# Simple vectorial Lie algebras in characteristic $2$ and their superizations

Sofiane Bouarroudj1, Pavel Grozman2, Alexei Lebedev2,
Dimitry Leites3, Irina Shchepochkina4 1New York University Abu Dhabi, Division
of Science and Mathematics, P.O. Box 129188, United Arab Emirates;
<EMAIL_ADDRESS>
2Equa Simulation AB, Stockholm, Sweden<EMAIL_ADDRESS><EMAIL_ADDRESS>
3Department of Mathematics, Stockholm University, Roslagsv. 101, Kräftriket
hus 6, SE-106 91 Stockholm, Sweden<EMAIL_ADDRESS>
4Independent University of Moscow, Bolshoj Vlasievsky per, dom 11, RU-119 002
Moscow, Russia<EMAIL_ADDRESS>
###### Abstract.

The list of simple finite dimensional Lie algebras over an algebraically
closed field of characteristic 2 is wider than that in other characteristics.
In particular, it contains desuperizations of modular analogs (whose
description is often far from being obvious) of complex simple vectorial Lie
superalgebras both exceptional and serial. We describe 15 Weisfeiler gradings
of the 5 exceptional families and one W-grading for each of 2 series (with 2
exceptional subseries) of analogs in characteristic 2 of simple complex Lie
algebras. For each of these Lie algebras obtained as desuperization of a Lie
superalgebra, and for the Shen Lie algebra (see arXiv:1307.1551), we establish
the number of parameters the unconstrained shearing vector depends on. Two of
the 15 exceptions are completely new.

One of the exceptional Lie algebras is a deform of the Lie algebra of
divergence-free vector fields; the deformed algebra is not isomorphic to the
initial one as might happen in positive characteristic even for deforms given
by cocycle of nontrivial cohomology class.

In characteristic 2, every simple Lie superalgebra can be obtained from a
simple Lie algebra by one of the two methods described in arXiv:1407.1695. The
simple Lie superalgebras, that can be obtained by these two methods from
simple Lie algebras we describe here, are new.

###### Key words and phrases:

modular vectorial Lie algebra, modular vectorial Lie superalgebra

###### 1991 Mathematics Subject Classification:

Primary 17B50, 17B20; Secondary 70F25

S.B. was partly supported by the grant AD 065 NYUAD

##  1\. Introduction: overview of the scenery

Hereafter ${\mathbb{K}}$ is an algebraically closed field of characteristic
$p>0$; in the main text, $p=2$ and all Lie algebras are finite-dimensional
unless otherwise stated. In Introductions we give a brief sketch of the
construction and main ideas; for background, see §3. For further details, see
[LS1, Leb]. All voluminous computations are performed with the help of
SuperLie package.

The letter $p$ is used to designate the parity function and the “momenta”
indeterminates; in this text it is impossible to confuse them; on one occasion
we denoted parity function by $\Pi$.

Recall that a given Lie algebra or superalgebra ${\mathfrak{g}}$ without
proper ideals and of dimension $>1$ is said to be simple; ${\mathfrak{g}}$ is
semisimple if its radical is zero. We say that ${\mathfrak{h}}$ is almost
simple if it can be sandwiched (nonstrictly) between a simple Lie superalgebra
${\mathfrak{s}}$ and the Lie superalgebra ${\mathfrak{der}}({\mathfrak{s}})$
of derivations of ${\mathfrak{s}}$, i.e.,
${\mathfrak{s}}\subset{\mathfrak{h}}\subset{\mathfrak{der}}({\mathfrak{s}})$.
For the functorial approach, see [BGL4].

###  1.1. What is a reasonable classification problem?

On several occasions P. Deligne told us (in our own words), see [LL]:

“Over ${\mathbb{K}}$, to classify ALL simple Lie (super)algebras and their
representations are, perhaps, not very reasonable problems, and definitely
very tough; investigate first the restricted case: it is related to geometry
and hence meaningful.”

Be as it may, to describe restricted Lie (super)algebras one often needs
nonrestricted ones…

In this paper we concentrate on simple Lie (super)algebras, the ones easier to
begin with, keeping in mind that algebras of the following types of are no
less important than simple ones:

$\bullet$ the ones of the the form ${\mathfrak{g}}(A)$, where $A$ is a Cartan
matrix (see [BGL1, BGLLS]);

$\bullet$ the result of the complete or partial generalized Cartan
prolongation, see Subsec. 3.4, 3.4.1;

$\bullet$ central extensions and algebras of derivations of the simple
Lie(super)algebras (see Subsec. 14.1.1, [BGLL1]);

$\bullet$ deforms of the simple Lie(super)algebras, see [BGL4].

The algebras of the above two types will be called relatives of the
corresponding simple Lie (super)algebra and each other.

###  1.2. Goal: classification of simple finite-dimensional Lie algebras over
${\mathbb{K}}$

In 1960s, Kostrikin and Shafarevich suggested a method for producing simple
finite-dimensional Lie algebras over ${\mathbb{K}}$, together with the final
list, for $p>7$. In the KSh-method, the initial ingredients for constructing
the simple modular Lie algebras are simple Lie algebras over ${\mathbb{C}}$ of
two types: finite-dimensional, and infinite-dimensional vectorial ones, the
elements of the latter ones being vector fields with polynomial coefficients.
From each of these ingredients one obtains a finite-dimensional modular Lie
algebra, which can sometimes be deformed. Among nontrivial deforms (the
results of deformations) and the other simple Lie algebras one selects
nonisomorphic algebras.

If $p>3$, the goal is reached, see [S, BGP].

Dzhumadildaev and Kostrikin suggested to simplify the KSh-procedure by
considering certain “standard” modular Lie algebras from the very beginning,
and deforming them; on the other hand, the stock of “standard” examples should
include, if $p<7$, certain nonsimple Lie algebras, see [KD, SkT1]. Until year
2000 or so it was believed that the initial KSh-method produces all simple Lie
algebras only if $p>5$, whereas the Melikyan algebras should be added to the
list if $p=5$. This belief was based on insufficient study of deformations and
too narrow choice of “standard” examples: Melikyan algebras are deforms of
Poisson Lie algebras, see [KD].

####  1.2.1. Deforms.

The abundance of deforms of simple Lie (super)algebras for $p>0$, especially
overwhelming for $p=2$, is, however, misleading due to the presence of
semitrivial deforms each of which is given by a cocycle representing a
nontrivial cohomology class but, though integrable, yielding a deform
isomorphic to the initial algebra. For a description of many semitrivial
deforms, see [BLLS1]. We say that a nontrivial deform which is not semitrivial
is a true deform.

####  1.2.2. How to select “standard” examples?.

A sharpening of the KSh-method suggested in [KD] was further simplified in
[Lei], where the stock of “standard” simple Lie algebras is diminished by
eliminating the vectorial simple Lie algebras from the input of the KSh-method
but adding one more procedure, previously used anyway, namely generalized
Cartan prolongation (complete or partial), see Subsec. 3.4, 3.4.1.

On the other hand, the stock of “standard” simple Lie algebras must be
enlarged with the examples found after [KD] was published: [GL4] for $p=3$,
and [SkT1, Leb, Ei, BGLLS], and this paper for $p=2$.

If $p=3$, the classification is conjectured. The examples given in [GL4], and
the ones obtained by the KSh-method, together with their deforms,
conjecturally exhaust the stock of “standard” examples; see also [BLW] where
an earlier claim concerning deforms is corrected.

If $p=2$, we are still completing the stock of “standard” examples.

####  1.2.3. Splitting the problem into smaller chunks.

All simple Lie algebras are of the two types: the root system of a “symmetric”
algebra contains the root $-\alpha$ of the same multiplicity as that of
$\alpha$ for any root $\alpha$; the algebras with root systems without this
property are said to be “lopsided”.

Symmetric algebras. A big quantity of “symmetric” simple Lie algebras consists
of algebras ${\mathfrak{g}}(A)$ with indecomposable Cartan matrix $A$ or a
“relative” of such an algebra of the form
${\mathfrak{g}}^{(i)}(A)/{\mathfrak{c}}$, where111This is much shorter and
more graphic than the correct notation $({\mathfrak{g}}(A))^{(i)}$; usually,
we will similarly place subscripts designating the degree, etc.
${\mathfrak{g}}^{(i)}(A)$ is the $i$th derived of ${\mathfrak{g}}(A)$ and
${\mathfrak{c}}$ is the center.

For any $p$, finite-dimensional Lie algebras ${\mathfrak{g}}(A)$ with
indecomposable Cartan matrix $A$, and their simple relatives, were classified
in [WK] with an omission; for corrections, see [Sk1, KWK], where no claim was
made that these were the only corrections needed; for the proof of this, a
classification of Lie superalgebras of the form ${\mathfrak{g}}(A)$ with
indecomposable Cartan matrix $A$, and their simple relatives, and precise
definitions of related notions, see [BGL1].

Lopsided algebras is a wild terrain a part of which we study here.

###  1.3. Super goal

Although Lie superalgebras appeared in topology in 1930s (over finite fields,
often over ${\mathbb{Z}}/2$), the understanding of their importance dawned
only in 1970s, thanks to their applications in physics. The problem “classify
simple Lie superalgebras” appeared on the agenda of researchers. Over
${\mathbb{C}}$, the finite-dimensional simple Lie superalgebras were
classified by several teams of researchers, see reviews [K2, Kapp]. The
classification of certain types of simple vectorial Lie superalgebras was
explicitly announced in [K2], together with a conjecture listing all primitive
vectorial Lie superalgebras; for the first counterexamples, see [Le2, ALSh].

A classification of the simple vectorial Lie superalgebras over ${\mathbb{C}}$
was implicitly announced when the first exceptional examples were given [Sh3,
Sh5, Sh14] and explicitly at a conference [LS0]. The claim of [K2] was being
corrected in [LS] (the correction contained the complete list of simple
vectorial algebras, bar one exception later described in [Sh5], and the method
of classification of simple ${\mathbb{Z}}$-graded Lie superalgebras of depth
1) and [K, CK1, CK1a, CK2, K10, CCK], where the proof in the case of
${\mathbb{Z}}$-grading compatible with parity was given; for further
corrections and proofs, see §13 and, ultimately, [LS1].

Although to prove classification of simple finite-dimensional Lie
superalgebras over ${\mathbb{K}}$ for $p$ “sufficiently big” (say, $>7$) will
be a more cumbersome and excruciating task than that for Lie algebras, the
answer (conjectural but doubtless) is obvious: to get restricted
superalgebras, take the obvious modular analogs of the complex simple Lie
superalgebras (both finite-dimensional and infinite-dimensional vectorial for
the shearing vector $\underline{N}=\underline{N}_{s}$, see (10)) passing to
the derived and quotients modulo center if needed; to get nonrestricted
superalgebras, consider true deforms, see Subsec. 1.2.1, of above-mentioned
analogs (for $\underline{N}$ unconstrained speaking about vectorial algebras).
For $p$ “small” the classification problem becomes more and more involved.
Nevertheless, in the 2 cases the classification is obtained:

$\bullet$ For any $p$, the super goal is reached for Lie superalgebras of the
form ${\mathfrak{g}}(A)$ with indecomposable Cartan matrices $A$, see [BGL1].
If ${\mathfrak{g}}={\mathfrak{g}}(A)$ is not simple, its “relative” of the
form ${\mathfrak{g}}^{(i)}/{\mathfrak{c}}$, where ${\mathfrak{g}}^{(i)}$ is
the $i$th derived of ${\mathfrak{g}}$ and ${\mathfrak{c}}$ its center, is
simple.

$\bullet$ Amazingly, the super goal is reached if $p=2$, see [BLLS2], with a
catch: modulo the classification of simple Lie algebras, i.e., without
explicit list of all examples. Here, we contribute new examples to a
conjectural list of “standard” simple Lie algebras, but our main result is an
explicit description of simple vectorial Lie algebras analogous to those over
${\mathbb{C}}$.

###  1.4. Getting Lie algebras from Lie superalgebras if $p=2$

Observe that since for any odd element $x\in{\mathfrak{g}}$ in any Lie
superalgebra ${\mathfrak{g}}$ over any field ${\mathbb{K}}$, we have
$[x,x]:=2x^{2}\in U({\mathfrak{g}})$. If $p=2$, then it is squaring
$x\longmapsto x^{2}$ for any odd $x\in{\mathfrak{g}}$ which, together with the
brackets of even elements with other elements, defines the multiplication in
any Lie superalgebra (for details, see Subsec. 3.1), while the bracket of odd
elements is the polarization of the squaring. Hence,

For $p=2$, every Lie superalgebra with the bracket as multiplication is a Lie
algebra.

