6)=\mathop{\oplus}_{i\geq-1}\ {\mathfrak{h}}(0|6)_{i}$.
Since ${\mathfrak{sl}}(4)\simeq{\mathfrak{o}}(6)$, we can identify
${\mathfrak{spe}}(4)_{0}$ with ${\mathfrak{h}}(0|6)_{0}$.

It is not difficult to see that the elements of degree $-1$ in the standard
gradings of ${\mathfrak{spe}}(4)$ and ${\mathfrak{h}}(0|6)$ constitute
isomorphic ${\mathfrak{sl}}(4)\simeq{\mathfrak{o}}(6)$-modules; moreover, it
is possible to embed ${\mathfrak{spe}}(4)_{1}$ into ${\mathfrak{h}}(0|6)_{1}$.

Sergeev’s extension ${\mathfrak{as}}$ is the result of the restriction to
${\mathfrak{spe}}(4)\subset{\mathfrak{h}}(0|6)$ of the cocycle that turns
${\mathfrak{h}}(0|6)$ into ${\mathfrak{po}}(0|6)$, see eq. (179). The
quantization ${\mathcal{Q}}$ deforms ${\mathfrak{po}}(0|6)$ into
${\mathfrak{gl}}(\Lambda(\xi))$; the through maps

$T_{\lambda}:{\mathfrak{as}}\longrightarrow{\mathfrak{po}}(0|6)\stackrel{{\scriptstyle{\mathcal{Q}}}}{{\longrightarrow}}{\mathfrak{gl}}(\Lambda(\xi))$

are representations of ${\mathfrak{as}}$ in the $4|4$-dimensional modules
$\mathop{\mathrm{spin}}\nolimits_{\lambda}$ isomorphic to each other for all
$\lambda\neq 0$, so we assume $\lambda=1$ and do not indicate it. The explicit
form of $T_{\lambda}$ is

(105) $T_{\lambda}\colon\text{\footnotesize$\begin{pmatrix}A~{}&B\cr
C&-A^{t}\end{pmatrix}$}+a\cdot
1_{4|4}\longmapsto\text{\footnotesize$\begin{pmatrix}A~{}&B-\lambda\widetilde{C}\cr
C&-A^{t}\end{pmatrix}$}+\lambda a\cdot 1_{4|4},$

where $\widetilde{C}$ is defined in the line under eq. (104). Clearly,
$T_{\lambda}$ is irreducible for any $\lambda$.

###  2.12. Deformations of the Buttin superalgebra (after [Ko1, LSh3])

As is clear from the definition of the Buttin bracket (and [G] from where it
went unnoticed), there is a regrading (namely, ${\mathfrak{b}}(n;n)$ given by
$\deg\xi_{i}=0,\deg q_{i}=1$ for all $i$) under which ${\mathfrak{b}}(n)$,
initially of depth 2, takes the form
${\mathfrak{g}}=\mathop{\oplus}_{i\geq-1}\ {\mathfrak{g}}_{i}$ with
${\mathfrak{g}}_{0}={\mathfrak{vect}}(0|n)$ and
${\mathfrak{g}}_{-1}\simeq\Pi({\mathbb{C}}[\xi])$. Now let us replace the
${\mathfrak{vect}}(0|n)$-module ${\mathfrak{g}}_{-1}$ of functions (with
inverted parity) by the rank-1 module (over the algebra of functions) of
$\lambda$-densities (also with inverted parity), i.e., set
${\mathfrak{g}}_{-1}\simeq\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|n))$,
where the ${\mathfrak{vect}}(0|n)$-action on the generator
$\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}^{\lambda}$ of the module
$\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|n)$ is given by the formula

(106)
$L_{D}(\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}^{\lambda})=\lambda\mathop{\text{\rm
div}}\nolimits
D\cdot\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}^{\lambda}\;\text{ and
}\;p(\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}^{\lambda})={\bar{1}}\text{~{}~{}when
considered in ${\mathfrak{g}}_{-1}$}.$

Define ${\mathfrak{b}}_{\lambda}(n;n)$ to be the Cartan prolong

(107)
${\mathfrak{b}}_{\lambda}(n;n):=({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}=(\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|n)),{\mathfrak{vect}}(0|n))_{*}.$

Clearly, this is a deform of ${\mathfrak{b}}(n;n)$. Kotchetkov called he
collection of these prolongs ${\mathfrak{b}}_{\lambda}(n;n)$ for all
$\lambda\in{\mathbb{C}}{\mathbb{P}}^{1}$ the main deformation, see [Ko1]; the
other deformations are called singular, see Theorem 2.12.1. Theorem. The
deformation of ${\mathfrak{b}}(n)$ with odd parameter is called quantization:
it is a splitting image of the quantization of the Poisson (super)algebra.

The deform ${\mathfrak{b}}_{\lambda}(n)$ of ${\mathfrak{b}}(n)$ is a regrading
of ${\mathfrak{b}}_{\lambda}(n;n)$ described as follows. Set

(108) ${\mathfrak{b}}_{a,b}(n):=\\{M_{f}\in{\mathfrak{m}}(n)\mid
a\;\mathop{\text{\rm div}}\nolimits
M_{f}=(-1)^{p(f)}2(a-bn)\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}\\}.$

Using the explicit form of the divergence of $M_{f}$ we see that

(109)
$\begin{array}[]{ll}{\mathfrak{b}}_{a,b}(n)&=\\{M_{f}\in{\mathfrak{m}}(n)\mid(bn-
aE)\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}=a\Delta f\\}=\\\ &\\{D\in{\mathfrak{vect}}(n|n+1)\mid
L_{D}(\mathop{\mathrm{{vol\/}}}\nolimits_{q,\xi,\tau}^{a}\alpha_{0}^{a-bn})=0\\},\text{~{}~{}where
$\Delta:=\mathop{\sum}_{i\leq n}\frac{\partial^{2}}{\partial
q_{i}\partial\xi_{i}}$ }.\end{array}$

It is subject to a direct verification that
${\mathfrak{b}}_{a,b}(n)\simeq{\mathfrak{b}}_{\lambda}(n)$ for
$\lambda=\frac{2a}{n(a-b)}$. This isomorphism shows that $\lambda$ actually
runs over ${\mathbb{C}}{\mathbb{P}}^{1}$, not ${\mathbb{C}}$, as one might
hastily think.

For future use, we will denote the operator that singles out
${\mathfrak{b}}_{\lambda}(n)$ in ${\mathfrak{m}}(n)$ as follows:

(110) $\mathop{\text{\rm div}}\nolimits_{\lambda}:=(bn-
aE)\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{0.25pt}{$\dfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}{\raisebox{-0.5pt}{$\tfrac{\partial
f}{\partial\tau}$}}-a\Delta\;\text{ for
$\lambda:=\mathchoice{\raisebox{0.25pt}{$\dfrac{2a}{n(a-b)}$}}{\raisebox{0.25pt}{$\dfrac{2a}{n(a-b)}$}}{\raisebox{-0.5pt}{$\tfrac{2a}{n(a-b)}$}}{\raisebox{-0.5pt}{$\tfrac{2a}{n(a-b)}$}}$
and $\Delta:=\mathop{\sum}\limits_{i\leq
n}\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial^{2}}{\partial
q_{i}\partial\xi_{i}}$}}{\raisebox{0.25pt}{$\dfrac{\partial^{2}}{\partial
q_{i}\partial\xi_{i}}$}}{\raisebox{-0.5pt}{$\tfrac{\partial^{2}}{\partial
q_{i}\partial\xi_{i}}$}}{\raisebox{-0.5pt}{$\tfrac{\partial^{2}}{\partial
q_{i}\partial\xi_{i}}$}}$}.$

As follows from the description of ${\mathfrak{vect}}(m|n)$-modules ([BL4])
and the simplicity criteria for ${\mathbb{Z}}$-graded Lie superalgebras
([K2]), the Lie superalgebras ${\mathfrak{b}}_{\lambda}(n)$ are simple for
$n>1$ and $\lambda\neq 0$, 1, $\infty$, see eq. (111). It is also clear that
the Lie superalgebras ${\mathfrak{b}}_{\lambda}(n)$ are non-isomorphic for
distinct values of $\lambda$, except for the isomorphisms already described in
list (23).

The Lie superalgebra ${\mathfrak{b}}(n)={\mathfrak{b}}_{0}(n)$ is not simple:
it has an $\varepsilon=(0|1)$-dimensional, center. At $\lambda=1$ and
$\infty$, the Lie superalgebras ${\mathfrak{b}}_{\lambda}(n)$ are not simple
either: each of them has an ideal of codimension $\varepsilon^{n+1}$ and
$\varepsilon^{n}$, respectively. The corresponding exact sequences are

(111)
$\begin{array}[]{c}0\longrightarrow{\mathbb{C}}M_{1}\longrightarrow{\mathfrak{b}}(n)\longrightarrow{\mathfrak{le}}(n)\longrightarrow
0,\\\
0\longrightarrow{\mathfrak{b}}_{1}^{\prime}(n)\longrightarrow{\mathfrak{b}}_{1}(n)\longrightarrow{\mathbb{C}}\cdot
M_{\xi_{1}\dots\xi_{n}}\longrightarrow 0,\\\
0\longrightarrow{\mathfrak{b}}_{\infty}^{\prime}(n)\longrightarrow{\mathfrak{b}}_{\infty}(n)\longrightarrow{\mathbb{C}}\cdot
M_{\tau\xi_{1}\dots\xi_{n}}\longrightarrow 0.\\\ \end{array}$

Clearly, at the exceptional values of $\lambda$, i.e., 0, 1, and $\infty$, the
deformations of ${\mathfrak{b}}_{\lambda}(n)$ should be investigated extra
carefully. As we will see immediately, this pays: at each of the exceptional
points we find additional deformations. The following is an Open problem:

What is the reason for the exceptional deformation at $\lambda=-1$?

Other exceptional values ($\lambda=\frac{1}{2}$ and $-\frac{3}{2}$) come from
the isomorphisms

${\mathfrak{b}}_{1/2}(2;2)\simeq{\mathfrak{h}}_{1/2}(2|2)={\mathfrak{h}}(2|2)$
and ${\mathfrak{h}}_{\lambda}(2|2)\simeq{\mathfrak{h}}_{-1-\lambda}(2|2)$, see
list (23),

and the existence of an even non-degenerate bilinear form on
$\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits^{1/2})$ which gives rise to a central
extension

$0\longrightarrow{\mathbb{C}}K_{1}\longrightarrow{\mathfrak{po}}(2|2)\longrightarrow{\mathfrak{h}}(2|2)\longrightarrow
0.$

The restriction of the well-known deformation ${\mathcal{Q}}$ — quantization —
of ${\mathfrak{po}}(2|2)$ to ${\mathfrak{h}}(2|2)$ explains the exceptional
deformation at $\lambda=\frac{1}{2}$. For the proof of the next theorem, see
§10.

######  Theorem (Deformations of ${\mathfrak{b}}_{\lambda}(n)$, after [Ko1,
LSh3]).

For ${\mathfrak{g}}={\mathfrak{b}}_{\lambda}(n)$, set
$H=H^{2}({\mathfrak{g}};{\mathfrak{g}})$ for brevity. Then,

_1)_ $\mathop{\mathrm{sdim}}\nolimits~{}H=(1|0)$ for
${\mathfrak{g}}={\mathfrak{b}}_{\lambda}(n)$, except for $\lambda=0$, $-1$,
$1$, $\infty$ for $n>2$.

_$1_{2}$)_ For $n=2$, in addition to the above,
$\mathop{\mathrm{sdim}}\nolimits~{}H\neq(1|0)$ at $\lambda=\frac{1}{2}$ and
$\lambda=-\frac{3}{2}$.

_2)_ At the exceptional values of $\lambda$ listed in heading $1)$, we have

$\mathop{\mathrm{sdim}}\nolimits~{}H=(2|0)$ at $\lambda=\pm 1$ and $n$ odd, or
$\lambda=\infty$ and $n$ even, or $n=2$ and $\lambda=\frac{1}{2}$ or
$\lambda=-\frac{3}{2}$.

$\mathop{\mathrm{sdim}}\nolimits~{}H=(1|1)$ at $\lambda=0$ or $\lambda=\infty$
and $n$ odd, or $\lambda=\pm 1$ and $n$ even.

Let $k=(k_{1},\dots,k_{n})$; we set $q^{k}=q_{1}^{k_{1}}\cdots q_{n}^{k_{n}}$
and $|k|=\sum k_{i}$. The corresponding cocycles $C$ are given by the
following non-zero values in terms of the generating functions $f$ and $g$,
where $d_{{\bar{1}}}(f)$ is the degree of $f$ with respect to odd
indeterminates only:

${\mathfrak{b}}_{\lambda}(n)$ | $p(C)$ | $C(f,g)$
---|---|---
${\mathfrak{b}}_{0}(n)$ | odd | $(-1)^{p(f)}(d_{{\bar{1}}}(f)-1)(d_{{\bar{1}}}(g)-1)fg$
${\mathfrak{b}}_{-1}(n)$ | $n+1\pmod{2}$ | $f=q^{k},\ g=q^{l}\longmapsto(4-|k|-|l|)q^{k+l}\xi_{1}\dots\xi_{n}+$
|  | $\tau\Delta(q^{k+l}\xi_{1}\dots\xi_{n})$
${\mathfrak{b}}_{1}(n)$ | $n+1\pmod{2}$ | $f=\xi_{1}\dots\xi_{n},\ g\longmapsto\begin{cases}(d_{{\bar{1}}}(g)-1)g,&\text{if $g\neq af$, $a\in{\mathbb{C}}$}\\\ 2(n-1)f,&\text{if $g=f$ and $n$ is even}\end{cases}$
${\mathfrak{b}}_{\infty}(n)$ | $n\pmod{2}$ | $f=\tau\xi_{1}\dots\xi_{n},\ g\longmapsto\begin{cases}(d_{{\bar{1}}}(g)-1)g,&\text{if $g\neq af$, $a\in{\mathbb{C}}$}\\\ 2f,&\text{if $g=f$ and $n$ is odd}\end{cases}$
${\mathfrak{b}}_{1/2}(2)$ | even |  induced by the quantization ${\mathcal{Q}}$ of ${\mathfrak{po}}$, see [Fu, Vo, Tyut]
$\lambda$ generic | even |  see (112), (113)

On
${\mathfrak{b}}_{1/2}(2)\simeq{\mathfrak{b}}_{-3/2}(2)\simeq{\mathfrak{h}}(2|2;1)$,
the cocycle $C$ is the one induced on
${\mathfrak{h}}(2|2)={\mathfrak{po}}(2|2)/{\mathbb{C}}\cdot K_{1}$ by the
usual deformation (quantization) of ${\mathfrak{po}}(2|2)$ with cocycle $Q$
applied to the nonstandard regrading ${\mathfrak{po}}(2|2;1)$ with subsequent
passage to the quotient modulo the center _(spanned by constants)_.

_3)_ The space $H$ is diagonalizable with respect to the Cartan subalgebra of
${\mathfrak{der}}~{}~{}{\mathfrak{g}}$. Let the cocycle $M$ corresponding to
the main deformation be one of the eigenvectors. Let $C$ be another
eigenvector in $H$, it determines a singular deformation from item $2)$. The
only cocycles $kM+lC$, where $k,l\in{\mathbb{C}}$, that can be extended to a
global deformation are those for $kl=0$, i.e., either $M$ or $C$.

All the singular deformations of the bracket $\\{-,-\\}_{old}$ in
${\mathfrak{b}}_{\lambda}(n)$, except the ones for $\lambda=\frac{1}{2}$ or
$-\frac{3}{2}$ and $n=2$, are linear in $\hbar$ even if $\hbar$ is even:

(112) $\\{f,g\\}_{\hbar}^{sing}=\\{f,g\\}_{old}+\hbar\cdot C(f,g)\text{ for
any }f,g\in{\mathfrak{b}}_{\lambda}(n).$

####  2.12.2. Remark: A relation with the Vey cocycle.

C. Roger observed that the singular deformation (quantization) of
${\mathfrak{b}}_{0}(n)={\mathfrak{b}}(n)$ is, up to a sign, the wedge product
of two 1-cocycles, the derivations $f\longmapsto(d_{{\bar{1}}}(f)-1)f$. He
also noted that the cocycle on
${\mathfrak{b}}_{1/2}(2)\simeq{\mathfrak{h}}(2|2;1)$ induced by the
quantization of ${\mathfrak{po}}(2|2)$ is a straightforward superization of
the well-known Vey’s cocycle [V].

Since the elements of ${\mathfrak{b}}_{\lambda}(n)$ are encoded by functions
(for us: polynomials or formal power series) in $\tau$, $q$ and $\xi$ subject
to one relation with an odd left-hand side which contains $\tau$, it seems
plausible that the bracket in ${\mathfrak{b}}_{\lambda}(n)$ can be, at least
for generic values of the parameter $\lambda$, expressed solely in terms of
$q$ and $\xi$. This is indeed the case, and here is an explicit formula (in
which $\\{f,g\\}_{B.b.}$ is the antibracket and $\Delta=\mathop{\sum}_{i\leq
n}\ \frac{\partial^{2}}{\partial q_{i}\partial\xi_{i}}$):

(113)
$\begin{array}[]{l}\\{f_{1},f_{2}\\}_{\lambda}^{main}=\\{f_{1},f_{2}\\}_{B.b.}+\lambda(c_{\lambda}(f_{1},f_{2})f_{1}\Delta
f_{2}+(-1)^{p(f_{1})}c_{\lambda}(f_{2},f_{1})(\Delta f_{1})f_{2}),\\\
\text{where $c_{\lambda}(f_{1},f_{2})=\frac{\deg f_{1}-2}{2+\lambda(\deg
f_{2}-n)}$},\end{array}$

and $\deg$ is computed with respect to the standard grading $\deg
q_{i}=\deg\xi_{i}=1$.

####  2.12.3. The exceptional nonstandard regradings
$\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}}$ and
$\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{h}}}$.

This is a regrading of ${\mathfrak{b}}_{a,b}(2)$ given by the formulas:

(114) $\deg\tau=0,\;\;\deg\xi_{1}=\deg\xi_{2}=-1,\;\;\deg q_{1}=\deg q_{2}=1.$

We have the following two cases:

(115) $\begin{array}[]{ll}1)&\text{\text@underline{$b=0$ or $a=b$}:
~{}${\mathfrak{b}}_{a,0}(2;\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}})\simeq{\mathfrak{le}}(2)$
and
${\mathfrak{b}}_{a,a}(2;\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}})\simeq{\mathfrak{b}}_{\infty}^{\prime}(2)$,}\\\
&\text{in particular, ${\mathfrak{g}}_{-2}=0$;}\\\
2)&\text{\text@underline{$a$ and $b$ generic}:
}{\mathfrak{g}}_{-2}=\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{\xi_{1}\xi_{2}})\text{
and
}{\mathfrak{g}}_{-1}=\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{\xi_{1}},\mathop{\mathrm{Le}}\nolimits_{\xi_{2}},\mathop{\mathrm{Le}}\nolimits_{Q_{1}},\mathop{\mathrm{Le}}\nolimits_{Q_{2}}),\end{array}$

where $Q_{1}=A\xi_{1}\xi_{2}q_{1}+B\tau\xi_{2}$,
$Q_{2}=A\xi_{1}\xi_{2}q_{2}-B\tau\xi_{1}$ with some coefficients $A$ and $B$
determined by $a$ and $b$. The bracket on ${\mathfrak{g}}_{-1}$ is determined
by the odd form $\omega=c\sum dQ_{i}d\xi_{i}$, so ${\mathfrak{g}}_{0}$ must be
contained in ${\mathfrak{m}}(2)_{0}$. Direct calculations show that
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{0}=4|4$ and

(116) ${\mathfrak{g}}_{0}={\mathfrak{spe}}(2)\oplus{\mathbb{C}}X,\text{ where
}X=\mathop{\mathrm{Le}}\nolimits_{a\tau+b\sum q_{i}\xi_{i}}.$

Indeed,

(117)
${\mathfrak{spe}}(2)_{0}\simeq{\mathfrak{sl}}(2)=\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{q_{1}\xi_{2}},\mathop{\mathrm{Le}}\nolimits_{q_{2}\xi_{1}},\mathop{\mathrm{Le}}\nolimits_{q_{1}\xi_{1}-q_{2}\xi_{2}}),\quad{\mathfrak{spe}}(2)_{-1}={\mathbb{C}}\cdot\mathop{\mathrm{Le}}\nolimits_{1},$

and

(118)
${\mathfrak{spe}}(2)_{1}={\mathbb{C}}\mathop{\mathrm{Le}}\nolimits_{\alpha\xi_{1}\xi_{2}P(q)+\beta\tau\Delta(\xi_{1}\xi_{2}P(q))},$

where $P(q)$ is a degree-2 monomial, $\alpha,\beta\in{\mathbb{C}}$ and
$\Delta=\sum\frac{\partial^{2}}{\partial q_{i}\partial\xi_{i}}$, see eq. (80).

The eigenvalues of $X$ on ${\mathfrak{g}}_{-1}$ are: $b-a$ on
$({\mathfrak{g}}_{-1})_{\bar{0}}$ and $b+a$ an
$({\mathfrak{g}}_{-1})_{\bar{1}}$. Therefore,

(119)
${\mathfrak{b}}_{a,b}(2;\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}})\simeq{\mathfrak{b}}_{-b,-a}(2)\simeq{\mathfrak{b}}_{b,a}(2).$

For $n>2$, as well as for ${\mathfrak{m}}(n)$ with $n>1$, similar regradings
are not of Weisfeiler type, as is not difficult to see.

The exceptional grading $\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}}$ of
${\mathfrak{b}}_{\lambda}(2)$ defines the exceptional grading
$\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{h}}}$ of the isomorphic algebra
${\mathfrak{h}}_{\lambda}(2|2)$, see eq. (23).

Thus, the exceptional regradings
$\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}}$ or
$\mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{h}}}$ do not provide us with new
W-graded vectorial algebras. Still, they are needed to interpret the
automorphisms (119).

###  2.13. The exceptional Lie subsuperalgebra ${\mathfrak{k}}{\mathfrak{as}}$
of ${\mathfrak{k}}(1|6)$

Like ${\mathfrak{vect}}(1|m;m)$, the Lie superalgebra
${\mathfrak{k}}{\mathfrak{as}}$ is not determined by its non-positive part and
requires a closer study. The Lie superalgebra
${\mathfrak{g}}={\mathfrak{k}}(1|2n)$ is generated by the elements of
${\mathbb{C}}[t,\xi_{1},\dots,\xi_{n},\eta_{1},\dots,\eta_{n}]$. The standard
${\mathbb{Z}}$-grading of ${\mathfrak{g}}$ is induced by the
${\mathbb{Z}}$-grading of ${\mathbb{C}}[t,\xi,\eta]$ given by

(120) $\deg K_{f}=\deg f-2,\text{ where }\deg
t=2,\quad\deg\xi_{i}=\deg\eta_{i}=1\ \ \text{for all $i$}.$

Clearly, in this grading, ${\mathfrak{g}}$ is of depth 2\. Let us consider the
functions that generate several first homogeneous components of
${\mathfrak{g}}=\mathop{\oplus}_{i\geq-2}\ {\mathfrak{g}}_{i}$:

(121)
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{component}&{\mathfrak{g}}_{-2}&{\mathfrak{g}}_{-1}&{\mathfrak{g}}_{0}&{\mathfrak{g}}_{1}\\\
\hline\cr\text{its
generators}&1&\Lambda^{1}(\xi,\eta)&\Lambda^{2}(\xi,\eta)\oplus{\mathbb{C}}\cdot
t&\Lambda^{3}(\xi,\eta)\oplus t\Lambda^{1}(\xi,\eta)\\\ \hline\cr\end{array}$

As one can prove directly, the component ${\mathfrak{g}}_{1}$ generates the
whole subalgebra ${\mathfrak{g}}_{+}=\mathop{\oplus}_{i>0}\
{\mathfrak{g}}_{i}$.

The component ${\mathfrak{g}}_{1}$ splits into two
${\mathfrak{g}}_{0}$-modules ${\mathfrak{g}}_{11}=\Lambda^{3}$ and
${\mathfrak{g}}_{12}=t\Lambda^{1}$. It is obvious that ${\mathfrak{g}}_{12}$
is always irreducible and the component ${\mathfrak{g}}_{11}$ is trivial for
$n=1$.

The partial Cartan prolongs of ${\mathfrak{g}}_{11}$ and ${\mathfrak{g}}_{12}$
are well known (here ${\mathfrak{d}}({\mathfrak{po}}(0|2n))$ is the Lie
superalgebra
${\mathfrak{der}}~{}({\mathfrak{po}}(0|2n))={\mathbb{C}}E\rtimes{\mathfrak{po}}(0|2n)$,
where $E=\sum\xi_{i}\partial_{\xi_{i}}+\sum\eta_{i}\partial_{\eta_{i}}$ is the
grading operator):

$\begin{matrix}({\mathfrak{g}}_{-}\oplus{\mathfrak{g}}_{0},{\mathfrak{g}}_{11})_{*}^{mk}\simeq{\mathfrak{po}}(0|2n)\oplus{\mathbb{C}}\cdot
K_{t}\simeq{\mathfrak{d}}({\mathfrak{po}}(0|2n));\\\
({\mathfrak{g}}_{-}\oplus{\mathfrak{g}}_{0},{\mathfrak{g}}_{12})_{*}^{mk}={\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}\oplus{\mathfrak{g}}_{0}\oplus{\mathfrak{g}}_{12}\oplus{\mathbb{C}}\cdot
K_{t^{2}}\simeq{\mathfrak{osp}}(2n|2).\end{matrix}$

Observe a remarkable property of ${\mathfrak{k}}(1|6)$: only for $2n=6$ the
component ${\mathfrak{g}}_{11}$ splits into 2 irreducible modules; we will
denote by ${\mathfrak{g}}_{11}^{\xi}$ the module containing
$\xi_{1}\xi_{2}\xi_{3}$, and by ${\mathfrak{g}}_{11}^{\eta}$ the module
containing $\eta_{1}\eta_{2}\eta_{3}$.