Let F be the functor forgetting the superstructure: we only use bracket and
never squaring.

To classify simple Lie superalgebras is a much more difficult task than to
classify simple Lie algebras of the same type: the former is based on the
latter and careful study of the representation theory of Lie algebras. In
[KL], however, it was suggested to reverse the process:

(1) To obtain simple Lie algebras for $p=2$, apply the functor F to
every simple Lie superalgebra.

There are two reasons to consider Lie superalgebras to obtain Lie algebras:

(a) Although at the time [KL] was written the classification of simple
vectorial Lie superalgebras over ${\mathbb{C}}$ was only conjectural, the list
of examples was wider than that of simple vectorial Lie algebras for $p=2$,
and was (and still is, as this paper shows) able to bring about new simple
examples. In [KL], the modular analog of one of the 15 exceptional simple
complex vectorial Lie superalgebras was desuperized yielding a new simple Lie
algebra.

(b) The methodical “frontal attack” on the classification for $p=2$ should be
much more excruciating, as is clear from results in [SkT1, GZ], than that used
for $p>3$ by Premet and Strade, or even than that due to Block and Wilson for
a simpler problem of classification of restricted simple Lie algebras for
$p>7$. So, a plausible stock of simple examples will be helpful. Besides, we
give interpretations of the algebras we consider; this is of independent
interest.

Here, after a long break, we continue exploring method 1) of (1). It provides
us with new examples of simple vectorial Lie algebras of the form
$\textbf{F}({\mathfrak{g}})$, where ${\mathfrak{g}}$ is a modular, indigenous
for $p=2$, version of a simple vectorial Lie superalgebra over ${\mathbb{C}}$.

####  1.4.1. Parameters of the Lie superalgebra that change under
desuperization.

The unconstrained shearing vector, see Subsec. 3.3.1, of the vectorial Lie
algebra $\textbf{F}({\mathfrak{g}})$ may depend on more parameters than the
shearing vector of ${\mathfrak{g}}$. For example, for
${\mathfrak{vect}}(a;\underline{N}|b)$, the vector $\underline{N}$ depends on
$a$ parameters, whereas for
$\textbf{F}({\mathfrak{vect}}(a;\underline{N}|b))={\mathfrak{vect}}(a+b;\widetilde{}\underline{N})$
the shearing vector $\widetilde{}\underline{N}=(\underline{N},1\dots,1)$ has
$a+b$ coordinates while the unconstrained shearing vector depends on $a+b$
parameters. The same applies to the desuperizations of Lie superalgebras of
series ${\mathfrak{h}}$, ${\mathfrak{k}}$, ${\mathfrak{m}}$,
${\mathfrak{le}}$.

Observe that divergence-free subalgebras of ${\mathfrak{h}}$ and
${\mathfrak{le}}$ are prolongs of traceless subalgebras of their 0th
components. Therefore,

constraints imposed on the shearing vector $\underline{N}$ corresponding to
the space of generating functions determine various divergence-free
subalgebras of ${\mathfrak{h}}(n;\underline{N})$ and
${\mathfrak{le}}(n;\underline{N})$.

The same abstract Lie superalgebra ${\mathfrak{g}}$ can be realized in several
ways as vectorial ${\mathfrak{g}}(a;\underline{N}|b;r)$, by means of
Weisfeiler filtrations or associated regradings $r$, see subsection 3.6. This
${\mathfrak{g}}(a;\underline{N}|b;r)$ can be interpreted as the (generalized)
Cartan prolong of the nonpositive222In the cases of partial prolong, the
elements of degree 1 are involved. part of ${\mathfrak{g}}$ in the
corresponding grading.

The Lie algebra obtained by desuperization might acquire new properties which
its namesakes for $p\neq 2$ do not have: for example, the Poisson Lie algebra
${\mathfrak{po}}(2n;\underline{N})$ has deforms
$\textbf{F}({\mathfrak{b}}_{\lambda}(n;n;\underline{N}))$ depending on a
parameter $\lambda\in{\mathbb{K}}P^{1}$.

##### Desuperization might turn distinct (types of) Lie superalgebras into one
(type of) Lie algebra

Examples: under F

the Lie superalgebras of types ${\mathfrak{k}}$ and ${\mathfrak{m}}$ in the
standard grading (18) turn into ${\mathfrak{k}}$;

the Lie superalgebras of types ${\mathfrak{h}}_{\Pi}$ and ${\mathfrak{le}}$ in
the standard grading (18) turn into ${\mathfrak{h}}_{\Pi}$.

###  1.5. A step aside: Volichenko algebras

The notion of Volichenko algebra333In memory of I. Volichenko who was the
first to study inhomogeneous subalgebras in Lie superalgebras., which is an
inhomogeneous relative parity subspace ${\mathfrak{h}}\subset{\mathfrak{g}}$
of a Lie superalgebra ${\mathfrak{g}}$ closed with respect to the superbracket
in ${\mathfrak{g}}$, was introduced in [KL]. For a classification of simple
(without any nontrivial ideals, both one-sided and two-sided) finite-
dimensional Volichenko algebras over ${\mathbb{C}}$ under a certain
(hopefully, inessential) technical assumption, and examples of certain
infinite-dimensional algebras, see [LSerg].

The results of [ILMS] suggested to look at the definition of the Lie
superalgebra for $p=2$, see sec. 3.1, more carefully. If one does this, it is
not difficult to deduce that

(2) $\textbf{if $p=2$, the Volichenko algebras are, actually, Lie algebras}.$

In [KL] and [ILMS] the fact (2) had not been understood.

##  2\. Introduction, cont. Our strategy, main results and open problems

###  2.1. Generalized Cartan prolongation as a main procedure for getting
vectorial Lie (super)algebras

In [Lei], it is shown how to obtain simple Lie algebras over ${\mathbb{C}}$ of
the two types of prime importance for the classification procedure over
${\mathbb{K}}$: finite-dimensional and vectorial by induction and using
(generalized, in particular, partial) Cartan prolongation, see Subsec. 3.4.
One thus first obtains all finite-dimensional Lie algebras (each of them has a
Cartan matrix); during the next step one obtains vectorial Lie algebras.

The same method works for $p$ sufficiently big, and, with new standard
examples added, for $p>2$. After that, there remain considerable technical
problems: describe deforms and classify nonisomorphic deforms.

Super versions of the classification for any characteristic is much more
complicated than its nonsuper counterpart, we have to supply the input with
several more types of algebras, but the main procedures are still the same:
generalized, especially partial, prolongations and deformations.

The super classification shares its features with the classification of simple
Lie algebras for $p=2$; more exactly, we have in mind the methods for
producing the list of simple examples, not the proof of its completeness.

Over ${\mathbb{C}}$, the following fact is well-known:

(3) given a simple Lie algebra of the form ${\mathfrak{g}}(A)$, and its
${\mathbb{Z}}$-grading, the generalized prolong of the nonpositive (with
respect to that grading) part of ${\mathfrak{g}}(A)$ is isomorphic to
${\mathfrak{g}}(A)$, bar two series of exceptions corresponding to certain
simplest gradings of the embedded algebras —
${\mathfrak{sl}}(n+1)\subset{\mathfrak{vect}}(n)$ and
${\mathfrak{sp}}(2n+2)\subset{\mathfrak{k}}(2n+1)$ — and the ambients are the
exceptional prolongs.

In several papers we considered simple Lie (super)algebras, their
${\mathbb{Z}}$-gradings either simplest, or, additionally, almost simplest if
$p=2$, and investigated the prolongs of the nonpositive parts.

Here we consider desuperizations of the analogs for $p=2$ of complex serial
and exceptional simple vectorial Lie superalgebras. Realization of a Lie
(super)algebra in terms of vector fields implies that ${\mathfrak{g}}$ is
endowed with a filtration, one of them, called Weisfeiler filtration, is most
important, see eq. (17). Associated with the filtration is the grading; the
Weisfeiler ones are briefly called the W-filtration and W-grading. We
investigate if it suffices to consider these algebras as abstract ones, not as
W-filtered or W-graded. In other words, for several W-gradings of vectorial
Lie algebras, we describe an analog of the fact (3): when the prolongs of the
nonpositive parts corresponding to nonisomorphic regradings produce regradings
of the same prolong, and when the prolongs of the nonpositive parts of
different regradings are nonisomorphic.

###  2.2. “Hidden supersymmetries” of Lie algebras

It is sometimes possible to endow the space of a given Lie algebra
${\mathfrak{g}}$ with (several) Lie superalgebra structures. For example, for
${\mathfrak{g}}={\mathfrak{sl}}(n)$ over any ground field, consider any
distribution of parities (of the pairs corresponding to positive and negative
simple roots) of the Chevalley generators; we thus get Lie superalgebras
${\mathfrak{sl}}(a|b)$ for $a+b=n$ in various supermatrix formats. The sets of
defining relations between the Chevalley generators corresponding to different
formats are different.

It is, clearly, possible to perform such changes of parities of (pairs of)
Chevalley generators for any simple Lie algebra but, except for
${\mathfrak{sl}}(n)$, the simple Lie algebras obtained by factorization modulo
the ideal of relations, see [BGL1], are infinite-dimensional unless $p=2$.

If $p=2$, the following fact is obvious (here $x\longmapsto x^{[2]}$ is the
restriction map and $x\longmapsto x^{2}$ is the squaring, cf. (7)):

(4) any classically444In [BLLS2], in addition to the classical restrictedness,
we introduced another, indigenous to $p=2$, versions of restrictedness, whose
meaning is yet unclear but since they pertain to classical and often used
algebras, e.g., ${\mathfrak{o}}(2n+1)$ and
${\mathfrak{h}}(2n+1;\underline{N}_{s})$, we believe that these new
“restrictednesses” are important. restricted and ${\mathbb{Z}}/2$-graded Lie
algebra ${\mathfrak{g}}={\mathfrak{g}}_{+}\oplus{\mathfrak{g}}_{-}$ can be
turned into a Lie superalgebra by setting $x^{2}:=x^{[2]}$ for any
$x\in{\mathfrak{g}}_{-}$.

In [BLLS2], a (rather unexpected) generalization of the possibility (4) is
described: every simple Lie algebra ${\mathfrak{g}}$ can be turned into a
simple Lie superalgebra by slightly enlarging its space if ${\mathfrak{g}}$ is
not restricted. This generalization, and a “queerification”, are the two
methods which from every simple Lie algebra produce simple Lie superalgebras,
and every simple Lie superalgebra can be obtained in one of these two ways as
proved in [BLLS2]. These two methods applied to the simple Lie algebras we
describe in this paper yield a huge quantity of new simple Lie superalgebras,
both serial and exceptional. We do not list them, this is a routine work
modulo a far from routine, see [KLL], job of describing all
${\mathbb{Z}}/2$-gradings of our newly found simple Lie algebras.

###  2.3. Our strategy

We consider the W-grading (of the desuperized $p=2$ version of the simple
vectorial Lie superalgebra over ${\mathbb{C}}$) for which the nonpositive
part, especially the 0th component, is most clear. Then we consider the
regradings described in Table (181), and perform generalized Cartan
prolongation of the nonpositive (or negative) part of the regraded algebra in
the hope to get a new simple Lie (super)algebra, as in (3).

When this approach is inapplicable since there is no visible analog, suitable
for $p=2$, of the Lie (super)algebras we worked with over ${\mathbb{C}}$, we
consider the description of the Lie superalgebras as the sum of its even and
odd parts, desuperize this description and only after this consider the
W-grading of the desuperization.

We begin with serial simple vectorial Lie superalgebras over ${\mathbb{C}}$ in
their standard W-grading.

Next, for each of the five exceptional simple vectorial Lie superalgebras over
${\mathbb{C}}$, we consider desuperizations of their 15 W-gradings.

###  2.4. Main results

1) We describe, as complete or partial Cartan prolongs, the modular $p=2$
analogs of

(a) The serial simple (for generic $\frac{a}{b}$) Lie algebras
${\mathfrak{po}}_{a,b}(2n;\widetilde{}\underline{N})$, their simple relatives
for $\frac{a}{b}=0$ and $\infty$, see (55), and also
$\widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\nu}(2^{n}-1;\widetilde{}\underline{N})$.
These simple Lie algebras are new.