Observe further, that
${\mathfrak{g}}_{0}={\mathfrak{c}}{\mathfrak{o}}(6)\simeq{\mathfrak{gl}}(4)$.
As ${\mathfrak{gl}}(4)$-modules, ${\mathfrak{g}}_{11}^{\xi}$ and
${\mathfrak{g}}_{11}^{\eta}$ are the symmetric squares
$S^{2}(\mathop{\mathrm{id}}\nolimits)$ and
$S^{2}(\mathop{\mathrm{id}}\nolimits^{*})$ of the tautological 4-dimensional
${\mathfrak{gl}}(4)$-module $\mathop{\mathrm{id}}\nolimits$ and its dual,
respectively.

######  Theorem (The partial Cartan prolong ${\mathfrak{k}}{\mathfrak{as}}$,
see [Sh5], [Sh14]).

The partial Cartan prolong
${\mathfrak{k}}{\mathfrak{as}}^{\xi}=({\mathfrak{g}}_{-}\oplus{\mathfrak{g}}_{0},{\mathfrak{g}}_{11}^{\xi}\oplus{\mathfrak{g}}_{12})_{*}^{mk}$
is infinite-dimensional and simple. It is isomorphic to
${\mathfrak{k}}{\mathfrak{as}}^{\eta}=({\mathfrak{g}}_{-}\oplus{\mathfrak{g}}_{0},{\mathfrak{g}}_{11}^{\eta}\oplus{\mathfrak{g}}_{12})_{*}^{mk}$.

Convention. When it does not matter which of the isomorphic algebras
${\mathfrak{k}}{\mathfrak{as}}^{\xi}\simeq{\mathfrak{k}}{\mathfrak{as}}^{\eta}$
to take, we will simply write ${\mathfrak{k}}{\mathfrak{as}}$, but by default
we are always dealing with ${\mathfrak{k}}{\mathfrak{as}}^{\xi}$.

###  2.14. The fifteen W-regradings of exceptional simple vectorial Lie
superalgebras

######  Theorem (The 15 W-regradings ([Sh14], [CK2])).

All W-regradings of the exceptional simple vectorial Lie superalgebras are the
regradings of their “standard” ambients listed in table $\eqref{1.10}$.

Thus, there are $5$ isomorphism classes of abstract Lie superalgebras
consisting of $15$ W-filtered Lie superalgebras. For a proof, see Section
8.1.4.

Table (125) lists the terms ${\mathfrak{g}}_{i}$ for $-2\leq i\leq 0$ of the
15 exceptional W-graded algebras.

Observe that none of the simple W-graded vectorial Lie superalgebras is of
depth $>3$ and only two algebras are of depth 3:
${\mathfrak{m}}{\mathfrak{b}}(4|5;K)$, for which

(122)
${\mathfrak{m}}{\mathfrak{b}}(4|5;K)_{-3}\simeq\Pi(\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}),$

and another one,
${\mathfrak{k}}{\mathfrak{le}}(9|6;CK)={\mathfrak{k}}{\mathfrak{le}}(9|11)$,
for which

(123)
${\mathfrak{k}}{\mathfrak{le}}(9|11)_{-3}\simeq\Pi(\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}\boxtimes{\mathbb{C}}[-3]),$

where ${\mathbb{C}}[-3]\simeq{\mathbb{C}}\,\mathbbmss{1}$ is the module over
${\mathbb{C}}\,\theta\partial_{\theta}\in{\mathfrak{vect}}(0|1)\subset{\mathfrak{k}}{\mathfrak{le}}(9|11)_{0}$,
such that $(\theta\partial_{\theta})\mathbbmss{1}=-3\mathbbmss{1}$. Define the
representation $T^{1/2}$ of ${\mathfrak{vect}}(0|n)$ by setting (cf. (47))

(124)
$T^{1/2}_{D}(f\mathop{\mathrm{{vol\/}}}\nolimits^{1/2})=(D(f)+(-1)^{p(f)p(D)}f\cdot\frac{1}{2}\mathop{\text{\rm
div}}\nolimits D)\mathop{\mathrm{{vol\/}}}\nolimits^{1/2}$

for any $D\in{\mathfrak{vect}}(0|n)$. In table (125), the star ${\star}$ marks
the Lie superalgebras ${\mathfrak{g}}$ of depth 3 whose components
${\mathfrak{g}}_{-3}$ are described in eqs. (122) and (123).

(125) ${\mathfrak{g}}$ | ${\mathfrak{g}}_{-2}$ | ${\mathfrak{g}}_{-1}$ | ${\mathfrak{g}}_{0}$ | $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{-}$
---|---|---|---|---
${\mathfrak{v}}{\mathfrak{le}}(4|3)$ | $-$ | $\Pi(\Lambda(3)/{\mathbb{C}}1)$ | ${\mathfrak{c}}({\mathfrak{vect}}(0|3))$ | $4|3$
${\mathfrak{v}}{\mathfrak{le}}(4|3;1)$ | ${\mathbb{C}}[-2]$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2;\Lambda(2))}$ | ${\mathfrak{c}}({\mathfrak{sl}}(2;\Lambda(2)))\ltimes T^{1/2}({\mathfrak{vect}}(0|2))$ | $5|4$
${\mathfrak{v}}{\mathfrak{le}}(4|3;K)$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(3)}\boxtimes{\mathbb{C}}[-2]$ |  $\Pi(\mathop{\mathrm{id}}\nolimits^{*}_{{\mathfrak{sl}}(3)}\boxtimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}\boxtimes{\mathbb{C}}[-1]$) | ${\mathfrak{sl}}(3)\oplus{\mathfrak{sl}}(2)\oplus{\mathbb{C}}z$ | $3|6$
${\mathfrak{v}}{\mathfrak{as}}(4|4)$ | $-$ | $\mathop{\mathrm{spin}}\nolimits$ | ${\mathfrak{as}}$ | $4|4$
${\mathfrak{k}}{\mathfrak{as}}$ | ${\mathbb{C}}[-2]$ | $\Pi(\mathop{\mathrm{id}}\nolimits)$ | ${\mathfrak{c}}{\mathfrak{o}}(6)$ | $1|6$
${\mathfrak{k}}{\mathfrak{as}}(;1\xi)$ | $\Lambda(1)$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}\boxtimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{gl}}(2;\Lambda(1))}$ | ${\mathfrak{sl}}(2)\oplus({\mathfrak{gl}}(2;\Lambda(1))\ltimes{\mathfrak{vect}}(0|1))$ | $5|5$
${\mathfrak{k}}{\mathfrak{as}}(;3\xi)$ | $-$ | $\Lambda(3)$ | $\Lambda(3)\oplus{\mathfrak{sl}}(1|3)$ | $4|4$
${\mathfrak{k}}{\mathfrak{as}}(;3\eta)$ | $-$ | $\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|3)$ | ${\mathfrak{c}}({\mathfrak{vect}}(0|3))$ | $4|3$
${\mathfrak{m}}{\mathfrak{b}}(4|5)$ | $\Pi({\mathbb{C}}[-2])$ | $\mathop{\mathrm{{Vol\/}}}\nolimits(0|3)^{1/2}$ | ${\mathfrak{c}}({\mathfrak{vect}}(0|3))$ | $4|5$
${\mathfrak{m}}{\mathfrak{b}}(4|5;1)$ | $\Lambda(2)/{\mathbb{C}}1$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2;\Lambda(2))}$ | ${\mathfrak{c}}({\mathfrak{sl}}(2;\Lambda(2))\ltimes T^{1/2}({\mathfrak{vect}}(0|2))$ | $5|6$
${\mathfrak{m}}{\mathfrak{b}}(4|5;K)^{\star}$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(3)}\boxtimes{\mathbb{C}}[-2]$ | $\Pi(\mathop{\mathrm{id}}\nolimits^{*}_{{\mathfrak{sl}}(3)}\boxtimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}\boxtimes{\mathbb{C}}[-1])$ | ${\mathfrak{sl}}(3)\oplus{\mathfrak{sl}}(2)\oplus{\mathbb{C}}z$ | $3|8$
${\mathfrak{k}}{\mathfrak{le}}(9|6)$ | ${\mathbb{C}}[-2]$ | $\Pi(T^{0}_{0}(\vec{0}))$ | ${\mathfrak{svect}}(0|4)_{3,4}$ | $9|6$
${\mathfrak{k}}{\mathfrak{le}}(9|6;2)$ | $\Pi(\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(1|3)})$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2;\Lambda(3))}$ | ${\mathfrak{sl}}(2;\Lambda(3))\ltimes{\mathfrak{sl}}(1|3)$ | $11|9$
${\mathfrak{k}}{\mathfrak{le}}(9|6;K)$ | $\mathop{\mathrm{id}}\nolimits$ | $\Pi(\Lambda^{2}(\mathop{\mathrm{id}}\nolimits^{*}))$ | ${\mathfrak{sl}}(5)$ | $5|10$
${\mathfrak{k}}{\mathfrak{le}}(9|6;CK)^{\star}$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(3;\Lambda(1))}^{*}$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}\boxtimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(3;\Lambda(1))}$ | ${\mathfrak{sl}}(2)\oplus\left({\mathfrak{sl}}(3;\Lambda(1))\ltimes{\mathfrak{vect}}(0|1)\right)$ | $9|11$

###  2.15. W-gradings of serial simple vectorial Lie superalgebras

######  Theorem (34 W-gradings).

Any Weisfeiler regrading ${\mathfrak{h}}$ of a given simple vectorial Lie
superalgebra is one of regradings from Tables (131), (132), (136) and (125).

For its proof, see Section 8.1.4. To facilitate the comparison of various
vectorial superalgebras, we offer the following table (131). The most
interesting phenomena occur for extremal values of $r$ and small values of the
superdimension $m|n$.

The central element $z\in{\mathfrak{g}}_{0}$ is supposed to be chosen so that
it acts on ${\mathfrak{g}}_{k}$ as $k\cdot\mathop{\mathrm{id}}\nolimits$.

Let $\Lambda(n):={\mathbb{C}}[\xi_{1},\dots,\xi_{n}]$ be the Grassmann
superalgebra generated by the elements $\xi_{i}$, each of degree 0. We denote
(recall Examples 2.10.1)

(126)
$\begin{array}[]{ll}{\mathfrak{vect}}(0|n)\text{-modules}:&\Lambda(0):={\mathbb{C}},\quad
T(\vec{0}):=\Lambda(n),\quad T_{0}(\vec{0}):=\Lambda(n)/{\mathbb{C}}\cdot
1,\\\
&\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|n):=\\{v\in\mathop{\mathrm{{Vol\/}}}\nolimits(0|n)\mid\int
v=0\\};\\\
{\mathfrak{svect}}(0|n)\text{-modules}:&T_{0}^{0}(\vec{0}):=\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|n)/{\mathbb{C}}\cdot
1\ .\end{array}$

Over ${\mathfrak{svect}}(0|n)$, it is convenient to consider
$\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|n)$ as a submodule of $\Lambda(n)$.

For the range of the parameter $r$ in tables (131) and (132), see table (99).
Certain values of $r$ (namely, $r=k-1$ for ${\mathfrak{k}}(1|2k)$, as well as
$r=n-1$ for ${\mathfrak{m}}(n)$ and its subalgebras) are excluded because, for
these values of $r$, the corresponding grading is not a W-grading: the
${\mathfrak{g}}_{0}$-module ${\mathfrak{g}}_{-1}$ is reducible.

For $N=5^{\prime},3^{\prime}$ and 9: The ${\mathfrak{g}}_{0}$-modules
${\mathfrak{g}}_{-1}$ are described as ${\mathfrak{vect}}(0|2)$-modules.

For $N=16$: we set $p(\mu)\equiv n\pmod{2}$, so $\mu$ can be an odd
indeterminate. The Lie superalgebras
$\widetilde{{\mathfrak{svect}}}_{\mu}(0|n)$ are isomorphic for non-zero values
of $\mu$; hence, so are the algebras
$\widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\mu}(2^{n-1}-1|2^{n-1})$. Hence,
for $n$ even, we can set $\mu=1$, whereas if $\mu$ is odd, we should consider
it as an additional indeterminate on which the coefficients depend.

For $N=1$, 3, etc.: the symbol
$\mathop{\mathrm{id}}\nolimits_{{\mathfrak{g}}}$ is short for
$\mathop{\mathrm{id}}\nolimits_{{\mathfrak{g}}(\mathop{\mathrm{sdim}}\nolimits;\Lambda(r))}$.

For $N=11$, etc.: the symbol
$\mathop{\mathrm{id}}\nolimits_{{\mathfrak{osp}}}\boxtimes{\mathbb{C}}[-1]$
denotes the tensor product of
$\mathop{\mathrm{id}}\nolimits_{{\mathfrak{osp}}(m-2r|2n;\Lambda(r))}$ by the
${\mathfrak{c}}$-module ${\mathbb{C}}[-1]$, where

${\mathfrak{c}}\subset{\mathfrak{c}}({\mathfrak{osp}}(m-2r|2n))\subset{\mathfrak{c}}({\mathfrak{osp}}(m-2r|2n;\Lambda(r))),$

with the natural action of ${\mathfrak{vect}}(0|r)$ in $\Lambda(r)$. Line $14$
should be similarly understood.