The divergence-free subalgebras
${\mathfrak{s}}{\mathfrak{h}}(2n;\widetilde{}\underline{N})$ of the Lie
algebra
${\mathfrak{h}}(2n;\underline{M})=\textbf{F}({\mathfrak{le}}(n;\underline{N}|n))$
of Hamiltonian vector fields, discovered in [Leb], are singled out here by
imposing constraints on $\underline{M}$.

(b) The generalized Cartan prolongs of nonpositive (or negative) parts of all
15 W-gradings, see Table 23.2, of the 5 exceptional simple vectorial Lie
algebras — desuperizations of $p=2$ analogs of the exceptional simple
vectorial Lie superalgebras over ${\mathbb{C}}$, see Subsec. 3.16. The simple
Lie algebras thus obtained are new, see Table 23.4, two of them,
${\mathfrak{v}}{\mathfrak{le}}(9;\underline{N})$ and
${\mathfrak{m}}{\mathfrak{b}}_{3}(11;\underline{N})$, are described from
another point of view in [BGLLS].

For $p\neq 2$, the 15 W-graded analogs of the exceptional simple vectorial Lie
superalgebras over ${\mathbb{C}}$ constitute, as abstract ones, 5 Lie
superalgebras; all of their finite-dimensional analogs for $p=2$ remain
isomorphic in the following cases:

(5)
$\begin{array}[]{l}{\mathfrak{v}}{\mathfrak{le}}(7;\widetilde{\underline{L}})\simeq{\mathfrak{v}}{\mathfrak{le}}(9;\widetilde{\underline{M}})\simeq\widetilde{{\mathfrak{v}}{\mathfrak{le}}}(9;\widetilde{}\underline{N});\\\
{\mathfrak{m}}{\mathfrak{b}}_{3}(9;\widetilde{\underline{L}})\simeq{\mathfrak{m}}{\mathfrak{b}}_{3}(11;\widetilde{\underline{M}})\simeq{\mathfrak{m}}{\mathfrak{b}}_{2}(11;\widetilde{}\underline{N});\\\
\widetilde{{\mathfrak{k}}{\mathfrak{le}}}(15;\widetilde{\underline{L}})\simeq{\mathfrak{k}}{\mathfrak{le}}_{3}(20;\widetilde{\underline{M}})\simeq{\mathfrak{k}}{\mathfrak{le}}_{2}(20;\widetilde{}\underline{N}).\end{array}$

For $p=2$, the analogs of the W-graded simple vectorial Lie superalgebra
${\mathfrak{kas}}$ over ${\mathbb{C}}$ are not simple; they contain a simple
ideal of codimension 1, the derived algebra. Moreover, instead of 4
W-regradings of one abstract Lie superalgebra there appear 3 new simple Lie
algebras, one of which has 2 W-regradings. Namely, let
$\dim\widetilde{}\underline{N}$ denote the dimension of the parameter space
the shearing vector depends on555The (non)critical coordinates of the shearing
vectors are established with computer’s aid in all cases by explicit
computation of the bases of the first several terms ${\mathfrak{g}}_{i}$.;
then we have:

(6)
$\begin{array}[]{ll}{\mathfrak{k}}{\mathfrak{as}}(7;\widetilde{\underline{L}})\simeq{\mathfrak{k}}{\mathfrak{as}}(8;\widetilde{\underline{M}}),&\dim\widetilde{\underline{L}}=\dim\widetilde{\underline{M}}=1;\\\
{\mathfrak{k}}{\mathfrak{as}}(10;\widetilde{}\underline{N}),&\dim\widetilde{}\underline{N}=2;\\\
\widetilde{{\mathfrak{k}}{\mathfrak{as}}}(7;\widetilde{}\underline{N}),&\dim\widetilde{}\underline{N}=4.\end{array}$

2) We show that all regradings of
${\mathfrak{m}}{\mathfrak{b}}_{3}(11;\underline{N})$ are regradings of a true
deform of ${\mathfrak{svect}}(5;\underline{N})$, see Subsec. 1.2.1.

Other results: We explicitly describe Shen’s version of both
${\mathfrak{g}}(2)$ and the Melikyan algebra as vectorial Lie algebra.

####  2.4.1. Open problems.

1) Do gradings of exceptional and serial vectorial Lie superalgebras other
than those we have considered in this paper yield new simple Lie algebras? The
fact (3) and result (6) suggest to inspect the other 32 W-gradings of serial
Lie superalgebras of 34 possible ones, see [LS1], Subsec. 1.16.

2) Unlike ${\mathfrak{vas}}$, associated with the spinor representation
([LSh]) of the only nontrivial central extension of ${\mathfrak{spe}}(n)$
existing for $p\neq 2,3$ only for $n=4$, there might exist several new simple
${\mathbb{Z}}$-graded vectorial Lie superalgebras ${\mathfrak{g}}$ with
${\mathfrak{g}}_{0}$ equal to some of the nontrivial central extensions of
various (existing only if $p=2$ or 3) analogs of ${\mathfrak{spe}}(n)$, see
[BGLL1]. Are there any?

3) Describe all ${\mathbb{Z}}/2$-grading of the newly found simple Lie
algebras to obtain new simple Lie superalgebras, see [BLLS2].

4) Is the simple Lie algebra of dimension 59 described in Subsec. 19.1 new?

##  3\. Notation and background

Here, we remind the basics and give modifications of the familiar formulas
needed under the passage from ${\mathbb{C}}$ to the fields of characteristic
2. In some formulas given for $p=2$, we retain signs convenient for comparison
with the cases where $p\neq 2$.

As is now customary, we denote the elements of ${\mathbb{Z}}/2$ by ${\bar{0}}$
and ${\bar{1}}$, to distinguish them from integers.

We denote by ${\mathfrak{c}}$ the center of a given Lie (super)algebra;
${\mathfrak{c}}({\mathfrak{g}})$ or
${\mathfrak{c}}{\mathfrak{g}}:={\mathfrak{c}}\oplus{\mathfrak{g}}$ denotes the
result of the trivial central extension of ${\mathfrak{g}}$. Let
${\mathfrak{g}}^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}k}:=[{\mathfrak{g}},[{\mathfrak{g}},\dots[{\mathfrak{g}},{\mathfrak{g}}]\dots]$,
the $k$-fold bracket. Let ${\mathfrak{a}}\ltimes{\mathfrak{b}}$ or
${\mathfrak{b}}\rtimes{\mathfrak{a}}$ denote the semidirect sum of modules
(algebras) in which ${\mathfrak{a}}$ is a submodule (ideal).

###  3.1. Definition of Lie superalgebras for $p=2$

For $p=2$, a Lie superalgebra is a superspace
${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}$ such
that ${\mathfrak{g}}_{\bar{0}}$ is a Lie algebra, ${\mathfrak{g}}_{\bar{1}}$
is a ${\mathfrak{g}}_{\bar{0}}$-module (made two-sided by symmetry) and with a
squaring (roughly speaking, the halved bracket) defined on
${\mathfrak{g}}_{\bar{1}}$

(7) $\begin{array}[]{c}x\longmapsto x^{2}\quad\text{such that
$(ax)^{2}=a^{2}x^{2}$ for any $x\in{\mathfrak{g}}_{\bar{1}}$ and
$a\in{\mathbb{K}}$, and}\\\ {}[x,y]:=(x+y)^{2}-x^{2}-y^{2}\text{~{}is a
bilinear form on ${\mathfrak{g}}_{\bar{1}}$ with values in
${\mathfrak{g}}_{\bar{0}}$.}\end{array}$

The Jacobi identity turns, if $p=2$, into the following two conditions unless
${\mathbb{K}}={\mathbb{Z}}/2$:

(8) $\begin{array}[]{l}~{}[x^{2},y]=[x,[x,y]]\text{~{}for
any~{}}x\in{\mathfrak{g}}_{\bar{1}},y\in{\mathfrak{g}}_{\bar{0}},\\\
~{}[x^{2},x]=0\;\text{ for any $x\in{\mathfrak{g}}_{\bar{1}}$.}\end{array}$

For ${\mathbb{K}}={\mathbb{Z}}/2$, the conditions (8) should be replaced with
the following one:

(9) $~{}[x^{2},y]=[x,[x,y]]\text{~{}for
any~{}}x\in{\mathfrak{g}}_{\bar{1}},y\in{\mathfrak{g}}.$

Because of the squaring, the definition of derived algebras changes. For any
Lie superalgebra ${\mathfrak{g}}$, set ${\mathfrak{g}}^{(0)}:={\mathfrak{g}}$;

$\begin{array}[]{l}{\mathfrak{g}}^{(1)}:=[{\mathfrak{g}},{\mathfrak{g}}]+\mathop{\mathrm{{Span}}}\nolimits\\{g^{2}\mid
g\in{\mathfrak{g}}_{\bar{1}}\\},\\\
{\mathfrak{g}}^{(i+1)}=({\mathfrak{g}}^{(i)})^{(1)}:=[{\mathfrak{g}}^{(i)},{\mathfrak{g}}^{(i)}]+\mathop{\mathrm{{Span}}}\nolimits\\{g^{2}\mid
g\in{\mathfrak{g}}^{(i)}_{\bar{1}}\\}\text{~{}~{} for any $i\geq
0$}.\end{array}$

An even linear map $r:{\mathfrak{g}}\longrightarrow{\mathfrak{gl}}(V)$ is said
to be a representation of the Lie superalgebra ${\mathfrak{g}}$ and $V$ is a
${\mathfrak{g}}$-module if

$\begin{array}[]{l}r([x,y])=[r(x),r(y)]\quad\text{ for any
$x,y\in{\mathfrak{g}}$;}\\\ r(x^{2})=(r(x))^{2}\text{~{}for any
$x\in{\mathfrak{g}}_{\bar{1}}$.}\end{array}$

###  3.2. Analogs of polynomials for $p>0$

Consider the supercommutative superalgebra ${\mathbb{C}}[x]$ of polynomials in
$a$ indeterminates $x=(x_{1},...,x_{a})$, for convenience let the first $m$
indeterminates be even and the rest $n$ ones odd ($m+n=a$). Among the bases of
${\mathbb{C}}[x]$ in which the structure constants are integers, there are two
canonical ones: the monomial one and the basis of divided powers constructed
as follows.

For any multi-index $\underline{r}=(r_{1},\ldots,r_{a})$, where
$r_{1},\dots,r_{m}\in{\mathbb{N}}$ and $r_{m+1},\dots,r_{n}\in\\{0,1\\}$, we
set

$u_{i}^{(r_{i})}:=\frac{x_{i}^{r_{i}}}{r_{i}!}\quad\text{and}\quad
u^{(\underline{r})}:=\prod\limits_{1\leq i\leq a}u_{i}^{(r_{i})}.$

These $u^{(\underline{r})}$ form an integer basis of ${\mathbb{C}}[x]$.
Clearly, we have

$\begin{array}[]{l}u^{(\underline{r})}\cdot
u^{(\underline{s})}=\mathop{\prod}\limits_{m+1\leq i\leq
n}\min(1,2-r_{i}-s_{i})\cdot(-1)^{\mathop{\sum}\limits_{m<i<j\leq
a}r_{j}s_{i}}\cdot\Big{(}\mathchoice{\raisebox{0.25pt}{$\genfrac{}{}{0.0pt}{0}{\underline{r}+\underline{s}}{\underline{r}}$}}{\raisebox{0.25pt}{$\genfrac{}{}{0.0pt}{0}{\underline{r}+\underline{s}}{\underline{r}}$}}{\raisebox{-0.5pt}{$\genfrac{}{}{0.0pt}{1}{\underline{r}+\underline{s}}{\underline{r}}$}}{\raisebox{-0.5pt}{$\genfrac{}{}{0.0pt}{1}{\underline{r}+\underline{s}}{\underline{r}}$}}\Big{)}u^{(\underline{r}+\underline{s})},\\\
\text{where}\quad\Big{(}\mathchoice{\raisebox{0.25pt}{$\genfrac{}{}{0.0pt}{0}{\underline{r}+\underline{s}}{\underline{r}}$}}{\raisebox{0.25pt}{$\genfrac{}{}{0.0pt}{0}{\underline{r}+\underline{s}}{\underline{r}}$}}{\raisebox{-0.5pt}{$\genfrac{}{}{0.0pt}{1}{\underline{r}+\underline{s}}{\underline{r}}$}}{\raisebox{-0.5pt}{$\genfrac{}{}{0.0pt}{1}{\underline{r}+\underline{s}}{\underline{r}}$}}\Big{)}:=\mathop{\prod}\limits_{1\leq
i\leq
m}\Big{(}\mathchoice{\raisebox{0.25pt}{$\genfrac{}{}{0.0pt}{0}{r_{i}+s_{i}}{r_{i}}$}}{\raisebox{0.25pt}{$\genfrac{}{}{0.0pt}{0}{r_{i}+s_{i}}{r_{i}}$}}{\raisebox{-0.5pt}{$\genfrac{}{}{0.0pt}{1}{r_{i}+s_{i}}{r_{i}}$}}{\raisebox{-0.5pt}{$\genfrac{}{}{0.0pt}{1}{r_{i}+s_{i}}{r_{i}}$}}\Big{)}.\end{array}$

In what follows, for clarity, we will write exponents of divided powers in
parentheses.