For $N=21$, 22: the terms “${\mathfrak{g}}_{-i}$” denote the superspace
isomorphic to the one in quotation marks, but with the action given by the
rules (129) and (130).

For $N=21-23$: denote the tautological $k|k$-dimensional
${\mathfrak{spe}}(k)$-module by $V$; let
$D=\mathop{\mathrm{diag}}\nolimits(1_{k},-1_{k})\in{\mathfrak{pe}}(k)$. Let
$W=V\otimes\Lambda(r)$ and $X\in{\mathfrak{vect}}(0|r)$. Let

(127) $\Xi=\xi_{1}\cdots\xi_{n}\in\Lambda(\xi_{1},\dots,\xi_{n}).$

Denote by $T^{r}$, where $r$ is integer, the representations of
${\mathfrak{vect}}(0|r)$ in $W$ in ${\mathfrak{spe}}(n-r)\otimes\Lambda(r)$
and ${\mathfrak{spe}}(n-r)\otimes\Lambda(r)$ given by the formula

(128) $T^{r}(X)=1\otimes
X+D\otimes\mathchoice{\raisebox{0.25pt}{$\dfrac{1}{n-r}$}}{\raisebox{0.25pt}{$\dfrac{1}{n-r}$}}{\raisebox{-0.5pt}{$\tfrac{1}{n-r}$}}{\raisebox{-0.5pt}{$\tfrac{1}{n-r}$}}\mathop{\text{\rm
div}}\nolimits X.$

${\mathfrak{g}}={\mathfrak{sle}}^{\prime}(n;r)_{0}$ for $r\neq n-2$. For
${\mathfrak{g}}_{0}$, we have:

(129) $\begin{array}[]{l}\text{${\mathfrak{vect}}(0|r)$ acts on the ideal
${\mathfrak{spe}}(n-r)\otimes``\Lambda(r)^{\prime\prime}$ via $T^{r}$, see
eq.~{}\eqref{1.3.4}};\\\ \text{any $X\otimes
f\in{\mathfrak{spe}}(n-r)\otimes\Lambda(r)$ acts in ${\mathfrak{g}}_{-1}$ as
$\mathop{\mathrm{id}}\nolimits\otimes f$ and in ${\mathfrak{g}}_{-2}$ as
$0$};\\\ \text{any $X\in{\mathfrak{vect}}(0|r)$ acts in ${\mathfrak{g}}_{-1}$
via $T^{r}$ and in ${\mathfrak{g}}_{-2}$ as $X$.}\end{array}$

${\mathfrak{g}}={\mathfrak{sle}}^{\prime}(n;n-2)$. For ${\mathfrak{g}}_{0}$,
we note that

${\mathfrak{spe}}(2)\simeq{\mathbb{C}}(\mathop{\mathrm{Le}}\nolimits_{q_{1}\xi_{1}-q_{2}\xi_{2}})\ltimes{\mathbb{C}}\mathop{\mathrm{Le}}\nolimits_{\xi_{1}\xi_{2}},$

whereas ${\mathfrak{g}}_{-2}$ and ${\mathfrak{g}}_{-1}$ are as above, for
$r<n-2$. Set
${\mathfrak{h}}={\mathbb{C}}(\mathop{\mathrm{Le}}\nolimits_{q_{1}\xi_{1}-q_{2}\xi_{2}})$.
In this case

(130)
${\mathfrak{g}}_{0}\simeq\underline{``\left({\mathfrak{h}}\otimes\Lambda(n-2)\ltimes{\mathbb{C}}\mathop{\mathrm{Le}}\nolimits_{\xi_{1}\xi_{2}}\otimes(\Lambda(n-2)\setminus{\mathbb{C}}\xi_{3}\cdots\xi_{n})\right)^{\prime\prime}}\ltimes
T^{1}({\mathfrak{vect}}(0|n-2)).$

The action of ${\mathfrak{vect}}(0|n-2)$, the quotient of ${\mathfrak{g}}_{0}$
modulo the underlined ideal in quotation marks in (130) is performed via
(129). In the subspace
$\xi_{1}\xi_{2}\otimes\Lambda(n-2)\subset{\mathfrak{g}}_{0}$, this action is
the same as in the space of volume forms. So we can consider everything,
except for the terms proportional to $\Xi$, see eq. (127), or, speaking
correctly, take the irreducible submodule of functions with integral 0.

For $N=27$, 32, 33: the terms “${\mathfrak{g}}_{i}$” denote the superspace
isomorphic to the one in quotation marks, but with the action given by eqs.
(129) and (134).

In the exceptional case $ar=bn$, i.e., $\lambda=\frac{2}{n-r}$, we see that
the ${\mathfrak{vect}}(0|r)$-action on the ideal
$({\mathfrak{c}}{\mathfrak{spe}}(n-r)\otimes\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
of ${\mathfrak{g}}_{0}$, and on ${\mathfrak{g}}_{-}$, is the same as for
${\mathfrak{sle}}^{\prime}$, see eq. (129).

In table (131), to save space in line 18 we denote the
${\mathfrak{pe}}(n-r;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$-module
$\mathop{\mathrm{id}}\nolimits_{{\mathfrak{pe}}(n-r;\Lambda(r))}$ by
$\mathop{\mathrm{id}}\nolimits_{{\mathfrak{pe}}}$, and we use similar
abbreviations in other lines.

Non-standard gradings in cases 1–23, with non-Weisfeiler gradings excluded,
are shown in tables (131), (132).

(131) $N$ | ${\mathfrak{g}}$ | ${\mathfrak{g}}_{-2}$ | ${\mathfrak{g}}_{-1}$ | ${\mathfrak{g}}_{0}$
---|---|---|---|---
$1$ | ${\mathfrak{vect}}(n|m;r)$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{gl}}}$ | ${\mathfrak{gl}}(n|m-r;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$2$ | ${\mathfrak{vect}}(1|m;m)$ | $-$ | $\Lambda(m)$ | $\Lambda(m)\ltimes{\mathfrak{vect}}(0|m)$
$3$ | ${\mathfrak{svect}}(n|m;r),\ n\neq 1$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}}$ | ${\mathfrak{sl}}(n|m-r;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$3^{\prime}$ | ${\mathfrak{svect}}(2|1)$ | $-$ | $\Pi(T_{0}(\vec{0}))$ | ${\mathfrak{sl}}(2|1)\simeq{\mathfrak{vect}}(0|2)$
$4$ | ${\mathfrak{svect}}^{\prime}(1|m;r),\ r\neq m$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}}\boxtimes_{\Lambda(r)}\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|r)$ | ${\mathfrak{sl}}(n|m-r;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$5$ | ${\mathfrak{svect}}^{\prime}(1|m;m)$ | $-$ | $\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|m)$ | $\Lambda(m)\ltimes{\mathfrak{svect}}(0|m)$
$5^{\prime}$ | ${\mathfrak{svect}}^{\prime}(1|2)$ | $-$ | $T_{0}(\vec{0})$ | ${\mathfrak{sl}}(1|2)\simeq{\mathfrak{vect}}(0|2)$
$6$ | ${\mathfrak{h}}(2n|m)$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{osp}}}$ | ${\mathfrak{osp}}(m|2n)$
$7$ | ${\mathfrak{h}}(2n|m;r),\ 2r<m$ | $T_{0}(\vec{0})$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{osp}}}$ | ${\mathfrak{osp}}(m-2r|2n;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$8$ | ${\mathfrak{h}}(2n|2r;r)$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sp}}}$ | ${\mathfrak{sp}}(2n;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$9$ | ${\mathfrak{h}}_{\lambda}(2|2)$ | $-$ | $\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|2))$ | ${\mathfrak{osp}}(2|2)\simeq{\mathfrak{vect}}(0|2)$
$10$ | ${\mathfrak{h}}_{\lambda}(2|2;1)$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sp}}}\boxtimes_{\Lambda(1)}\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|1)$ | ${\mathfrak{sp}}(2;\Lambda(1))\ltimes{\mathfrak{vect}}(0|1)$

(132) $N$ | ${\mathfrak{g}}$ | ${\mathfrak{g}}_{-2}$ | ${\mathfrak{g}}_{-1}$ | ${\mathfrak{g}}_{0}$
---|---|---|---|---
$11$ | $\begin{matrix}{\mathfrak{k}}(2n+1|m;r)\ \text{for $r\neq k-1$}\\\ \text{if $m=2k$ and $n=0$}\end{matrix}$ | $\Lambda(r)\boxtimes{\mathbb{C}}[-2]$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{osp}}}\boxtimes{\mathbb{C}}[-1]$ | ${\mathfrak{c}}{\mathfrak{osp}}(m-2r|2n;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$12$ | ${\mathfrak{k}}(1|2m;m)$ | $-$ | $\Lambda(m)$ | $\Lambda(m)\ltimes{\mathfrak{vect}}(0|m)$
$13$ | ${\mathfrak{k}}(1|2m+1;m)$ | $\Lambda(m)$ | $\Pi(\Lambda(m))$ | $\Lambda(m)\ltimes{\mathfrak{vect}}(0|m)$
$14$ | ${\mathfrak{m}}(n;r),\ r\neq n$ | $\Pi(\Lambda(r))\boxtimes{\mathbb{C}}[-2]$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{pe}}}$ | ${\mathfrak{c}}{\mathfrak{pe}}(n-r;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$15$ | ${\mathfrak{m}}(n;n)$ | $-$ | $\Pi(\Lambda(n))$ | $\Lambda(n)\ltimes{\mathfrak{vect}}(0|n)$
$16$ | $\widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\mu}(2^{n-1}-1|2^{n-1})$ | $-$ | $\frac{\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits(0|n))}{{\mathbb{C}}(1+\mu\xi_{1}\dots\xi_{n})\mathop{\mathrm{{vol\/}}}\nolimits(\xi)}$ | $\widetilde{{\mathfrak{svect}}}_{\mu}(0|n)$
$17$ | ${\mathfrak{le}}(n)$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{pe}}}$ | ${\mathfrak{pe}}(n)$
$18$ | ${\mathfrak{le}}(n;r)$ | $\Pi(T_{0}(\vec{0}))$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{pe}}}$ | ${\mathfrak{pe}}(n-r;\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$19$ | ${\mathfrak{le}}(n;n)$ | $-$ | $\Pi(T_{0}(\vec{0}))$ | ${\mathfrak{vect}}(0|n)$
$20$ | ${\mathfrak{sle}}^{\prime}(n)$ | $-$ | $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{spe}}}$ | ${\mathfrak{spe}}(n)$
$21$ | ${\mathfrak{sle}}^{\prime}(n;r),\ r\neq n-2,n$ |  “$\Pi(T_{0}(\vec{0}))$” |  “$\mathop{\mathrm{id}}\nolimits\boxtimes\Lambda(r)$” | $({\mathfrak{spe}}(n-r)\otimes\Lambda(r)^{\prime})\ltimes T^{r}({\mathfrak{vect}}(0|r))$
$22$ | ${\mathfrak{sle}}^{\prime}(n;n-2)$ |  “$\Pi(T_{0}(\vec{0}))$” |  “$\mathop{\mathrm{id}}\nolimits\boxtimes\Lambda(r)$” |  see eq. (130)
$23$ | ${\mathfrak{sle}}^{\prime}(n;n)$ | $-$ | $\Pi(T^{0}_{0}(\vec{0}))$ | ${\mathfrak{svect}}(0|n)$

For $N=24-34$: 1) We assume that
$\lambda=\mathchoice{\raisebox{0.25pt}{$\dfrac{2a}{n(a-b)}$}}{\raisebox{0.25pt}{$\dfrac{2a}{n(a-b)}$}}{\raisebox{-0.5pt}{$\tfrac{2a}{n(a-b)}$}}{\raisebox{-0.5pt}{$\tfrac{2a}{n(a-b)}$}}\neq
0$, 1, $\infty$; these three exceptional cases (corresponding to the “drop-
outs” ${\mathfrak{le}}(n)$, ${\mathfrak{b}}_{1}^{\prime}(n)$ and
${\mathfrak{b}}^{\prime}_{\infty}(n)$, respectively) are considered
separately.

2) The irreducibility condition of the ${\mathfrak{g}}_{0}$-module
${\mathfrak{g}}_{-1}$ for ${\mathfrak{g}}={\mathfrak{b}}^{\prime}_{\infty}$
excludes $r=n-1$.

3) The case where $r=n-2$ is extra exceptional, so in table (131) we assume,
unless otherwise specified, that

(133) $0<r<n-2;\;\;\text{ additionally $a\neq b$ and $(a,b)\neq\alpha(n,n-2)$
for any $\alpha\in{\mathbb{C}}$.}$

For $N=24-27$: we consider ${\mathfrak{b}}_{a,b}(n;r)$ for $0<r<n-2$ and $ar-
bn\neq 0$; in particular, this excludes
${\mathfrak{b}}^{\prime}_{\infty}(n;n)={\mathfrak{b}}^{\prime}_{a,a}(n;n)$ and
${\mathfrak{b}}^{\prime}_{1}(n;n-2)={\mathfrak{b}}^{\prime}_{n,n-2}(n;n-2)$.

If $z$ is the central element of ${\mathfrak{c}}{\mathfrak{spe}}(n-r)$ that
acts on ${\mathfrak{g}}_{-1}$ as $-\mathop{\mathrm{id}}\nolimits$, then

(134) $z\otimes\psi\text{ acts on ${\mathfrak{g}}_{-1}$ as
$-\mathop{\mathrm{id}}\nolimits\otimes\psi$, and on ${\mathfrak{g}}_{-2}$ as
$-2\mathop{\mathrm{id}}\nolimits\otimes\psi$.}$

Set

(135) $c:=\mathchoice{\raisebox{0.25pt}{$\dfrac{a}{ar-
bn}$}}{\raisebox{0.25pt}{$\dfrac{a}{ar-bn}$}}{\raisebox{-0.5pt}{$\tfrac{a}{ar-
bn}$}}{\raisebox{-0.5pt}{$\tfrac{a}{ar-bn}$}}.$

Let $a\cdot\mathop{\mathrm{str}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits$
be the representation of
${\mathfrak{pe}}(n-r)={\mathfrak{s}}{\mathfrak{pe}}(n-r)\ltimes{\mathbb{C}}D$,
which is $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{s}}{\mathfrak{pe}}(n-r)}$
on ${\mathfrak{s}}{\mathfrak{pe}}(n-r)$ and sends
$D=\mathop{\mathrm{diag}}\nolimits(1_{n-r},-1_{n-r})\in{\mathfrak{pe}}(n-r)$
to $2a\cdot\mathop{\mathrm{id}}\nolimits$.

Non-standard gradings in cases 24–34 are shown in table (136).

(136) $24$ | ${\mathfrak{b}}_{\lambda}(n)$ | $\Pi({\mathbb{C}}[-2])$ | $\mathop{\mathrm{id}}\nolimits$ | ${\mathfrak{spe}}(n)\ltimes{\mathbb{C}}(az+bd)$
---|---|---|---|---
$25$ | $\begin{array}[]{l}{\mathfrak{b}}_{\lambda}(n;r),\\\ r<n-2\end{array}$ | $\Pi((-c)\mathop{\mathrm{str}}\nolimits)\boxtimes\mathop{\mathrm{{Vol\/}}}\nolimits^{2c}(0|r)$ | $\Big{(}\mskip-3.0mu\Big{(}\mskip-3.0mu\Big{(}\mskip-3.0mu\mskip-3.0mu-\frac{c}{2}\Big{)}\mathop{\mathrm{str}}\nolimits\Big{)}\otimes\mathop{\mathrm{id}}\nolimits\Big{)}\boxtimes\mathop{\mathrm{{Vol\/}}}\nolimits^{c}(0|r)$ |  $({\mathfrak{pe}}(n{-}r)\otimes\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$26$ | ${\mathfrak{b}}_{\lambda}(n;n)$ | — | $\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}(0|n))$ | ${\mathfrak{vect}}(0|n)$
$27$ | $\begin{array}[]{l}{\mathfrak{b}}_{2/(n-r)}(n;r),\cr r<n-2\end{array}$ | $\Pi({\mathbb{C}})\boxtimes\Lambda(r)$ |  $\mathop{\mathrm{id}}\nolimits\boxtimes``\Lambda(r)$” |  ${\mathfrak{c}}{\mathfrak{pe}}(n-r)\otimes``\Lambda(r)$” $\ltimes T^{r}({\mathfrak{vect}}(0|r))$
$28$ | ${\mathfrak{b}}^{\prime}_{\infty}(n)$ | $\Pi({\mathbb{C}})$ | $\mathop{\mathrm{id}}\nolimits$ | ${\mathfrak{spe}}(n)_{a,a}$
$29$ | $\begin{array}[]{l}{\mathfrak{b}}^{\prime}_{\infty}(n;r),\cr r<n-2\end{array}$ | $\Pi({\mathbb{C}})\boxtimes\Lambda(r)$ | $\mathop{\mathrm{id}}\nolimits\boxtimes\Lambda(r)$ | $(({\mathfrak{spe}}(n-r)_{a,a})\otimes\Lambda(r))\ltimes{\mathfrak{vect}}(0|r)$
$30$ | ${\mathfrak{b}}^{\prime}_{\infty}(n;n)$ | — | $\Pi(\Lambda(n))$ | $(\Lambda(n)\setminus{\mathbb{C}}\cdot\Xi)\ltimes{\mathfrak{svect}}(0|n)$
$31$ | ${\mathfrak{b}}^{\prime}_{1}(n)$ | $\Pi({\mathbb{C}})$ | $\mathop{\mathrm{id}}\nolimits$ | ${\mathfrak{spe}}(n)_{n,n-2}$
$32$ | $\begin{array}[]{l}{\mathfrak{b}}^{\prime}_{1}(n;r)\cr r<n-2\end{array}$ |  “$\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|r))$” |  $\mathop{\mathrm{id}}\nolimits\boxtimes``\Lambda(r)$” | $(({\mathfrak{spe}}(n{-}r)_{n,n-2})\otimes``\Lambda(r)^{\prime\prime})\ltimes T^{r}({\mathfrak{vect}}(0|r))$
$33$ | ${\mathfrak{b}}^{\prime}_{1}(n;n-2)$ |  “$\Pi(T_{0}(\vec{0}))$” |  $\mathop{\mathrm{id}}\nolimits\boxtimes``\Lambda(r)$” |  $(({\mathfrak{spe}}(n{-}r)_{n,n-2})\otimes``\Lambda(r)$”) as in ${N=32}$ with ${\mathfrak{c}}{\mathfrak{spe}}(2)$ instead of ${\mathfrak{spe}}(2)$
$34$ | ${\mathfrak{b}}^{\prime}_{1}(n;n)$ | — | $\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|n))$ | ${\mathfrak{vect}}(0|n)$

###  2.16. Grozman’s theorems and a description of ${\mathfrak{g}}$ as
${\mathfrak{g}}_{{\bar{0}}}\oplus{\mathfrak{g}}_{{\bar{1}}}$

It is convenient to describe the Lie superalgebra ${\mathfrak{g}}$ of twisted
multi-vector fields as a ${\mathfrak{vect}}$-module. Similarly, in [CK2] the
exceptional algebras are described as
${\mathfrak{g}}={\mathfrak{g}}_{{\bar{0}}}\oplus{\mathfrak{g}}_{{\bar{1}}}$.
For most of the series, such description is of little value because each
homogeneous component ${\mathfrak{g}}_{{\bar{0}}}$ and
${\mathfrak{g}}_{{\bar{1}}}$ has a complicated structure. For the exceptional
simple Lie superalgebras, the situation is totally different!

Observe that apart from being beautiful, such a description is useful for
constructing simple Volichenko algebras (inhomogeneous with respect to parity
subalgebras of Lie superalgebras), cf. [LS], and simple Lie algebras in
characteristic 2.

In relation to this, we recall a theorem of Grozman. He completely described
bilinear differential operators acting in spaces of tensor fields on any
$m$-dimensional manifold $M$ and invariant under all changes of coordinates.
Since the problem is local, we can assume that $M\simeq{\mathbb{C}}^{m}$, see
[BL4]. Miraculously, “most” of the first-order invariant operators determines
a Lie superalgebra on its domain of definition. Some of these superalgebras
turn out to be very close to simple. In the constructions below we use some of
these invariant operators.

Let $\rho$ be an irreducible representation of the group $GL(n)$ in a finite-
dimensional vector space $V$ and let $\lambda=(\lambda_{1},\dots,\lambda_{n})$
be its lowest weight. A tensor field of type $\rho$ (or of type $V$: depending
on the emphasis) on an $n$-dimensional connected manifold $M$ is any section
$t$ of the locally trivial vector bundle over $M$ with fiber $V$ such that,
under the change of coordinates,

$t(y(x))=\rho\left({\frac{\partial{y}}{\partial{x}}}\right)t(x).$

The space of tensor field of type $\rho$ or $V$ will be denoted by $T(\rho)$,
or $T(V)$, or even $T(\lambda)$. Examples:

$\begin{array}[]{l}T(\mu,\dots,\mu)=\mathop{\mathrm{{Vol\/}}}\nolimits^{\mu},\text{~{}~{}the
space of $\mu$-densities};\\\ \text{$\Omega^{r}:=T(0,\dots,0,1,\dots,1)$, with
$r$ symbols 1, is the space of differential $r$-forms},\\\ \text{in
particular, $T(0)=\Omega^{0}$ is the space of functions also denoted by
${\mathcal{F}}$};\\\ {\mathfrak{vect}}(n)=T(-1,0,\dots,0);\\\
L^{r}:=T(-1,\dots,-1,0,\dots,0)\text{~{}~{}with symbols 1, is the space of
$r$-vector fields,}\\\ \text{the $r$th exterior power (over the algebra of
functions ${\mathcal{F}}=\Omega^{0}$) of ${\mathfrak{vect}}(n)$}.\\\
\end{array}$

The spaces of twisted $r$-forms and twisted $r$-vector fields with twist $\mu$
are defined to be, respectively,

$\Omega^{r}_{\mu}=\Omega^{r}\otimes_{\Omega^{0}}\mathop{\mathrm{{Vol\/}}}\nolimits^{\mu}\text{
and
}L^{r}_{\mu}=L^{r}\otimes_{\Omega^{0}}\mathop{\mathrm{{Vol\/}}}\nolimits^{\mu}.$

Obviously, $L^{r}_{\mu}\simeq\Omega^{n-r}_{\mu-1}$ and
$\mathop{\mathrm{{Vol\/}}}\nolimits^{1}=\Omega^{n}$.

The following statements are excerpts from Grozman’s difficult result [G]. To
describe one of the operators, $P_{4}$, we need the Nijenhuis bracket, defined
by the formula

$\begin{array}[]{l}\omega^{k}\otimes\xi,\omega^{l}\otimes\eta\longmapsto(\omega^{k}\wedge\omega^{l})\otimes[\xi,\eta]\
+\\\ +\left(\omega^{k}\wedge
L_{\xi}(\omega^{l})+(-1)^{k}d\omega^{k}\wedge\iota_{\xi}(\omega^{l})\right)\otimes\eta+\left(-L_{\eta}(\omega^{k})\wedge\omega^{l}+(-1)^{l}\iota_{\eta}(\omega^{k})\wedge
d\omega^{l})\right)\otimes\xi,\end{array}$

where $\iota_{X}$ is the inner product by the field $X$ and $L_{X}$ is the Lie
derivative along $X$. The Nijenhuis bracket admits the following
interpretation which makes its invariance manifest: the operator
$D:(\Omega^{k}\otimes_{\Omega^{0}}{\mathfrak{vect}}(M),\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}})\longrightarrow\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}$ given by the formula

$\begin{gathered}D(\omega^{k}\otimes\xi,\omega):=d\left(\omega^{k}\wedge\iota_{\xi}(\omega)+(-1)^{k}\omega^{k}\wedge\iota_{\xi}(d\omega)\right)=d\omega^{k}\wedge\iota_{\xi}(\omega)+(-1)^{k}\omega^{k}\wedge
L_{\xi}(\omega)\end{gathered}$

is, for a fixed $\omega^{k}\otimes\xi$, a superderivation of the
supercommutative superalgebra $\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}$, and the Nijenhuis bracket is just the
supercommutator of these superderivations. So we can identify
$\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\otimes_{\Omega^{0}}{\mathfrak{vect}}(M)$ with the Lie
subsuperalgebra $C(d)\subset{\mathfrak{vect}}(\widehat{M})$, where
$\widehat{M}$ is the supermanifold $(M,\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(M))$, i.e., $C(d)$ is the centralizer of the exterior
differential on $\Omega^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(M))$:

$C(d)=\\{D\in{\mathfrak{vect}}(\widehat{M})\mid[D,d]=0\\}.$

Dualizations. To any mapping $F:T(V)\longrightarrow T(W)$ there corresponds
the dual mapping $F^{*}:(T(W))^{*}\longrightarrow(T(V))^{*}$. If we consider
tensors with compact support, so integration can be performed, we can identify

$(T(V))^{*}\simeq
T(V^{*})\otimes_{\Omega^{0}}\mathop{\mathrm{{Vol\/}}}\nolimits\simeq
T(V^{*}\otimes\mathop{\mathrm{str}}\nolimits),$

where $\mathop{\mathrm{str}}\nolimits$ denotes the 1-dimensional
${\mathfrak{gl}}$-module given by the trace (or supertrace, if $M$ is a
supermanifold). The formal dual is defined to be

$(T(V))^{*}:=T(V^{*}\otimes\mathop{\mathrm{str}}\nolimits).$

Given a bilinear mapping $F:T(V_{1})\otimes T(V_{2})\longrightarrow T(W)$, it
is possible to dualize it with respect to each argument:

$\begin{array}[]{l}F^{*1}:T(W^{*}\otimes\mathop{\mathrm{str}}\nolimits)\otimes
T(V_{2})\longrightarrow T(V_{1}^{*}\otimes\mathop{\mathrm{str}}\nolimits),\\\
F^{*2}:T(V_{1})\otimes
T(W^{*}\otimes\mathop{\mathrm{str}}\nolimits)\longrightarrow
T(V_{2}^{*}\otimes\mathop{\mathrm{str}}\nolimits).\end{array}$

######  Theorem (_[G]_).

The invariant under arbitrary changes of variables irreducible differential
bilinear operators $D:T(\rho_{1})\otimes T(\rho_{2})\longrightarrow
T(\rho_{3})$ of order $1$ are, up to dualizations and permutation of
arguments, only the following ones:

${\bf P_{1}}:\Omega^{r}\otimes T(\rho_{2})\longrightarrow
T(\rho_{3}),\qquad(w,t)\longmapsto Z(dw,t),$

where $Z$ is an order-$0$ operator, the extension of the projection
$\rho_{1}\otimes\rho_{2}\longrightarrow\rho_{3}$ onto any of the irreducible
components;

$\begin{gathered}{\bf P_{2}}:{\mathfrak{vect}}\otimes T(\rho)\longrightarrow
T(\rho),\quad\text{ \text@underline{the Lie derivative}};\\\ {\bf
P_{3}}:T(S^{p}(\mathop{\mathrm{id}}\nolimits^{*}))\otimes
T(S^{q}(\mathop{\mathrm{id}}\nolimits^{*}))\longrightarrow
T(S^{p+q-1}(\mathop{\mathrm{id}}\nolimits^{*})),\quad\text{
\text@underline{the Poisson bracket}};\end{gathered}$

${\bf P_{4}}$: On manifolds, the bracket in $C(d)$ is called the  Nijenhuis
bracket. This bracket is a linear combination of operators $P_{1}$,
$P_{1}^{*1}$, their composition with the permutation operator $T(V)\otimes
T(W)\longrightarrow T(W)\otimes T(V)$, and a new, irreducible, operator
denoted $P_{4}$;

${\bf
P_{5}}:\Omega^{p}\otimes\Omega^{q}\longrightarrow\Omega^{p+q+1};\quad\omega_{1},\
\omega_{2}\longmapsto(-1)^{p(\omega_{1})}a(d\omega_{1}\cdot\omega_{2})+b(\omega_{1}d\omega_{2}),\text{
where }a,b\in{\mathbb{C}};$

${\bf
P_{6}}:\Omega^{p}_{\mu}\otimes\Omega^{q}_{\nu}\longrightarrow\Omega^{p+q+1}_{\mu+\nu}$,
where $|\mu|^{2}+|\nu|^{2}\neq 0$ and $p+q<n$ is given by the formula

$\omega_{1}\mathop{\mathrm{{vol\/}}}\nolimits^{\mu},\
\omega_{2}\mathop{\mathrm{{vol\/}}}\nolimits^{\nu}\longmapsto\left(\nu(-1)^{p(\omega_{1})}d\omega_{1}\cdot\omega_{2}-\mu\omega_{1}d\omega_{2}\right)\mathop{\mathrm{{vol\/}}}\nolimits^{\mu+\nu};$
${\bf P_{7}}:L^{p}\otimes L^{q}\longrightarrow L^{p+q-1}\quad\text{ the
\text@underline{Schouten bracket}};$

${\bf P_{8}}:L^{p}_{\mu}\otimes L^{q}_{\nu}\longrightarrow
L^{p+q-1}_{\mu+\nu}$, a deformation of the Schouten bracket given by the next
formula on manifolds for $p+q\leq n$, and on supermanifolds of superdimension
$n|1$ for $p,q\in{\mathbb{C}}$:

(137) $\begin{array}[]{l}X\mathop{\mathrm{{vol\/}}}\nolimits^{\mu},\
Y\mathop{\mathrm{{vol\/}}}\nolimits^{\nu}\longmapsto\\\
\left((\nu-1)(\mu+\nu-1)\mathop{\text{\rm div}}\nolimits X\cdot
Y+(-1)^{p(X)}(\mu-1)(\mu+\nu-1)X\mathop{\text{\rm div}}\nolimits
Y-(\mu-1)(\nu-1)\mathop{\text{\rm
div}}\nolimits(XY)\right)\mathop{\mathrm{{vol\/}}}\nolimits^{\mu+\nu},\end{array}$

where the divergence of a multi-vector field $f$ is defined to be

$\mathop{\text{\rm div}}\nolimits(f)=\mathop{\sum}\limits_{i\leq
n}\mathchoice{\raisebox{0.25pt}{$\dfrac{\partial^{2}f}{\partial_{x_{i}}\partial_{\check{x}_{i}}}$}}{\raisebox{0.25pt}{$\dfrac{\partial^{2}f}{\partial_{x_{i}}\partial_{\check{x}_{i}}}$}}{\raisebox{-0.5pt}{$\tfrac{\partial^{2}f}{\partial_{x_{i}}\partial_{\check{x}_{i}}}$}}{\raisebox{-0.5pt}{$\tfrac{\partial^{2}f}{\partial_{x_{i}}\partial_{\check{x}_{i}}}$}}$

in local coordinates $(x,\check{x})$ on the supermanifold $\check{M}$
associated to the sheaf of sections of the exterior algebra of the tangent
bundle on any supermanifold $M$; here the checked coordinates on $\check{M}$
are

$\theta_{i}=\Pi\left({\frac{\partial}{\partial{x_{i}}}}\right)=\check{x}_{i},\emph{~{}~{}(see
Remark \ref{rem2.3.7})}.$

######  Theorem (Lie and associative (super)algebras determined by invariant
operators, [G]).

The following natural invariant operators determine associative or Lie
superalgebras on their domains:

$P_{3}$, $P_{4}$, $P_{7}$, and $P_{8}$ define Lie superalgebras. The vectorial
Lie superalgebra $C(d)$ is, however, not transitive.

$P_{5}$: For $ab=0$ and, of course, $|a|^{2}+|b|^{2}\neq 0$, it determines an
associative superalgebra structure.

For $a=b\neq 0$, it determines the structure of a nilpotent Lie superalgebra
on $\Pi(\Omega^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}})$.

The bracket given by $d\omega_{1}\cdot\omega_{2}-\omega_{1}d\omega_{2}$
determines a Lie algebra structure on the space $\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}$.

$P_{6}$ multiplied by $\dfrac{\mu-\nu}{\mu\nu}$ determines structures of
nilpotent Lie superalgebras on the superspaces

$\Pi\left(\mathop{\oplus}\limits_{\lambda\in{\mathbb{C}}}\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\otimes\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}/d\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\right),\quad\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}_{+}=\Pi(\mathop{\oplus}\limits_{\lambda>0;\;\lambda\in{\mathbb{R}}}\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\otimes\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}),\quad\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}_{-}=\Pi(\mathop{\oplus}\limits_{\lambda<0;\;\lambda\in{\mathbb{R}}}\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\otimes\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}).$

$P_{6}$ multiplied by $\dfrac{\mu-\nu}{\mu+\nu}$ determines a nilpotent Lie
superalgebra structure on

$\Pi\left(d\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\bigoplus\mathop{\oplus}\limits_{\lambda\neq
0;\;\lambda\in{\mathbb{R}}}\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}\otimes\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}\right).$

$P_{8}$ multiplied by $\dfrac{1}{\mu\nu}$ determines a nilpotent Lie
superalgebra structures on $\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}_{+}$ and $\Omega^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}_{-}$.

##### Remarks

The claims of this Theorem are breezily formulated in the short note [G], here
they are formulated more explicitly. Their proof is done by a direct
verification.

The operator $P_{8}$ is a deformation of the Schouten bracket we considered in
${\mathfrak{b}}_{\lambda}(n)$, so this simple Lie superalgebra could have been
described 20 years earlier than in [LSh3] if more attention was paid to
Grozman’s results.

The operator $P_{6}$ is the “Fourier transform” of the Schouten bracket.

The divergence of multi-vector fields is the “Fourier transform” of the
exterior derivation.

Grozman proved that there are no invariant differential operators of degree
$>3$, while the operators of degree 2 and 3 are compositions of degree-1
operators, with one exceptional degree-3 operator $Gz$ on the line now bearing
Grozman’s name. For a version of Grozman’s theorem over fields of
characteristic $p>0$, see [BoLe]. For the extension of Grozman’s theorem to
non-differential operators on the line, see [IoMa].

###  2.17. Weisfeiler-type gradings with infinite-dimensional components

It is sometimes convenient to consider gradings of our vectorial Lie
superalgebras which lead them out of the GLAPG class, but still preserve the
finiteness of the depth and preserve the two characteristic features of the
W-gradings:

(138) $1)$ for these grading $\oplus{\mathfrak{g}}_{i}$, the subalgebra
${\mathfrak{g}}_{\geq 0}:=\mathop{\oplus}_{i\geq 0}\ {\mathfrak{g}}_{i}$ is
maximal; $2)$ the ${\mathfrak{g}}_{0}$-module ${\mathfrak{g}}_{-1}$ is
irreducible. Namely, we set $\deg x_{i}=0$ for $r$ even indeterminates $x$; we
denote this grading $\bar{r}$.

For the exceptional simple vectorial Lie superalgebras, and
${\mathfrak{b}}_{\lambda}(n;\bar{n})$, the multiplications are described as
follows:

${\mathfrak{g}}={\mathfrak{k}}{\mathfrak{le}}(5|10)$: here,
${\mathfrak{g}}_{\bar{0}}={\mathfrak{svect}}(5|0)\simeq d\Omega^{3}(5|0)$ and
${\mathfrak{g}}_{\bar{1}}=\Pi(d\Omega^{1}(5|0))$ with the natural
${\mathfrak{g}}_{\bar{0}}$-action on ${\mathfrak{g}}_{\bar{1}}$ and the
bracket of any two odd elements is given by their product. Since
${\mathfrak{svect}}(5|0)$ preserves the volume element
$\mathop{\mathrm{{vol\/}}}\nolimits$ and all its powers, we can make, for any
permutation $(ijklm)$ of $(12345)$, the following identification

$dx_{i}\wedge dx_{j}\wedge dx_{k}\wedge dx_{l}:=dx_{i}\wedge dx_{j}\wedge
dx_{k}\wedge
dx_{l}\otimes\mathop{\mathrm{{vol\/}}}\nolimits^{-1}=\mathop{\mathrm{sign}}\nolimits(ijklm)\partial_{x_{m}}.$

${\mathfrak{g}}={\mathfrak{v}}{\mathfrak{as}}(4|4)$: here,
${\mathfrak{g}}_{\bar{0}}={\mathfrak{vect}}(4|0)$ and
${\mathfrak{g}}_{\bar{1}}=\Omega^{1}(4|0)\otimes_{\Omega^{0}(4|0)}\mathop{\mathrm{{Vol\/}}}\nolimits^{-1/2}(4|0)$
with the natural ${\mathfrak{g}}_{\bar{0}}$-action on
${\mathfrak{g}}_{\bar{1}}$ and the bracket of odd elements is given by

$\left[\mathchoice{\raisebox{0.25pt}{$\dfrac{\omega_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{0.25pt}{$\dfrac{\omega_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}},\mathchoice{\raisebox{0.25pt}{$\dfrac{\omega_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{0.25pt}{$\dfrac{\omega_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}\right]=\mathchoice{\raisebox{0.25pt}{$\dfrac{d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{0.25pt}{$\dfrac{d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}},$

where we identify

$\mathchoice{\raisebox{0.25pt}{$\dfrac{dx_{i}\wedge dx_{j}\wedge
dx_{k}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{0.25pt}{$\dfrac{dx_{i}\wedge
dx_{j}\wedge
dx_{k}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{dx_{i}\wedge
dx_{j}\wedge
dx_{k}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{dx_{i}\wedge
dx_{j}\wedge
dx_{k}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}=\mathop{\mathrm{sign}}\nolimits(ijkl)\partial_{x_{l}}\text{
for any permutation $(ijkl)$ of $(1234)$}.$

${\mathfrak{g}}={\mathfrak{v}}{\mathfrak{le}}(3|6)$: here,
${\mathfrak{g}}_{\bar{0}}={\mathfrak{vect}}(3|0)\oplus{\mathfrak{sl}}(2)^{(1)}_{\geq
0}$, where ${\mathfrak{g}}^{(1)}_{\geq
0}={\mathfrak{g}}\otimes{\mathbb{C}}[x_{1},x_{2},x_{3}]$, and

${\mathfrak{g}}_{\bar{1}}=\left(\Omega^{1}(3|0)\otimes_{\Omega^{0}(3|0)}\mathop{\mathrm{{Vol\/}}}\nolimits^{-1/2}(3|0)\right)\boxtimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)^{(1)}_{\geq
0}},\text{ with the natural ${\mathfrak{g}}_{\bar{0}}$-action on
${\mathfrak{g}}_{\bar{1}}$.}$