Over any field ${\mathbb{K}}$ of characteristic $p>0$, we consider the
supercommutative superalgebra

${\mathcal{O}}(a;\underline{N}):={\mathbb{K}}[u;\underline{N}]:=\mathop{\mathrm{{Span}}}\nolimits_{{\mathbb{K}}}\left(u^{(\underline{r})}\mid
r_{i}\begin{cases}<p^{N_{i}}&\text{for $i\leq m$}\\\ =0\text{ or 1}&\text{for
$i>m$}\end{cases}\right),$

where $\underline{N}=(N_{1},...,N_{m},1,\dots,1)$ is the shearing vector with
$N_{i}\in{\mathbb{N}}\cup\infty$ (we assume that $p^{\infty}=\infty$). The
algebra ${\mathcal{O}}(a;\underline{N})={\mathbb{K}}[u;\underline{N}]$,
sometimes briefly denoted ${\mathcal{F}}$, is called the algebras of divided
powers.

Important particular cases:

(10)
$\underline{N}_{s}:=(1,\dots,1)\text{~{}~{}and~{}~{}}\underline{N}_{\infty}:=(\infty,\dots,\infty).$

Clearly,
${\mathcal{O}}(a;\underline{N}_{s})={\mathbb{K}}[u;\underline{N}_{s}]$ is the
algebra of truncated polynomials. Only ${\mathbb{K}}[u;\underline{N}_{s}]$ is
indeed generated by the indeterminates declared. The list of generators of
${\mathbb{K}}[u;\underline{N}]$ consists of $u_{i}^{(p^{k_{i}})}$ for all $i$
and all $k_{i}$ such that $1\leq k_{i}<N_{i}$ when $u_{i}$ are even.

###  3.3. Vectorial Lie algebras and superalgebras

Since any derivation $D$ of a given algebra is determined by the values of $D$
on the generators, we see that the Lie algebra
${\mathfrak{der}}({\mathcal{O}}(m;\underline{N}))$ of all derivations of
${\mathcal{O}}(m;\underline{N})$ has more than $m$ functional parameters
(coefficients of the analogs of partial derivatives) if $N_{i}\neq 1$ for at
least one $i$. The distinguished666These derivatives are sometimes called
special which is an ill-chosen term in view of the fact that the Lie
(super)algebra of divergence-free vector field is called special, and hence
all its elements are special. partial derivatives are defined by the formula

$\partial_{i}(u_{j}^{(k)})=\delta_{ij}u_{j}^{(k-1)}\;\text{ for all
$k<p^{N_{j}}$}.$

The Lie algebra of all derivations
${\mathfrak{der}}({\mathcal{O}}(m;\underline{N}))$ turns out to be not so
interesting (because it is not transitive, see [Dr]) as its Lie subalgebra of
its distinguished derivations

${\mathfrak{vect}}(m;\underline{N})=\left\\{\sum f_{i}\partial_{i}\mid
f_{i}\in{\mathcal{O}}(m;\underline{N})\right\\}$

often denoted $W(m;\underline{N})$ in honor of Witt who was the first to study
it.

Superization is immediate. The Lie superalgebra
${\mathfrak{vect}}(n;\underline{N}|m)$, and its subalgebras, are called the
vectorial Lie superalgebras. The symbol ${\mathfrak{g}}(a|b)$ denotes the Lie
superalgebra with given name ${\mathfrak{g}}$ realized by vector fields on the
space of superdimension $a|b$ and endowed with the standard grading (e.g., for
${\mathfrak{vect}}$ the degrees of all indeterminates are equal to 1; for the
contact series the “time” indeterminate is of degree 2); this realization is
taken as a point of reference for regradings governed by the parameter $r$ and
usually (for details, see [Sh14, LS]) denoting the number of odd
indeterminates of degree 0. The regraded Lie superalgebra is denoted by
${\mathfrak{g}}(a|b;r)$. In the standard grading ($r=0$) the parameter $r$ is
not indicated.

The ${\mathfrak{vect}}(m;\underline{N}|n)$-module of “functions” is sometimes
abbreviated to $\mathop{\mathrm{id}}\nolimits$, it is the identity or
tautological module over ${\mathfrak{vect}}(m;\underline{N}|n;r)$ and its
subalgebras with the same negative part. For brevity this module is often
shorthanded as

(11) ${\mathcal{F}}:={\mathcal{O}}(m;\underline{N}|n;r).$

####  3.3.1. Critical coordinates and unconstrained shearing vectors.

The coordinate of the shearing vector $\underline{N}$ is said to be critical
if it can not take however big value. Obviously, all the coordinates of
$\underline{N}$ concerning the odd indeterminates are critical; in what
follows the notion is only applied to the even indeterminates. The shearing
vector without any restrictions on its coordinates is said to be of
unconstrained form; we denote it $\underline{N}^{u}$.

######  Conjecture.

If the value of the unconstrained coordinate $\underline{N}_{i}$ can be $>1$,
then it can take any value except for the coordinate corresponding to one
indeterminate for ${\mathfrak{vle}}$, see (87) and (90).

###  3.4. The (generalized) Cartan prolongation

Let ${\mathfrak{g}}_{-}=\mathop{\oplus}\limits_{-d\leq
i\leq-1}{\mathfrak{g}}_{i}$ be a nilpotent ${\mathbb{Z}}$-graded Lie algebra
and ${\mathfrak{g}}_{0}$ a Lie subalgebra of the Lie algebra
${\mathfrak{der}}_{0}{\mathfrak{g}}$ of ${\mathbb{Z}}$-grading-preserving
derivations. Let

(12) $DS^{k}$ be the operation of rising to the $k$th divided symmetric power
and $DS^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}:=\oplus_{k}DS^{k}$.

Define the natural maps $i$, $j$, and the $(k,\underline{N})$th prolong of the
pair $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})$ by setting

$\begin{array}[]{l}i\colon
DS^{k+1}({\mathfrak{g}}_{-})^{*}\otimes{\mathfrak{g}}_{-}\longrightarrow
DS^{k}({\mathfrak{g}}_{-})^{*}\otimes{\mathfrak{g}}_{-}^{*}\otimes{\mathfrak{g}}_{-};\\\
j\colon DS^{k}({\mathfrak{g}}_{-})^{*}\otimes{\mathfrak{g}}_{0}\longrightarrow
DS^{k}({\mathfrak{g}}_{-})^{*}\otimes{\mathfrak{g}}_{-}^{*}\otimes{\mathfrak{g}}_{-};\\\
{\mathfrak{g}}_{k,\underline{N}}:=\left(j(DS^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}}_{-})^{*}\otimes{\mathfrak{g}}_{0})\cap
i(DS^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}}_{-})^{*}\otimes{\mathfrak{g}}_{-})\right)_{k,\underline{N}},\end{array}$

where the subscript $k$ in the right hand side singles out the component of
degree $k$.

Superization of the prolongation construction is immediate.

Set
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*,\underline{N}}:=\mathop{\oplus}\limits_{i\geq-d}{\mathfrak{g}}_{i,\underline{N}}$;
then, as is easy to verify,
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*,\underline{N}}$ is a Lie
(super)algebra. Provided ${\mathfrak{g}}_{0}$ acts on ${\mathfrak{g}}_{-1}$
without kernel, $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{*,\underline{N}}$ is
a subalgebra of ${\mathfrak{vect}}(m;\underline{N}|n)$ for
$m|n=\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{-}$.

####  3.4.1. A partial Cartan prolong: prolongation of a positive part.

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

(13)
${\mathfrak{h}}_{2}=\\{D\in{\mathfrak{g}}_{2}\mid[D,{\mathfrak{g}}_{-1}]\in{\mathfrak{h}}_{1}\\}.$

The terms ${\mathfrak{h}}_{i}$, where $i>2$, are similarly defined. Set
${\mathfrak{h}}_{i}={\mathfrak{g}}_{i}$ for $i\leq 0$ and
${\mathfrak{h}}_{*}=\oplus{\mathfrak{h}}_{i}$.

Examples: ${\mathfrak{vect}}(1|n;n)$ 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] is another example.

####  3.4.2. The serial simple vectorial Lie superalgebras over ${\mathbb{C}}$
as prolongs.

Since we only need the vectorial Lie superalgebras considered as abstract, not
realized by vector fields, we may consider their simplest filtrations with the
smallest codimension of their maximal subalgebras.

The central in ${\mathfrak{g}}_{0}$ element $z\in{\mathfrak{g}}_{0}$ is
supposed to be chosen so that it acts on ${\mathfrak{g}}_{k}$ as
$k\cdot\mathop{\mathrm{id}}\nolimits$. The symbol
$\mathop{\mathrm{id}}\nolimits$ denotes not only the identity operator but the
tautological module $V$ over the linear Lie superalgebra
${\mathfrak{g}}\subset{\mathfrak{gl}}(V)$; sometimes we write
$\mathop{\mathrm{id}}\nolimits_{\mathfrak{g}}:=V$ for clarity.

###  3.5. Traces and divergencies on vectorial Lie superalgebras

On any Lie algebra ${\mathfrak{g}}$ over a field ${\mathbb{K}}$, a trace is
any linear map
$\mathop{\mathrm{tr}}\nolimits:{\mathfrak{g}}\longrightarrow{\mathbb{K}}$ such
that

(14) $\mathop{\mathrm{tr}}\nolimits({\mathfrak{g}}^{(1)})=0.$

Let now ${\mathfrak{g}}$ be a ${\mathbb{Z}}$-graded vectorial Lie algebra with
${\mathfrak{g}}_{-}:=\mathop{\oplus}\limits_{i<0}{\mathfrak{g}}_{i}$ generated
by ${\mathfrak{g}}_{-1}$, and let $\mathop{\mathrm{tr}}\nolimits$ be a trace
on ${\mathfrak{g}}_{0}$. Recall that any ${\mathbb{Z}}$-grading of a given
vectorial Lie algebra is given by degrees of the indeterminates, so the space
of functions is also ${\mathbb{Z}}$-graded. The divergence is an
$\mathop{\mathrm{ad}}\nolimits_{{\mathfrak{g}}_{-1}}$-invariant prolongation
$\mathop{\text{\rm div}}\nolimits:{\mathfrak{g}}\longrightarrow{\mathcal{F}}$,
see (11), of the trace satisfying the following conditions:

$\begin{array}[]{l}\mathop{\text{\rm
div}}\nolimits:{\mathfrak{g}}\longrightarrow{\mathcal{F}}\text{~{}~{}
preserves the degree, i.e., $\deg\mathop{\text{\rm div}}\nolimits=0$};\\\
X_{i}(\mathop{\text{\rm div}}\nolimits D)=\mathop{\text{\rm
div}}\nolimits[X_{i},D]\text{~{}~{} for all elements $X_{i}$ that span
${\mathfrak{g}}_{-1}$};\\\ \mathop{\text{\rm
div}}\nolimits|_{{\mathfrak{g}}_{0}}=\mathop{\mathrm{tr}}\nolimits;\\\
\mathop{\text{\rm div}}\nolimits|_{{\mathfrak{g}}_{-}}=0.\end{array}$

Let ${\mathcal{F}}$ be the superalgebra of “functions” in $u$. We denote by
$\mathop{\mathrm{{Vol\/}}}\nolimits_{u}:={\mathcal{F}}^{*}$ the dual
${\mathfrak{vect}}(n;\underline{N}|m)$-module; as ${\mathcal{F}}$-module it is
generated by the volume element $\mathop{\mathrm{{vol\/}}}\nolimits_{u}=1^{*}$
with fixed indeterminates (“coordinates”) $u$ which we often do not indicate.
The rank 1 ${\mathcal{F}}$-module of weighted $\lambda$-densities is
$\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}$; the
${\mathfrak{vect}}(n;\underline{N}|m)$-action on
$\mathop{\mathrm{{Vol\/}}}\nolimits^{\lambda}$ is given by the Lie derivative

(15)
$L_{D}(f\mathop{\mathrm{{vol\/}}}\nolimits_{u}^{\lambda})=\left(D(f)+(-1)^{p(f)p(D)}\lambda
f\mathop{\text{\rm
div}}\nolimits(D)\right)\mathop{\mathrm{{vol\/}}}\nolimits_{u}^{\lambda}.$

By construction, the special Lie algebra
${\mathfrak{s}}{\mathfrak{g}}:=\mathop{\mathrm{Ker}}\nolimits\mathop{\text{\rm
div}}\nolimits|_{{\mathfrak{g}}}$ of divergence-free elements of
${\mathfrak{g}}$ is the complete prolong of
$({\mathfrak{g}}_{-},\mathop{\mathrm{Ker}}\nolimits\mathop{\mathrm{tr}}\nolimits|_{{\mathfrak{g}}_{0}})$;
e.g., ${\mathfrak{svect}}(n;\underline{N}|m)$ is the prolong of
${\mathfrak{sl}}(n|m)$.

The ${\mathfrak{svect}}(0|m)$-module $\mathop{\mathrm{{Vol\/}}}\nolimits(0|m)$
contains a submodule of codimension 1 which we often identify with a submodule
of ${\mathcal{O}}(0|m)$:

(16)
$\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|m):=\\{f\mathop{\mathrm{{vol\/}}}\nolimits\mid\int
f\mathop{\mathrm{{vol\/}}}\nolimits=0\\},\ \
T_{0}^{0}:=\mathop{\mathrm{{Vol\/}}}\nolimits_{0}(0|4)/{\mathbb{K}}\cdot\mathop{\mathrm{{vol\/}}}\nolimits.$

If there are several traces on ${\mathfrak{g}}_{0}$, and hence divergences on
${\mathfrak{g}}$, there are several types of special subalgebras, and we need
an individual name for each.

If ${\mathfrak{g}}$ is a Lie superalgebra, then the linear functional
$\mathop{\mathrm{tr}}\nolimits$ satisfying condition (14) is often called, for
emphasis, supertrace and denoted $\mathop{\mathrm{str}}\nolimits$. If we were
consistent, we should, accordingly, use the term superdivergence but instead
we drop the preface “super” in both cases.

###  3.6. Weisfeiler gradings

For vectorial Lie superalgebras, the invariant notion is filtration, not
grading. For $p=0$, the Weisfeiler filtrations were used in description of
infinite dimensional Lie (super)algebras ${\mathcal{L}}$ by selecting a
maximal subalgebra ${\mathcal{L}}_{0}$ of finite codimension; for the simple
vectorial Lie algebra, there is only one such ${\mathcal{L}}_{0}$. (Dealing
with finite dimensional algebras for $p>0$, we can confine ourselves to
maximal subalgebras of least codimension, or “almost least”.)

Let ${\mathcal{L}}_{-1}$ be a minimal ${\mathcal{L}}_{0}$-invariant subspace
strictly containing ${\mathcal{L}}_{0}$; for $i\geq 1$, set:

${\mathcal{L}}_{-i-1}=[{\mathcal{L}}_{-1},{\mathcal{L}}_{-i}]+{\mathcal{L}}_{-i}\;\text{
and
}\;{\mathcal{L}}_{i}=\\{D\in{\mathcal{L}}_{i-1}\mid[D,{\mathcal{L}}_{-1}]\subset{\mathcal{L}}_{i-1}\\}.$

We thus get a filtration:

(17)
${\mathcal{L}}={\mathcal{L}}_{-d}\supset{\mathcal{L}}_{-d+1}\supset\dots\supset{\mathcal{L}}_{0}\supset{\mathcal{L}}_{1}\supset\dots$

The $d$ in (17) is called the depth of ${\mathcal{L}}$, and of the associated
Weisfeiler-graded Lie superalgebra
${\mathfrak{g}}=\mathop{\oplus}\limits_{-d\leq i}{\mathfrak{g}}_{i}$, where
${\mathfrak{g}}_{i}={\mathcal{L}}_{i}/{\mathcal{L}}_{i+1}$. We will be briefly
say W-graded and W-filtered.

For the list of simple W-graded vectorial Lie superalgebras
${\mathfrak{g}}=\mathop{\oplus}\limits_{-d\leq i}{\mathfrak{g}}_{i}$ over
${\mathbb{C}}$, see [LS] reproduced in Tables 23.1 and 23.2.

The ${\mathbb{Z}}$-gradings of vectorial Lie superalgebras are defined by the
vector $\vec{r}$ of degrees of the indeterminates, but this vector can be
shorthanded for W-gradings to a number $r$, or a symbol. Let the
indeterminates $u_{i}$ be even, the $\theta_{j}$ odd. In (18), we consider
${\mathfrak{k}}(2n+1|m)$ preserving the Pfaff equation
$\widetilde{\alpha}_{1}(X)=0$ for $X\in{\mathfrak{vect}}(2n+1|m)$; here
$u=(t;p,q)$, $\theta=(\xi,\eta)$ for $m$ even, etc., see (19). For the series
${\mathfrak{m}}$, the indeterminates are denoted as in (23).

In (18), the “standard” realizations correspond to $r=0$, they are marked by
an asterisk $(*)$. For $r=0$, the codimension of ${\mathcal{L}}_{0}$ is the
smallest.

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

###  3.7. Natural generators of vectorial Lie superalgebras

Let ${\mathfrak{g}}=\oplus{\mathfrak{g}}_{i}$ be a Weisfeiler grading of a
given simple vectorial Lie superalgebra. We see that ${\mathfrak{g}}_{-1}$ is
an irreducible ${\mathfrak{g}}_{0}$-module with highest weight vector $H$, and
${\mathfrak{g}}_{1}$ is the direct sum of indecomposable (sometimes,
irreducible) ${\mathfrak{g}}_{0}$-modules with lowest weight vectors $v_{i}$.

Over ${\mathbb{C}}$, and over ${\mathbb{K}}$ for
$\underline{N}=\underline{N}_{s}$, the simple Lie superalgebra
${\mathfrak{g}}$ is generated (bar a few exceptions) by the generators of
${\mathfrak{g}}_{0}$, the vector $H$, and the $v_{i}$. (For other values of
$\underline{N}$, we have to add the ${\mathfrak{g}}_{0}$-lowest weight vectors
$v_{j}^{k}\in{\mathfrak{g}}_{j}$ for some $j>1$ to the above generators; these
cases are not considered.) So we have to describe the generators of
${\mathfrak{g}}_{0}$, or rather its quotient modulo center.

If ${\mathfrak{g}}_{0}$ is of the form ${\mathfrak{g}}(A)$ or its “relative”,
we select its Chevalley generators, see [BGL1].

If ${\mathfrak{g}}_{0}$ is an almost simple “lopsided” (in particular, of type
${\mathfrak{pe}}$, ${\mathfrak{spe}}$) but ${\mathbb{Z}}$-graded Lie
superalgebra, we apply the above described procedure to it: first, take its
0th components and its generators, then the highest and lowest weight vectors
in its components of degree $\pm 1$, etc.

If ${\mathfrak{g}}_{0}$ is semisimple of the form
${\mathfrak{s}}\otimes\Lambda(r)\ltimes{\mathfrak{vect}}(0|k)$, where
${\mathfrak{s}}$ is almost simple, then we take the already described
generators of ${\mathfrak{vect}}(0|k)$ and apply the above procedures to
${\mathfrak{s}}$.

For a list of defining relations for many simple Lie superalgebras over
${\mathbb{C}}$, and their relatives, see [GL1, GLP].

###  3.8. Forgetting the superstructure if $p=2$

Applying F to the known serial vectorial Lie superalgebras yields Table 23.1;
these Lie algebras are not necessarily simple, but their simple relatives
(subquotients) are derived by the routine procedure; here
$\widetilde{}\underline{N}$ is the vector $\underline{N}$ with each extra
coordinate equal to 1; the unconstrained form $\widetilde{}\underline{N}^{u}$
of $\widetilde{}\underline{N}$ may have fewer critical coordinates than
$\widetilde{}\underline{N}$, or even none at all.

###  3.9. Generating functions over ${\mathbb{C}}$

A laconic way to describe ${\mathfrak{k}}$, ${\mathfrak{m}}$ and their
subalgebras is via generating functions.

Odd contact form $\alpha_{1}$, where

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

For any $f\in{\mathbb{C}}[t,p,q,\theta]$, set:

(20)
$K_{f}=(2-E)(f){\frac{\partial}{\partial{t}}}-H_{f}+{\frac{\partial{f}}{\partial{t}}}E,$

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 vector field

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

with Hamiltonian $f$ that preserves $d\widetilde{\alpha}_{1}$, where

(22) $\widetilde{\alpha}_{1}=dt+\mathop{\sum}\limits_{1\leq i\leq
n}(p_{i}dq_{i}-q_{i}dp_{i})\ +\mathop{\sum}\limits_{1\leq j\leq
m}\theta_{j}d\theta_{j}.$

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

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

Even contact form $\alpha_{0}$, where

(23)
$\begin{array}[]{c}\alpha_{0}=d\tau+\mathop{\sum}\limits_{i}(\xi_{i}dq_{i}+q_{i}d\xi_{i}).\end{array}$

For any $f\in{\mathbb{C}}[q,\xi,\tau]$, set:

(24)
$\begin{array}[]{l}M_{f}=(2-E)(f){\frac{\partial}{\partial{\tau}}}-\mathop{\mathrm{Le}}\nolimits_{f}-(-1)^{p(f)}{\frac{\partial{f}}{\partial{\tau}}}E,\text{~{}~{}where
$E=\mathop{\sum}\limits_{i}y_{i}{\frac{\partial}{\partial{y_{i}}}}$ and where
$y=(q,\xi)$};\\\ \mathop{\mathrm{Le}}\nolimits_{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).\end{array}$

Since (here $L_{D}$ is the Lie derivative along the vector field $D$)

(25)
$\begin{array}[]{l}L_{K_{f}}(\alpha_{1})=2{\frac{\partial{f}}{\partial{t}}}\alpha_{1}=K_{1}(f)\alpha_{1},\\\
L_{M_{f}}(\alpha_{0})=-(-1)^{p(f)}2{\frac{\partial{f}}{\partial{\tau}}}\alpha_{0}=-(-1)^{p(f)}M_{1}(f)\alpha_{0},\end{array}$

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

$p(\mathop{\mathrm{Le}}\nolimits_{f})=p(M_{f})=p(f)+{\bar{1}}.$

Contact brackets, Poisson bracket, antibracket (Buttin bracket): To the
(super)commutators $[K_{f},K_{g}]$ and $[M_{f},M_{g}]$ there correspond
contact brackets of the generating functions:

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

The explicit expressions for the contact brackets are as follows. Let us first
define the brackets on functions that do not depend on $t$ (resp. $\tau$).