Recall that $\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(2)}$ denotes the
irreducible ${\mathfrak{sl}}(2)$-module $L^{1}$ with highest weight 1; its
tensor square splits into the symmetric square $L^{2}\simeq{\mathfrak{sl}}(2)$
and the exterior square — the trivial module $L^{0}$; accordingly, denote by
$v_{1}\wedge v_{2}$ and $v_{1}\bullet v_{2}$ the projections of $v_{1}\otimes
v_{2}\in L^{1}\otimes L^{1}$ onto the anti-symmetric and symmetric components,
respectively. For any $\omega_{1},\omega_{2}\in\Omega^{1}$ and $v_{1},v_{2}\in
L^{1}$, we set

$\left[\mathchoice{\raisebox{0.25pt}{$\dfrac{\omega_{1}\otimes
v_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{0.25pt}{$\dfrac{\omega_{1}\otimes
v_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{1}\otimes
v_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{1}\otimes
v_{1}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}},\mathchoice{\raisebox{0.25pt}{$\dfrac{\omega_{2}\otimes
v_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{0.25pt}{$\dfrac{\omega_{2}\otimes
v_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{2}\otimes
v_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}{\raisebox{-0.5pt}{$\tfrac{\omega_{2}\otimes
v_{2}}{\sqrt{\mathop{\mathrm{{vol\/}}}\nolimits}}$}}\right]=\mathchoice{\raisebox{0.25pt}{$\dfrac{(\omega_{1}\wedge\omega_{2})\otimes(v_{1}\wedge
v_{2})+(d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2})\otimes(v_{1}\bullet
v_{2})}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{0.25pt}{$\dfrac{(\omega_{1}\wedge\omega_{2})\otimes(v_{1}\wedge
v_{2})+(d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2})\otimes(v_{1}\bullet
v_{2})}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{(\omega_{1}\wedge\omega_{2})\otimes(v_{1}\wedge
v_{2})+(d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2})\otimes(v_{1}\bullet
v_{2})}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{(\omega_{1}\wedge\omega_{2})\otimes(v_{1}\wedge
v_{2})+(d\omega_{1}\wedge\omega_{2}+\omega_{1}\wedge
d\omega_{2})\otimes(v_{1}\bullet
v_{2})}{\mathop{\mathrm{{vol\/}}}\nolimits}$}},$

where we naturally identify $\Omega^{0}$ with
$\Omega^{3}\otimes_{\Omega^{0}}\mathop{\mathrm{{Vol\/}}}\nolimits^{-1}$; we
also naturally identify
$\Omega^{2}\otimes_{\Omega^{0}}\mathop{\mathrm{{Vol\/}}}\nolimits^{-1}$ with
${\mathfrak{vect}}(3|0)$ by setting

$\mathchoice{\raisebox{0.25pt}{$\dfrac{dx_{i}\wedge
dx_{j}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{0.25pt}{$\dfrac{dx_{i}\wedge
dx_{j}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{dx_{i}\wedge
dx_{j}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}{\raisebox{-0.5pt}{$\tfrac{dx_{i}\wedge
dx_{j}}{\mathop{\mathrm{{vol\/}}}\nolimits}$}}=\mathop{\mathrm{sign}}\nolimits(ijk)\partial_{x_{k}}\text{
for any permutation $(ijk)$ of $(123)$}.$

${\mathfrak{g}}={\mathfrak{b}}_{\lambda}(n;\bar{n})$: We have

${\mathfrak{g}}_{i}=\left(\Pi(\Lambda^{i+1}_{\Omega^{0}}({\mathfrak{vect}}(n|0))\right)\otimes\mathop{\mathrm{{Vol\/}}}\nolimits^{i\lambda}$

for $i=-1,0,\dots,n-1$. The multiplication is given by Grozman’s operator
$P_{8}$, see (137).

Consider the case where $n=2$ more closely. Clearly, one can interchange
${\mathfrak{g}}_{-1}$ and ${\mathfrak{g}}_{1}$; this possibility explains the
isomorphism

(139)
${\mathfrak{b}}_{\lambda}(2;\bar{2})\simeq{\mathfrak{b}}_{1-\lambda}(2;\bar{2}),$

and hence the mysterious isomorphism
${\mathfrak{h}}_{\lambda}(2|2)\simeq{\mathfrak{h}}_{-1-\lambda}(2|2)$
mentioned in (23).

In particular, we have additional outer automorphisms of
${\mathfrak{b}}_{1/2}(2;\bar{2})$, whereas for
$\lambda=\mathchoice{\raisebox{0.25pt}{$\dfrac{1}{2}$}}{\raisebox{0.25pt}{$\dfrac{1}{2}$}}{\raisebox{-0.5pt}{$\tfrac{1}{2}$}}{\raisebox{-0.5pt}{$\tfrac{1}{2}$}}$
and
$\lambda=-\mathchoice{\raisebox{0.25pt}{$\dfrac{3}{2}$}}{\raisebox{0.25pt}{$\dfrac{3}{2}$}}{\raisebox{-0.5pt}{$\tfrac{3}{2}$}}{\raisebox{-0.5pt}{$\tfrac{3}{2}$}}$
there are nontrivial central extensions of ${\mathfrak{h}}_{\lambda}(2|2)$.

Therefore, for $n=2$, to the exceptional values of $\lambda=0$, 1 and $\infty$
existing for any $n>2$, we should add $\lambda=-1$ (for which
${\mathfrak{g}}_{1}\simeq\mathop{\mathrm{{Vol\/}}}\nolimits$, and hence
${\mathfrak{g}}$ is not simple) and $\lambda=2$.

${\mathfrak{g}}={\mathfrak{m}}{\mathfrak{b}}(3|8)$: See §11.

###  2.18. On structures preserved

In the space ${\mathfrak{g}}_{-1}=\Lambda(n)$, the Lie superalgebra
${\mathfrak{g}}_{0}={\mathfrak{vect}}(0|n)$ preserves the tensor
$\mathop{\mathrm{mult}}\nolimits$ of valence $(2,1)$ that defines the
multiplication. If ${\mathfrak{g}}_{-1}=\Lambda(n)/{\mathbb{C}}\cdot 1$, then
the structure preserved is the tensor
$\mathop{\mathrm{mult}}\nolimits^{\prime}$ that defines multiplication on the
maximal ideal of $\Lambda(n)$.

The exceptional simple vectorial Lie superalgebras ${\mathfrak{g}}$ preserve
various distributions described in [KLS]. The structures preserved by the
serial simple Lie superalgebras are described by the structures preserved by
${\mathfrak{g}}_{0}$ on ${\mathfrak{g}}_{-1}$. They are as follows:

(1) The tensor products $B\otimes\mathop{\mathrm{mult}}\nolimits$ of a
bilinear (or a volume) form $B$ preserved (perhaps, conformally, up to
multiplication by a scalar) in the fiber of the vector bundle over a
$0|r$-dimensional supermanifold on which the structure governed by
$\mathop{\mathrm{mult}}\nolimits$ is preserved,

(2) The tensor $\mathop{\mathrm{mult}}\nolimits^{\prime}$, or
$\mathop{\mathrm{mult}}\nolimits$ twisted by a $\lambda$-density. Observe that
the volume element $B$ can be not only
$\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}$, but also
$(1+\alpha\xi_{1}\ldots\xi_{n})\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}$.

The structures of other types, namely certain pseudodifferential forms
preserved by ${\mathfrak{b}}_{\lambda}(n)$, are already described.

####  2.18.1. Remark: On Shmelev’s interpretations of
${\mathfrak{h}}_{\lambda}(2|2)\simeq{\mathfrak{b}}_{\lambda}(2;2)$

It seemed that ${\mathfrak{b}}_{\lambda}(2)$ had one more realization: G.
Shmelev interpreted
${\mathfrak{h}}_{\lambda}(2|2)\simeq{\mathfrak{b}}_{\lambda}(2;2)$ as a
deformation ${\mathfrak{h}}_{\lambda}(2|2)$ of the Lie superalgebra
${\mathfrak{h}}(2|2)$ of Hamiltonian vector fields as preserving the
pseudodifferential form

$d\eta^{\mathchoice{\raisebox{0.25pt}{$\dfrac{2\lambda-1}{1-\lambda}$}}{\raisebox{0.25pt}{$\dfrac{2\lambda-1}{1-\lambda}$}}{\raisebox{-0.5pt}{$\tfrac{2\lambda-1}{1-\lambda}$}}{\raisebox{-0.5pt}{$\tfrac{2\lambda-1}{1-\lambda}$}}}\left(\lambda
dqdp+(1-\lambda)d\xi d\eta\right);$

he also described ${\mathfrak{h}}_{\lambda}(2|2)$ as preserving the
pseudodifferential form

$d\eta^{\mathchoice{\raisebox{0.25pt}{$\dfrac{1}{\lambda}$}}{\raisebox{0.25pt}{$\dfrac{1}{\lambda}$}}{\raisebox{-0.5pt}{$\tfrac{1}{\lambda}$}}{\raisebox{-0.5pt}{$\tfrac{1}{\lambda}$}}-2}\left(dqdp+d\xi
d\eta\right).$

These interpretations are cited (but not used) in [ALSh, Po1], but
verification shows that Shmelev made a mistake calculating Lie derivatives, so
these interpretations are wrong.

##  3\. Step 2. Criterion of infinite dimensionality of the Cartan prolong

In this section, we superize Wilson’s proof of the following
Guillemin–Quillen–Sternberg theorem.

######  Theorem (Main technical theorem).

Let ${\mathfrak{g}}=\mathop{\oplus}_{i\geq-d}\ {\mathfrak{g}}_{i}$ be an
infinite-dimensional transitive Lie algebra. Then, there exists an element
$g\in{\mathfrak{g}}_{0}$ such that
$\mathop{\mathrm{rk}}\nolimits(\mathop{\mathrm{ad}}\nolimits_{g})|_{{\mathfrak{g}}_{-1}}=1$.

The converse statement is obvious: if such $g$ exists, then, clearly, the
Cartan prolong of $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})$ is infinite-
dimensional. Indeed, in some basis, $g$ takes the form $x\partial_{y}$ and all
the elements $x^{n}\partial_{y}$ belong to the Cartan prolong.

###  3.2. A reformulation of Theorem 3.1. Theorem

Let ${\mathcal{A}}=\mathop{\oplus}_{i\leq 0}\ {\mathcal{A}}_{i}$ be a
commutative ${\mathbb{Z}}$-graded algebra with unit generated by
$A:={\mathcal{A}}_{-1}$ and let $M=\mathop{\oplus}_{i\geq k}\ M_{i}$ be a
${\mathbb{Z}}$-graded ${\mathcal{A}}$-module. (In the case we are interested
in, ${\mathcal{A}}=S^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}}_{-1})$ and $M={\mathfrak{g}}$ .) We
will say that $M$ is a transitive module if

$Am\neq 0~{}~{}\text{ for any non-zero }\ m\in M_{i}\text{ and any }i\geq
k+1.$

Notice that each element $m\in M$ determines a linear mapping

$m:A\longrightarrow M,\quad a\longmapsto am.$

Clearly, $M$ is transitive if and only if there exists an $i\geq k+1$ and a
non-zero $m\in M_{i}$ such that $\ker m\neq A$.

Now it is clear that Theorem 3.1. Theorem can be reformulated in the following
equivalent form that is more convenient for us.

######  Theorem (Main technical theorem, a version).

Let $M=\mathop{\oplus}_{i\geq k}\ M_{i}$ be a ${\mathbb{Z}}$-graded infinite-
dimensional transitive ${\mathcal{A}}$-module. Then, there exists $m\in
M_{k+1}$ such that ${\mathop{\mathrm{codim}}\nolimits\ker m=1}$; hence,
$\mathop{\mathrm{rk}}\nolimits~{}m=1$.

###### Proof.

Without loss of generality we can assume that $k=-1$, i.e.,
$M=\mathop{\oplus}_{i\geq-1}\ M_{i}$ and $m\in M_{0}$.

$1)$ Since ${\mathcal{A}}$ is commutative, $\ker am\supset\ker m$ for any
$a\in A$ and $m\in M$.

$2)$ Set $r_{i}=\max_{m\in M_{i}}\ \\{\dim\ker m\\}$. By the above,
$r_{i+1}\leq r_{i}$; hence, the sequence $r_{i}$ stabilizes:
$r_{N}=r_{N+1}=\dots:=r$ for some $N$. Due to transitivity, $r<\dim A$, so it
remains to prove that $r=\dim A-1$.

$3)$ We will say that $m$ is maximally degenerate if $\dim\ker m=r$.

By step $1)$, if $a\in A$ and $m\in M_{n}$ for $n\geq N+1$ is maximally
degenerate, then $am$ is maximally degenerate too and $\ker am=\ker m$.

$4)$ Let $m_{0}\in M_{N+l}$ for $l\geq 0$ be a maximally degenerate element.
For any $n\geq N$, set

$V_{n}(m_{0}):=\\{m\in M_{n}\mid\ker m\supset\ker m_{0}\\}.$

It is clear that any non-zero element $m\in V_{n}(m_{0})$ is maximally
degenerate and $\ker m=\ker m_{0}$.