The Poisson bracket $\\{\cdot,\cdot\\}_{P.b.}$ (in the realization with the
form $\widetilde{\omega}_{0}$) is given by the equation

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

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

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

The Buttin777C. Buttin was the first to publish that the Schouten bracket
satisfies super Jacobi identity. bracket $\\{\cdot,\cdot\\}_{B.b.}$,
discovered by Schouten and initially known as the Schouten bracket, is very
popular in physics under the name antibracket, see [GPS]. It is given by the
formula

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

In terms of the Poisson and Buttin brackets, respectively, the contact
brackets are as follows:

(29)
$\begin{array}[]{l}\\{f,g\\}_{k.b.}=\displaystyle(2-E)(f){\frac{\partial{g}}{\partial{t}}}-{\frac{\partial{f}}{\partial{t}}}(2-E)(g)-\\{f,g\\}_{P.b.}\\\
\\{f,g\\}_{m.b.}=\displaystyle(2-E)(f){\frac{\partial{g}}{\partial{\tau}}}+(-1)^{p(f)}{\frac{\partial{f}}{\partial{\tau}}}(2-E)(g)-\\{f,g\\}_{B.b.}.\end{array}$

The Lie superalgebras of Hamiltonian vector fields (or Hamiltonian
superalgebra) and its special subalgebra (defined only if $n=0$) are

(30)
$\begin{array}[]{ll}{\mathfrak{h}}(2n|m)&=\\{D\in{\mathfrak{vect}}(2n|m)\mid
L_{D}\omega_{0}=0,\text{~{}~{}where $\omega_{0}:=d\alpha_{1}$}\\}\simeq\\\
&\\{D\in{\mathfrak{vect}}(2n|m)\mid
L_{D}\widetilde{\omega}_{0}=0,\text{~{}~{}where
$\widetilde{\omega}_{0}:=d\widetilde{\alpha}_{1}$}\\},\\\
{\mathfrak{h}}^{(1)}(m)&=\\{H_{f}\in{\mathfrak{h}}(0|m)\mid\int
f\mathop{\mathrm{{vol\/}}}\nolimits=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 [Le2], and its special subalgebra:

(31) $\begin{array}[]{l}{\mathfrak{le}}(n)=\\{D\in{\mathfrak{vect}}(n|n)\mid
L_{D}\omega_{1}=0,\text{~{}~{}where $\omega_{1}:=d\alpha_{0}$}\\},\\\
{\mathfrak{sle}}(n)=\\{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):

(32)
$\begin{array}[]{rcl}{\mathfrak{k}}(2n+1|m)&\cong&\mathop{\mathrm{{Span}}}\nolimits(K_{f}\mid
f\in{\mathbb{C}}[t,p,q,\xi]),\\\
{\mathfrak{po}}(2n|m)&\cong&\mathop{\mathrm{{Span}}}\nolimits(K_{f}\mid
f\in{\mathbb{C}}[t,p,q,\xi]\text{~{}~{} such
that~{}~{}}{\frac{\partial{f}}{\partial{t}}}=0)\cong{\mathbb{C}}[p,q,\xi],\\\
{\mathfrak{h}}(2n|m)&\cong&\mathop{\mathrm{{Span}}}\nolimits(H_{f}\mid
f\in{\mathbb{C}}[p,q,\xi])\simeq{\mathbb{C}}[p,q,\xi]/{\mathbb{C}}\cdot 1;\\\
{\mathfrak{m}}(n)&\cong&\mathop{\mathrm{{Span}}}\nolimits(M_{f}\mid
f\in{\mathbb{C}}[\tau,q,\xi]),\\\
{\mathfrak{b}}(n)&\cong&\mathop{\mathrm{{Span}}}\nolimits(M_{f}\mid
f\in{\mathbb{C}}[\tau,q,\xi]),\text{~{}~{} such
that~{}~{}}{\frac{\partial{f}}{\partial{\tau}}}=0)\cong\Pi({\mathbb{C}}[q,\xi]),\\\
{\mathfrak{le}}(n)&\cong&\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{f}\mid
f\in{\mathbb{C}}[q,\xi])\cong\Pi({\mathbb{C}}[q,\xi]/{\mathbb{C}}\cdot
1).\end{array}$

We have

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

###  3.10. Divergence-free subalgebras

Since, as is easy to calculate,

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

it follows that the divergence-free subalgebra of the contact Lie superalgebra
either coincides with it (for $m=2n+2$) or is the Poisson superalgebra. For
the pericontact series, the situation is more interesting: the divergence free
subalgebra is simple and new (as compared with the above list).

Since

(34) $\mathop{\text{\rm div}}\nolimits
M_{f}=(-1)^{p(f)}2\left((1-E){\frac{\partial{f}}{\partial{\tau}}}-\mathop{\sum}\limits_{i\leq
n}\frac{\partial^{2}f}{\partial q_{i}\partial\xi_{i}}\right),$

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

(35)
${\mathfrak{sm}}(n)=\mathop{\mathrm{{Span}}}\nolimits\left(M_{f}\in{\mathfrak{m}}(n)\mid(1-E){\frac{\partial{f}}{\partial{\tau}}}=\mathop{\sum}\limits_{i\leq
n}\frac{\partial^{2}f}{\partial q_{i}\partial\xi_{i}}\right).$

In particular,

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

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

The Lie superalgebras ${\mathfrak{sle}}(n)$, ${\mathfrak{s}}{\mathfrak{b}}(n)$
and ${\mathfrak{svect}}(1|n)$ have traceless ideals
${\mathfrak{sle}}^{(1)}(n)$, ${\mathfrak{s}}{\mathfrak{b}}^{(1)}(n)$ and
${\mathfrak{svect}}^{(1)}(n)$ of codimension 1; they are defined from the
exact sequences

(36)
$\begin{array}[]{c}0\longrightarrow{\mathfrak{sle}}^{(1)}(n)\longrightarrow{\mathfrak{sle}}(n)\longrightarrow{\mathbb{C}}\cdot\mathop{\mathrm{Le}}\nolimits_{\xi_{1}\dots\xi_{n}}\longrightarrow
0,\\\
0\longrightarrow{\mathfrak{s}}{\mathfrak{b}}^{(1)}(n)\longrightarrow{\mathfrak{s}}{\mathfrak{b}}(n)\longrightarrow{\mathbb{C}}\cdot
M_{\xi_{1}\dots\xi_{n}}\longrightarrow 0,\\\ \displaystyle
0\longrightarrow{\mathfrak{svect}}^{(1)}(n)\longrightarrow{\mathfrak{svect}}(1|n)\longrightarrow{\mathbb{C}}\cdot\xi_{1}\dots\xi_{n}\partial_{t}\longrightarrow
0.\end{array}$

####  3.10.1. A deform of the series ${\mathfrak{b}}$.

For an explicit form of $M_{f}$, see Subsec. 3.9. Set

${\mathfrak{b}}_{a,b}(n)=\left\\{M_{f}\in{\mathfrak{m}}(n)\mid
a\;\mathop{\text{\rm div}}\nolimits
M_{f}=(-1)^{p(f)}2(a-bn){\frac{\partial{f}}{\partial{\tau}}}\right\\}.$

We denote the operator that singles out ${\mathfrak{b}}_{\lambda}(n)$ in
${\mathfrak{m}}(n)$ as follows, cf. (15):

$\mathop{\text{\rm div}}\nolimits_{\lambda}=(bn-
aE){\frac{\partial}{\partial{\tau}}}-a\Delta,\;\text{ for
$\lambda=\frac{2a}{n(a-b)}$ and $\Delta=\mathop{\sum}\limits_{i\leq
n}\frac{\partial^{2}}{\partial q_{i}\partial\xi_{i}}$}.$

Taking into account the explicit form of the divergence of $M_{f}$ we get

(37)
$\begin{array}[]{ll}{\mathfrak{b}}_{a,b}(n)&=\\{M_{f}\in{\mathfrak{m}}(n)\mid(bn-
aE){\frac{\partial{f}}{\partial{\tau}}}=a\Delta f\\}=\\\
&\\{D\in{\mathfrak{vect}}(n|n+1)\mid
L_{D}(\mathop{\mathrm{{vol\/}}}\nolimits_{q,\xi,\tau}^{a}\alpha_{0}^{a-bn})=0\\}.\end{array}$

It is subject to a direct verification that
${\mathfrak{b}}_{a,b}(n)\simeq{\mathfrak{b}}_{\lambda}(n)$ for
$\lambda=\frac{2a}{n(a-b)}\in{\mathbb{C}}P^{1}$. Obviously, if
$\lambda=0,1,\infty$, the structure of ${\mathfrak{b}}_{\lambda}(n)$ differs
from the other members of the parametric family: the following exact sequences
single out simple Lie superalgebras (the quotient ${\mathfrak{le}}(n)$ and
ideals, the first derived subalgebras):

(38)
$\begin{array}[]{c}0\longrightarrow{\mathbb{C}}M_{1}\longrightarrow{\mathfrak{b}}(n)\longrightarrow{\mathfrak{le}}(n)\longrightarrow
0,\\\
0\longrightarrow{\mathfrak{b}}_{1}^{(1)}(n)\longrightarrow{\mathfrak{b}}_{1}(n)\longrightarrow{\mathbb{C}}\cdot
M_{\xi_{1}\dots\xi_{n}}\longrightarrow 0,\\\
0\longrightarrow{\mathfrak{b}}_{\infty}^{(1)}(n)\longrightarrow{\mathfrak{b}}_{\infty}(n)\longrightarrow{\mathbb{C}}\cdot
M_{\tau\xi_{1}\dots\xi_{n}}\longrightarrow 0.\\\ \end{array}$

The Lie superalgebras ${\mathfrak{b}}_{\lambda}(n)$ can be further deformed at
certain points $\lambda$, see [LSh].

###  3.11. Passage from ${\mathbb{C}}$ to ${\mathbb{K}}$

Here we have collected answers to several questions that stunned us while we
were writing this paper. We hope that even the simplest of these answers will
help the reader familiar with representations of Lie algebra over
${\mathbb{C}}$ but with no experience of working with characteristic $p$. For
$p=2$, several of our definitions are new.

####  3.11.1. The Lie (super)algebras preserving symmetric nondegenerate
bilinear forms ${\mathcal{B}}$.

We denote the Gram matrix of the bilinear form ${\mathcal{B}}$ also by
${\mathcal{B}}$, let ${\mathfrak{aut}}({\mathcal{B}})$ be the Lie
(super)algebras preserving ${\mathcal{B}}$. If ${\mathcal{B}}$ is odd and the
superspace on which it is defined is of superdimension $n|n$, we write
${\mathfrak{pe}}_{B}(n)$ instead of ${\mathfrak{aut}}({\mathcal{B}})$.

##### Let $p\neq 2$ and ${\mathfrak{g}}={\mathfrak{pe}}_{B}(n)$

If the form ${\mathcal{B}}$ is in canonical shape ${\mathcal{B}}=\Pi_{n|n}$,
then ${\mathfrak{g}}$ consists of the supermatrices of the form

(39) $X=\begin{pmatrix}A&B\\\ C&-A^{t}\end{pmatrix},\text{~{}~{}where $B$ is
symmetric and $C$ antisymmetric}.$

Clearly, $\mathop{\mathrm{str}}\nolimits X=2\mathop{\mathrm{tr}}\nolimits A$.
We also have ${\mathfrak{g}}^{(1)}={\mathfrak{spe}}(n)$, i.e., is of
codimension 1 and singled out by the condition $\mathop{\mathrm{str}}\nolimits
X=0$, which is equivalent to $\mathop{\mathrm{tr}}\nolimits A=0$.

The Lie superalgebra ${\mathfrak{le}}(n;\underline{N}|n)$ is, by definition,
the Cartan prolong
$(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}(n))_{*,\underline{N}}$.

Over ${\mathbb{C}}$, there is no shearing vector, and
${\mathfrak{le}}(n):={\mathfrak{le}}(n|n)$ is spanned by the elements
$\mathop{\mathrm{Le}}\nolimits_{f}$, where $f\in{\mathbb{C}}[q,\xi]$.

If $p>2$, the elements of
${\mathcal{O}}(q;\underline{N}|\xi)\oplus\mathop{\mathrm{{Span}}}\nolimits(q_{i}^{(p^{\underline{N}_{i}})}\mid
N_{i}<\infty)$, or ${\mathcal{O}}[q;\underline{N}_{\infty}|\xi]$ for
$\underline{N}=\underline{N}_{\infty}$, see (10), generate
${\mathfrak{le}}(n;\underline{N})$. If $N_{i}<\infty$ for at least one $i$,
the the additional part Irreg does not change while the regular part is of the
form looking alike for any $p>2$:

(40)
$\text{Reg}=\mathop{\mathrm{{Span}}}\nolimits\left(\mathop{\mathrm{Le}}\nolimits_{f}\mid
f\in{\mathcal{O}}(q;\underline{N}|\xi)\right)\oplus\mathop{\mathrm{{Span}}}\nolimits(q_{i}^{(2^{\underline{N}_{i}})}\mid
N_{i}<\infty),\quad\text{Irreg}=\mathop{\mathrm{{Span}}}\nolimits(\xi_{i}\partial_{u_{i}})_{i=1}^{n}.$

The prolong
$(\mathop{\mathrm{id}}\nolimits,{\mathfrak{spe}}(n))_{*,\underline{N}}$ is
singled out by the condition

$\mathop{\text{\rm
div}}\nolimits\mathop{\mathrm{Le}}\nolimits_{f}=0\Longleftrightarrow\Delta
f=0,\text{~{} where ~{}}\Delta=\mathop{\sum}\limits_{i\leq
n}\frac{\partial^{2}}{\partial q_{i}\partial\xi_{i}}.$

The operator $\Delta$ is, therefore, the Cartan prolong of the supertrace
expressed as an operator acting on the space of generating functions.

##### Modifications in the above description for $p=2$

If $p=2$, the analogs of symplectic and periplectic Lie (super)algebras accrue
additional elements: If the matrix of the bilinear form ${\mathcal{B}}$ is
$\Pi_{2n}$ (resp. $\Pi_{n|n}$), then ${\mathfrak{aut}}({\mathcal{B}})$
consists of the (super)matrices of the form

(41) $X=\begin{pmatrix}A&B\\\ C&A^{t}\end{pmatrix},\text{~{}~{}where $B$ and
$C$ are symmetric, $A\in{\mathfrak{gl}}(n)$}.$

Denote these general Lie (super)algebras

(42)
${\mathfrak{aut}}_{gen}({\mathcal{B}})=\begin{cases}{\mathfrak{o}}_{gen}(2n)&\text{for
${\mathcal{B}}=\Pi_{2n}$}\\\
{\mathfrak{pe}}_{gen}(n)&\text{for$\Pi_{n|n}$}.\end{cases}$

Let

(43) $ZD\text{~{}~{}denote the space of symmetric matrices with zeros on their
main diagonals.}$

The derived Lie (super)algebra ${\mathfrak{aut}}^{(1)}({\mathcal{B}})$
consists of the (super)matrices of the form (41), where $B,C\in ZD$. In other
words, these Lie (super)algebras resemble the orthogonal Lie algebras.

On these Lie (super)algebras ${\mathfrak{aut}}^{(1)}({\mathcal{B}})$ the
following (super)trace (half-trace) is defined:

(44) ${\text{htr}}:\begin{pmatrix}A&B\\\
C&A^{t}\end{pmatrix}\longrightarrow\mathop{\mathrm{tr}}\nolimits A.$

The traceless Lie sub(super)algebra of ${\mathfrak{aut}}^{(1)}({\mathcal{B}})$
is isomorphic to ${\mathfrak{aut}}^{(2)}({\mathcal{B}})$.

There is, however, an algebra $\widetilde{\mathfrak{aut}}({\mathcal{B}})$,
such that
${\mathfrak{aut}}^{(1)}({\mathcal{B}})\subset\widetilde{\mathfrak{aut}}({\mathcal{B}})\subset{\mathfrak{aut}}({\mathcal{B}})$,
consisting of (super)matrices of the form (41), where $B\in ZD$, and any
symmetric $C$ (or isomorphic to it version with $C\in ZD$, and any symmetric
$B$). We suggest to denote as follows:

(45)
$\begin{array}[]{l}\widetilde{\mathfrak{aut}}({\mathcal{B}})=\begin{cases}\textbf{F}({\mathfrak{pe}})(2n)&\text{for
${\mathcal{B}}$ even},\\\ {\mathfrak{pe}}(n)&\text{for ${\mathcal{B}}$
odd},\end{cases}\\\
{\mathfrak{aut}}^{(1)}({\mathcal{B}})=\\{X\in\widetilde{\mathfrak{aut}}({\mathcal{B}})\mid\mathop{\mathrm{tr}}\nolimits
A=0\\}=\begin{cases}\textbf{F}({\mathfrak{spe}})(2n)&\text{for ${\mathcal{B}}$
even},\\\ {\mathfrak{spe}}(n)&\text{for ${\mathcal{B}}$
odd}.\end{cases}\end{array}$

###  3.12. Central extensions

There is only one nontrivial central extension of ${\mathfrak{spe}}(n)$ for
${p\neq 2,3}$. It was discovered by A. Sergeev, so is denoted
${\mathfrak{as}}$ and exists only for $n=4$; for details, see [BGLL1].

###  3.13. Prolongs

The Lie superalgebra ${\mathfrak{pe}}_{gen}(n)$ is larger than
${\mathfrak{pe}}(n)$: both $B$ and $C$ in ${\mathfrak{pe}}_{gen}(n)$ are
symmetric, see (41). Observe that
${\mathfrak{pe}}_{gen}(n)\subset{\mathfrak{sl}}(n|n)$. Denote

(46)
${\mathfrak{le}}_{gen}(n;\underline{N}|n):=(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}_{gen}(n))_{*,\underline{N}}.$

Clearly, if $\underline{N}=\underline{N}_{\infty}$ see (10), then
${\mathfrak{le}}_{gen}(n;\underline{N}|n)$ consists of the regular part Reg
and an additional part Irreg, cf. eq (40)

$\text{Reg}=\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{f}\mid
f\in{\mathcal{O}}[q;\underline{N}|\xi]),\quad\text{Irreg}=\mathop{\mathrm{{Span}}}\nolimits(B_{i}:=\xi_{i}\partial_{u_{i}})_{i=1}^{n}.$

The part Irreg corresponds to the nonexisting generating functions
$\xi_{i}^{2}$. Clearly, ${\mathfrak{le}}_{gen}(n;\underline{N}|n)$ is
contained in ${\mathfrak{svect}}(n;\underline{N}|n)$, and therefore coincides
with ${\mathfrak{sle}}_{gen}(n;\underline{N}|n)$.

The only difference between ${\mathfrak{le}}_{gen}(n;\underline{N}|n)$ and
${\mathfrak{le}}(n;\underline{N}|n)$ is constituted by the space Irreg. These
nonexisting generating functions $\xi_{i}^{(2)}$ generate linear vector fields
corresponding to the diagonal elements of the matrices $B$ in (41), same as
the $q_{i}^{(2)}$ generate linear vector fields corresponding to the diagonal
elements of the matrices $C$ in (41), but these two sets of elements are
different in their nature: there are no elements of degree $>0$ in
$(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}_{gen})_{*,\underline{N}}$
whose brackets with ${\mathfrak{g}}_{-1}$ give the $B_{i}$.

The direct $p=2$ analog of the complex superalgebra ${\mathfrak{pe}}(n)$ is
the Lie superalgebra consisting of the supermatrices of the form (41) with
$B\in ZD$. Its commutant ${\mathfrak{pe}}(n)^{(1)}$ is of codimension 1 and
singled out in ${\mathfrak{pe}}(n)$ by the condition ${\text{htr}}=0$.

The Cartan prolong
$(\mathop{\mathrm{id}}\nolimits,{\mathfrak{pe}}(n))_{*,\underline{N}}$
consists of the regular part only, and therefore looks the same for any $p>0$.
The Cartan prolong
$(\mathop{\mathrm{id}}\nolimits,({\mathfrak{pe}}(n))^{(1)})_{*,\underline{N}}$
is singled out in ${\mathfrak{le}}(n;\underline{N}|n)$ by the following
condition in terms of generating functions: $\Delta(f)=0$.

Thus, the correct direct analogs of the complex Lie superalgebras
${\mathfrak{sle}}(n)$ and ${\mathfrak{spe}}(n)$ are
$(\mathop{\mathrm{id}}\nolimits,({\mathfrak{pe}}(n))^{(1)})_{*,\underline{N}}$
and ${\mathfrak{pe}}(n)^{(1)}$, respectively.

In [Leb], there were considered ${\mathfrak{g}}={\mathfrak{pe}}(n)$, the
derived algebras ${\mathfrak{g}}^{(1)}$ and ${\mathfrak{g}}^{(2)}$, and the
Cartan prolongs of these derived as ${\mathfrak{g}}_{0}$ with
${\mathfrak{g}}_{-1}=\mathop{\mathrm{id}}\nolimits$. Clearly,
${\mathfrak{g}}^{(1)}$ consists of supermatrices of the form (41) with zero-
diagonal matrices $B$ and $C$, whereas ${\mathfrak{g}}^{(2)}$ is singled out
of ${\mathfrak{g}}^{(1)}$ by the condition ${\text{htr}}=0$. The Cartan
prolongs of each of these Lie superalgebras only have the regular parts:

(47)
$\begin{array}[]{l}(\mathop{\mathrm{id}}\nolimits,{\mathfrak{g}}^{(1)})_{*,\underline{N}}=\mathop{\mathrm{{Span}}}\nolimits\left(\mathop{\mathrm{Le}}\nolimits_{f}\mid
f\in{\mathcal{O}}(q;\underline{N}_{s}|\xi)\right);\\\
(\mathop{\mathrm{id}}\nolimits,{\mathfrak{g}}^{(2)})_{*,\underline{N}}=\mathop{\mathrm{{Span}}}\nolimits\left(\mathop{\mathrm{Le}}\nolimits_{f}\mid
f\in{\mathcal{O}}(q;\underline{N}_{s}|\xi)\text{~{}~{}and~{}~{}}\Delta
f=0\right).\end{array}$

Let a bilinear form $B$ be defined on a superspace $V$; let $\textbf{F}(B)$ be
the same form considered on $\textbf{F}(V)$. Let
${\mathfrak{h}}_{B}(a;\underline{N}|b)$ denote the Hamiltonian Lie
superalgebra — the Cartan prolong of the ortho-orthogonal Lie superalgebra
${\mathfrak{o}}{\mathfrak{o}}_{B}(a|b)$ preserving the nondegenerate form $B$;
its desuperization is
${\mathfrak{h}}_{\textbf{F}(B)}(a+b;\widetilde{}\underline{N})$, where
$\widetilde{}\underline{N}$ has no critical coordinates.

####  3.13.1. Divergence-free subalgebras of series ${\mathfrak{h}}$ and
${\mathfrak{le}}$.

These subalgebras are prolongations of subalgebras of linear components of
${\mathfrak{h}}$ and ${\mathfrak{le}}$ consisting of traceless subalgebras.
Observe, that if $p=2$, these subalgebras, generated by (linear combinations
of) quadratic monomials, can be determined by imposing constraints on the
shearing vector.

##### Remark

For $\underline{N}$ with $N_{i}<\infty$ for all $i$ and $p=2$, the Lie
superalgebra ${\mathfrak{le}}^{(1)}(n;\underline{N}|n)$ is spanned by the
elements $f\in{\mathcal{O}}(q;\underline{N}|\xi)$, whereas each of the
“virtual” generating functions $q_{i}^{(2^{\underline{N}_{i}})}$ determines an
outer derivation of ${\mathfrak{le}}^{(1)}(n;\underline{N}|n)$.

####  3.13.2. ${\mathfrak{svect}}_{a,b}(0|n)$.

The notation ${\mathfrak{svect}}_{a,b}(0|n)$ is a shorthand for
${\mathfrak{svect}}(0|n)\ltimes{\mathbb{K}}(az+bd)$, where $d$ is the operator
that determines the standard ${\mathbb{Z}}$-grading of ${\mathfrak{g}}$, i.e.,
$d=\sum\xi_{i}\partial_{\xi_{i}}$, while $z$ is an element generating the
trivial center commuting with ${\mathfrak{svect}}(0|n)\ltimes{\mathbb{K}}\cdot
d$.

####  3.13.3. ${\mathfrak{spe}}_{a,b}(n)$.

For $p=0$, the meaning of ${\mathfrak{spe}}_{a,b}(n)$ is similar to that of
${\mathfrak{svect}}_{a,b}(0|n)$, but with
$d=\mathop{\mathrm{diag}}\nolimits(1_{n},-1_{n})$. To define the analog of
${\mathfrak{spe}}_{a,b}(n)$ for $p=2$, see line $N=7$ in Table 23.1, observe
that the codimension of ${\mathfrak{spe}}(n)$ in ${\mathfrak{m}}_{0}$ is equal
to 2. So, to pass from ${\mathfrak{spe}}(n)$ to ${\mathfrak{m}}_{0}$, we have
to add certain two elements, whereas to pass to ${\mathfrak{spe}}_{a,b}$ we
have to add a linear combination of these elements with coefficients $a$ and
$b$. The question is: “can we single out these elements in a canonical way?”

For $p=0$, we managed. The identity operator (in matrix realization) is one of
these elements. How to select the other element? There is no distinguished
element in ${\mathfrak{pe}}(n)$ that does not lie in ${\mathfrak{spe}}(n)$.
But, if $p=0$, there is an element
$\mathop{\mathrm{diag}}\nolimits(1_{n},-1_{n})$ corresponding to a “most
symmetric” generating function $\sum q_{i}\xi_{i}$.