$5)$ The following lemma is crucial in the next item. Actually, this lemma is
the key point of the whole proof of Theorem 3.2.1. Theorem.

######  Lemma (A useful lemma).

Let $V$ and $W$ be finite-dimensional vector spaces, and let
${U\subset\mathop{\mathrm{Hom}}\nolimits(V,W)}$ be a subspace such that any
non-zero homomorphism $u\in U$ is a monomorphism. Then,

(140) $\dim U\leq\dim W-\dim V+1.$

###### Proof.

(This elegant proof due to J. Bernstein replaces an original elementary but a
bit cumbersome proof.) Let $P(X)$ be the projective space corresponding to a
vector space $X$.

Since the mapping ${U\otimes V\longrightarrow W}$ is a monomorphism for any
non-zero $u\in U$, there exists a mapping ${i:P(U)\times P(V)\longrightarrow
P(W)}$ which for each fixed $a\in P(U)$ and $b\in P(V)$ induces embeddings
${i_{1}:a\times P(V)\longrightarrow P(W)}$ and ${i_{2}:P(U)\times
b\longrightarrow P(W)}$.

Let $h\in H^{2}(P(W))$, $h_{1}\in H^{2}(P(U))$ and $h_{2}\in H^{2}(P(V))$ be
the generators of the respective cohomology rings. Since $i_{1}$ and $i_{2}$
are embeddings, it follows that $i^{*}(h)=h_{1}+h_{2}$.

On the other hand, the only relations $h$, $h_{1}$ and $h_{2}$ satisfy are

$h^{\dim W+1}=0,\quad h_{1}^{\dim U+1}=0,\quad h_{2}^{\dim V+1}=0.$

Hence, replacing $h$ with $i^{*}(h)=h_{1}+h_{2}$ we see that the first
equation, i.e., $h^{\dim W+1}=0$, is a new relation only if the inequality
(140) holds. ∎

$6)$ Each $m\in V_{n}(m_{0})$ determines a monomorphism $A/\ker
m_{0}\longrightarrow V_{n-1}(m_{0})$. By Lemma 3.2.1a. Lemma we have

$\begin{array}[]{l}\dim V_{n}(m_{0})\leq\dim V_{n-1}(m_{0})-(\dim
A-r)+1\Longrightarrow\\\ \dim V_{n-1}(m_{0})\geq\dim V_{n}(m_{0})+(\dim
A-r)-1.\end{array}$

$7)$ Suppose $r<\dim A-1$. Then, by step $6)$, $\dim V_{n-1}(m_{0})\geq\dim
V_{n}(m_{0})+1$. Let us take $m_{0}\in M_{N+s}$, where $s>0$ is arbitrary.
Then, $V_{N+s}(m_{0})\neq 0$; hence,

$\dim M_{n}\geq\dim V_{N}(m_{0})\geq V_{N+1}(m_{0})+1\geq\cdots\geq\dim
V_{N+s}(m_{0})+s\geq 1+s.$

Since $\dim M_{N}<\infty$, we get a contradiction. Thus, $r=\dim A-1$.

$8)$ So for some $s\geq N$, we get an element $m_{0}\in M_{s}$ with
$\mathop{\mathrm{codim}}\nolimits\ker m_{0}=1$. By transitivity there are
$a_{1},\ldots,a_{s}\in A$ such that the element $m=a_{1}\cdots a_{s}m_{0}\in
M_{0}$ is non-zero. Due to step $1)$, $\ker m\supset\ker m_{0}$ and, by
transitivity, $\ker m\neq A$. Then, $\ker m=\ker m_{0}$ and $m$ is the sought-
for element. ∎

###  3.3. A superization of Wilson’s theorem

Let ${\mathcal{A}}=\mathop{\oplus}_{i\leq-1}\ {\mathcal{A}}_{i}$ be the
supercommutative superalgebra generated by $A:={\mathcal{A}}_{-1}$. Let $\
M\mathop{\oplus}_{i\geq-1}\ M_{i}$ be an ${\mathcal{A}}$-module. As above, we
will say that $M$ is transitive if $\ker m\neq 0$ for any non-zero $m\in
M_{i}$, where $i\geq 0$.

######  Theorem (A superization of Wilson’s theorem).

In any infinite-dimensional transitive graded ${\mathcal{A}}$-module $M$,
there exists an element $m\in M_{0}$ such that
$\mathop{\mathrm{codim}}\nolimits\ker m=1|0$.

###### Proof.

Let $\mathop{\mathrm{sdim}}\nolimits A_{\bar{1}}=0|n$.

$1)$ Due to supercommutativity of ${\mathcal{A}}$, we have

(141) $a^{2}m=0~{}~{}\text{ for any $a\in A_{\bar{1}}$ and $m\in M$};$ (142)
$a_{1}\cdots a_{i}m=0~{}~{}\text{ for any }m\in M\text{ and
}a_{1},\ldots,a_{i}\in A_{{\bar{1}}}\text{ if }i>n.$

$2)$ In view of condition (142) and the transitivity of $M$, we see that
$A_{\bar{0}}m\neq 0$ for any $m\in M_{i}$ with $i\geq n$.

$3)$ Let ${\mathcal{A}}_{0}\subset{\mathcal{A}}$ be the subalgebra generated
by $A_{{\bar{0}}}$. Let $\widetilde{M}=M/\oplus_{-1\leq i\leq n-1}\ M_{i}$.
Let us consider $\widetilde{M}$ as an ${\mathcal{A}}_{0}$-module and apply
Theorem 3.2.1. Theorem to $\widetilde{M}$. We get an element $m\in M_{n+1}$
such that $\dim(\ker m\cap A_{{\bar{0}}})=\dim A_{{\bar{0}}}-1$. If $\ker
m\cap A_{{\bar{1}}}\neq A_{{\bar{1}}}$, then there exists $a_{1}\in
A_{{\bar{1}}}$ such that $a_{1}\notin\ker m$. Then, $a_{1}m\neq 0$ and, thanks
to condition (141), we see that $\ker a_{1}m\supset\ker
m\oplus{\mathbb{C}}a_{1}$.

Similarly, for $i\leq n$, there exist $a_{1},\dots,a_{i}$ such that

$m_{1}=a_{1}\cdots a_{i}m\neq 0\text{~{}~{} and $\ker m_{1}\supset\ker
m+A_{{\bar{1}}}$.}$

Notice that $m_{1}\in M_{s}$, where $s=n+1-i>0$. Hence, $\ker m_{1}\neq A$;
thus, $\mathop{\mathrm{codim}}\nolimits\ker m_{1}=1|0$. Now we can complete
the proof in the same way as in step $8)$ of the proof of Theorem 3.2.1.
Theorem. ∎

######  Corollary (Existence of a rank-1 operator with even covector).

Let ${\mathfrak{g}}=\mathop{\oplus}_{i\geq-1}\ {\mathfrak{g}}_{i}$ be a
transitive infinite-dimensional ${\mathbb{Z}}$-graded Lie superalgebra with
$\dim{\mathfrak{g}}_{i}<\infty$ for all $i$. Then, there exists $v^{*}\otimes
w\in{\mathfrak{g}}_{0}$, where $v\in({\mathfrak{g}}_{-1})_{{\bar{0}}}$ and
$w\in{\mathfrak{g}}_{-1}$.

##  4\. Step 3. Main reduction: the three cases

######  Lemma (Corollaries of simplicity).

If ${\mathfrak{g}}_{*}=\mathop{\oplus}_{-1\leq i\leq k}\ {\mathfrak{g}}_{i}$,
where $k\geq 1$, is simple, then

_1)_ $[{\mathfrak{g}}_{-1},{\mathfrak{g}}_{1}]={\mathfrak{g}}_{0}$;

_2)_ the representation of ${\mathfrak{g}}_{0}$ in ${\mathfrak{g}}_{-1}$ is
irreducible, i.e., ${\mathfrak{g}}_{\geq}=\mathop{\oplus}_{i\geq 0}\
{\mathfrak{g}}_{i}$ is a maximal Lie subsuperalgebra;

_3)_ the representation of ${\mathfrak{g}}_{0}$ in ${\mathfrak{g}}_{-1}$ is
faithful.

###### Proof.

When one of the conditions $1)-3)$ is violated, it is easy to find an ideal in
${\mathfrak{g}}_{*}$.

If 1) fails, take
${\mathfrak{i}}={\mathfrak{g}}_{-1}\oplus[{\mathfrak{g}}_{-1},{\mathfrak{g}}_{1}]\oplus\left(\mathop{\oplus}_{i>0}\
{\mathfrak{g}}_{i}\right)$.

If 2) or 3) fails, take ${\mathfrak{i}}=\mathop{\oplus}_{n\geq 0}\
{\mathfrak{i}}_{n}$, where the ${\mathfrak{i}}_{n}$ are recursively defined
(here $\rho$ is the ${\mathfrak{g}}_{0}$-action on ${\mathfrak{g}}_{-1}$) by

$\begin{array}[]{l}{\mathfrak{i}}_{-1}=\begin{cases}\text{a
${\mathfrak{g}}_{0}$-invariant subsuperspace},&\text{in case $2)$},\\\
0,&\text{in case $3)$};\end{cases}\\\
{\mathfrak{i}}_{0}=\begin{cases}{\mathfrak{g}}_{0},&\text{in case $2)$},\\\
\ker\rho,&\text{in case $3)$};\end{cases}\\\
{\mathfrak{i}}_{n}=\\{g\in{\mathfrak{g}}_{n}\mid[{\mathfrak{g}}_{-1},g]\subset{\mathfrak{i}}_{n-1}\\}\quad\text{
for }\quad n\geq\begin{cases}0,&\text{in case $2)$},\\\ 1,&\text{in case
$3)$}.\end{cases}\end{array}$

The Jacobi identity implies the inclusion

$[{\mathfrak{g}}_{-1},[{\mathfrak{g}}_{k},{\mathfrak{i}}_{n}]]\subset[{\mathfrak{g}}_{k-1},{\mathfrak{i}}_{n}]+[{\mathfrak{g}}_{k},[{\mathfrak{g}}_{-1},{\mathfrak{i}}_{n}]];$

the double induction on $k$ and $n$ shows that ${\mathfrak{i}}$ is an ideal in
${\mathfrak{g}}_{*}$. ∎

The conditions 2) and 3) of Lemma 4.1. Lemma allow us to think of
${\mathfrak{g}}_{0}$ as of an irreducible linear Lie superalgebra, i.e., as a
subalgebra of ${\mathfrak{gl}}(V)$, where $V={\mathfrak{g}}_{-1}$, such that
the tautological ${\mathfrak{gl}}(V)$-module $V$ is irreducible.

####  4.1.1. Notation. The four cases

Consider the following four cases:

1) Let $V_{1}$ and $V_{2}$ be linear superspaces, $V=V_{1}\otimes V_{2}$. The
tensor product of the ${\mathfrak{gl}}(V_{i})$-actions on $V_{1}\otimes V_{2}$
gives us a representation of
${\mathfrak{gl}}(V_{1})\oplus{\mathfrak{gl}}(V_{2})$:

$\rho(A,B)=A\otimes 1_{V_{2}}+1_{V_{1}}\otimes B\quad\text{ for any
}\;A\in{\mathfrak{gl}}(V_{1}),\ \ B\in{\mathfrak{gl}}(V_{2}).$

It is clear that $\rho$ has a $1$-dimensional kernel ${\mathbb{C}}(1_{V_{1}},\
-1_{V_{2}})$. Denote:

${\mathfrak{gl}}(V_{1})\bigodot{\mathfrak{gl}}(V_{2}):=\rho({\mathfrak{gl}}(V_{1})\oplus{\mathfrak{gl}}(V_{2})).$

2) The space of all operators in a superspace $V$ can be endowed with the
following two structures: that of the Lie superalgebra ${\mathfrak{gl}}(V)$
and that of the associative superalgebra $\mathop{\mathrm{Mat}}\nolimits(V)$.
We will carefully distinguish them. (Recall that we endow with the superscript
L (resp., SL) the Lie algebra (resp., Lie superalgebra) constructed on the
space of an associative algebra (resp., superalgebra) with the dot product
replaced by the bracket (resp., superbracket). In particular,
$\mathop{\mathrm{Mat}}\nolimits(V)^{L}={\mathfrak{gl}}(V)$. Various
superstructures on $V$ induce non-isomorphic, in general, superstructures on
both $\mathop{\mathrm{Mat}}\nolimits(V)$ and the associated Lie superalgebra
${\mathfrak{gl}}(V):=\mathop{\mathrm{Mat}}\nolimits(V)^{SL}$, as is clear in
the supermatrix format.)

Let $V_{0}$ be a $(1,1)$-dimensional superspace; define the two associative
subsuperalgebras:

$\begin{array}[]{l}{{\mathcal{A}}}\subset\mathop{\mathrm{Mat}}\nolimits(1|1)=\mathop{\mathrm{Mat}}\nolimits(V_{0})=\mathop{\mathrm{{Span}}}\nolimits\left\\{\text{\footnotesize$\begin{pmatrix}a&b\\\