Under the passage from $p=0$ to $p>2$ this “most symmetric” element lies in
${\mathfrak{pe}}(n)$ if $p|n$ and the choice of the second element becomes a
matter of taste.

If $p=2$, the situation becomes completely miserable. Now, the restriction of
$M_{\sum q_{i}\xi_{i}}$ onto ${\mathfrak{m}}_{-1}$ not only lies in
${\mathfrak{spe}}(n)$ for $n$ even, it coincides with the identity operator.
So, in this case, there is no distinguished operator not lying in
${\mathfrak{spe}}(n)$. What to do?

We suggest to consider the elements of ${\mathfrak{m}}_{0}$ as operators
acting not just on ${\mathfrak{m}}_{-1}$, but on the whole
${\mathfrak{m}}_{-}$. If ${\mathfrak{m}}_{0}$ is thus understood, there are
two well-defined linear forms that single out ${\mathfrak{spe}}(n)$ in
${\mathfrak{m}}_{0}$. Indeed, any operator $A\in{\mathfrak{m}}_{0}$ is of the
form $A=A_{-1}\oplus A_{-2}$, where $A_{i}=A|_{{\mathfrak{m}}_{i}}$ and
$A_{-2}=\lambda(A)\cdot\mathop{\mathrm{id}}\nolimits$. The two distinguished
linear forms are $\lambda$ and $\mu$, where $\mu(A)={\text{htr}}(A_{-1})$ (for
$p>2$, we can set $\mu(A)=\mathop{\mathrm{str}}\nolimits(A_{-1})$).

Now, ${\mathfrak{spe}}(n)$ is singled out by conditions $\lambda(A)=\mu(A)=0$,
while

(48)
${\mathfrak{spe}}_{a,b}(n)=\\{X\in{\mathfrak{pe}}(n)\mid(a\mu+b\lambda)(X)=0\\}.$

###  3.14. On ${\mathfrak{m}}$ and ${\mathfrak{b}}$

To pass from ${\mathfrak{b}}(n;\underline{N}|n)$ to
${\mathfrak{m}}(n;\underline{N}|n+1)$, we have to add to
${\mathfrak{b}}(n)_{0}={\mathfrak{pe}}(n)$ the central element; it will serve
as a grading operator of the prolong. We see that ${\mathfrak{m}}$ is the
generalized Cartan prolong of
$({\mathfrak{b}}(n)_{-},{\mathfrak{c}}{\mathfrak{b}}(n)_{0})$.

The commutant of ${\mathfrak{m}}(n;\underline{N}|n+1)_{0}$ is the same as that
of ${\mathfrak{b}}(n)_{0}={\mathfrak{pe}}(n)$, so is of codimension 2. Hence
there are two traces on ${\mathfrak{m}}(n;\underline{N}|n+1)_{0}$, and
therefore there are two divergences on ${\mathfrak{m}}$. One of them is given
by the operator

(49) $\partial_{\tau},\text{~{}~{}more
precisely~{}}D_{\tau}:=\partial_{\tau}\circ\mathop{\mathrm{sign}}\nolimits,\text{~{}~{}i.e.,~{}~{}}D_{\tau}(f)=(-1)^{p(f)}\partial_{\tau}(f)\text{~{}~{}for
any~{}}f\in{\mathcal{O}}(q;\underline{N}|\xi)$

since this should be the map commuting (not supercommuting) with
${\mathfrak{m}}_{-}$, see [Shch]. The condition $D_{\tau}(f)=0$, i.e., just
$\partial_{\tau}(f)=0$ singles out precisely ${\mathfrak{b}}(n)$.

####  3.14.1. ${\mathfrak{sb}}$.

The definition of ${\mathfrak{sb}}(n;\underline{N})$ is the same for any
characteristic $p$ (in terms of generating “functions” from an appropriate
space ${\mathcal{F}}$, see (11)):

(50)
${\mathfrak{sb}}(n;\underline{N})=\mathop{\mathrm{{Span}}}\nolimits(f\in{\mathcal{F}}\mid\Delta(f)=0).$

####  3.14.2. ${\mathfrak{b}}_{a,b}$ for $p=2$.

The other trace on ${\mathfrak{m}}_{0}$ is htr. On ${\mathfrak{le}}$, its
prolong was the operator $\Delta$. But $\Delta$ does not commute with the
whole ${\mathfrak{m}}_{-}$. To obtain the ${\mathfrak{m}}_{-}$-invariant
prolong of this trace on ${\mathfrak{m}}_{0}$, we have to express htr in terms
of the operators commuting with ${\mathfrak{m}}_{-}$ ($Y$-type vectors in
terms of [Shch]). Taking ${\mathfrak{m}}_{-}$ spanned by the elements

${\mathfrak{m}}_{-2}={\mathbb{K}}\cdot\partial_{\tau},\quad{\mathfrak{m}}_{-1}=\mathop{\mathrm{{Span}}}\nolimits(\partial_{q_{i}}+\xi_{i}\partial_{\tau},\;\partial_{\xi_{i}})_{i=1}^{n},$

we see that the operators commuting with ${\mathfrak{m}}_{-}$ are spanned by

$\partial_{\tau},\quad\partial_{q_{i}},\quad\partial_{\xi_{i}}+q_{i}\partial_{\tau}.$

In terms of these operators the vector field $M_{f}$ takes the form:

(51)
$M_{f}=f\partial_{\tau}+\mathop{\sum}\limits_{i}\Big{(}\partial_{q_{i}}(f)(\partial_{\xi_{i}}+q_{i}\partial_{\tau})+(\partial_{\xi_{i}}+q_{i}\partial_{\tau})(f)\partial_{q_{i}}\Big{)}$

and the invariant prolong of htr takes the form:

(52)
$\Delta^{\mathfrak{m}}(f)=\mathop{\sum}\limits_{i}((\partial_{\xi_{i}}+q_{i}\partial_{\tau})\partial_{q_{i}}(f)=\Delta(f)+E_{q}\partial_{\tau}(f),\text{~{}
where ~{}}E_{q}=\mathop{\sum}\limits_{i}q_{i}\partial_{q_{i}}.$

The condition $\Delta^{\mathfrak{m}}(f)=0$ singles out the $p=2$ analog of
${\mathfrak{sm}}$, whereas the condition

(53)
$b\partial_{\tau}(f)+a\Delta^{\mathfrak{m}}(f)=(b+aE_{q})\partial_{\tau}(f)+a\Delta(f)=0$

singles out the $p=2$ analog of ${\mathfrak{b}}_{a,b}$, cf. (37).

Setting
${\mathfrak{po}}_{a,b}(2n;\widetilde{}\underline{N}):=\textbf{F}({\mathfrak{b}}_{a,b}(n;\underline{N}))$
we single out a subalgebra in the Lie algebras of contact vector fields having
no analogs for $p\neq 2$.

Let us figure out how the parameter $\lambda$ of the regrading
${\mathfrak{po}}_{\lambda}(2n;\underline{N}):=\textbf{F}({\mathfrak{b}}_{\lambda}(n;\underline{N}))$
depends on parameters $a,b$ above; for summary, see $N=6,7$ in Table 23.1. The
space of ${\mathfrak{b}}_{a,b}(n;\underline{N}_{\infty})$ consists of vector
fields (51) satisfying eq. (53); the regrading

$\deg\tau=\deg q_{i}=1,\ \ \deg\xi_{i}=0\text{~{}~{}for all $i$}$

turns ${\mathfrak{b}}_{a,b}(n;\underline{N}_{\infty})$ into the Lie
superalgebra whose 0th component is isomorphic to ${\mathfrak{vect}}(0|n)$ and
the $(-1)$st component is isomorphic to the ${\mathfrak{vect}}(0|n)$-module of
weighted $\lambda$-densities. To express $\lambda$ in terms of parameters
$a,b$, take an element in the 0th component of
${\mathfrak{b}}_{a,b}(n;\underline{N}_{\infty})$ not lying in
${\mathfrak{b}}(n;\underline{N}_{\infty})$ and see how it acts on $M_{1}$. Let
$f=\alpha\tau+\beta q_{1}\xi_{1}$. Eq. (53) implies $\alpha b+\beta a$, so we
can take $f=a\tau+bq_{1}\xi_{1}$. Observe that $M_{bq_{1}\xi_{1}}$ acts on the
$(-1)$st component as the vector field $D=b\xi_{1}\partial_{\xi_{1}}$ and

(54)
${}[M_{a\tau},M_{1}]=[a\tau\partial_{\tau},\partial_{\tau}]=a\partial_{\tau}=\frac{a}{b}(\mathop{\text{\rm
div}}\nolimits D)M_{1},\text{~{}~{}hence $\lambda=\frac{a}{b}$}.$

The $p=2$ version of eq. (38) are the following exact sequences that single
out the simple Lie superalgebras (recall that
${\mathfrak{b}}(n;\underline{N})={\mathfrak{b}}_{\lambda}(n;\underline{N})$
for $\lambda=0$):

(55)
$\begin{array}[]{c}0\longrightarrow{\mathbb{K}}M_{1}\longrightarrow{\mathfrak{b}}(n;\underline{N})\longrightarrow{\mathfrak{le}}(n;\underline{N})\longrightarrow
0,\\\
0\longrightarrow{\mathfrak{b}}_{1}^{(1)}(n;\underline{N})\longrightarrow{\mathfrak{b}}_{1}(n;\underline{N})\longrightarrow{\mathbb{K}}\cdot
M_{\xi_{1}\dots\xi_{n}}\longrightarrow 0,\\\
0\longrightarrow{\mathfrak{b}}_{\infty}^{(1)}(n;\underline{N})\longrightarrow{\mathfrak{b}}_{\infty}(n;\underline{N})\longrightarrow{\mathbb{K}}\cdot
M_{\tau\xi_{1}\dots\xi_{n}}\longrightarrow 0.\\\ \end{array}$

###  3.15. On ${\mathfrak{k}}$ and ${\mathfrak{po}}$

Observe that although ${\mathfrak{h}}_{I}(n;\underline{N})$ is well defined,
there is no Lie algebra ${\mathfrak{po}}_{I}(n;\underline{N})$. Indeed, the
bracket should be antisymmetric, i.e., alternate, while
$\\{x_{i},x_{i}\\}_{I}=1$, not 0. Only
${\mathfrak{h}}_{\Pi}(2n;\underline{N})$ has it; this nontrivial central
extension is a correct direct analog of the complex Poisson Lie (super)algebra
${\mathfrak{po}}(2n|0)$.

To pass from ${\mathfrak{po}}(0|n)$ to ${\mathfrak{k}}(1;\underline{N}|2n)$,
we have to add, as a direct summand, a central element to
${\mathfrak{po}}(n)_{0}={\mathfrak{o}}_{\Pi}^{(1)}(n)$; it will act on the
prolong as a grading operator. We see that the generalized Cartan prolong of
the pair
$({\mathfrak{po}}(0|n)_{-},{\mathfrak{c}}{\mathfrak{o}}_{\Pi}^{(1)}(n))$ is
${\mathfrak{k}}(1;\underline{N}|2n)$.

The commutant of ${\mathfrak{k}}_{0}$ is the same as that of
${\mathfrak{po}}_{0}(n)={\mathfrak{o}}_{\Pi}^{(1)}(n)$, so is of codimension
2. Thus, there are two traces on ${\mathfrak{k}}_{0}$, and hence there are two
divergences on ${\mathfrak{k}}(1;\underline{N}|2n)$, same as on
${\mathfrak{m}}$.

######  Theorem ([Leb]).

Let the parity function be denoted by the same symbol as the parity functor
$\Pi$. Assume that

$\Pi(x_{0})=\Pi(x_{1})=\dots=\Pi(x_{n_{{\bar{0}}}})=\bar{0},\qquad\Pi(x_{n_{{\bar{0}}}+1})=\dots=\Pi(x_{n})=\bar{1}.$

The following is the normal form of the odd contact form on a superspace:

$dt+\mathop{\sum}\limits_{1\leq i\leq
k}p_{i}dq_{i}+\mathop{\sum}\limits_{1\leq i\leq
l}\xi_{i}d\eta_{i}\begin{cases}&\text{for $n_{{\bar{0}}}=2k$ and
$n_{{\bar{1}}}=2l$,}\\\ +\theta d\theta&\text{for $n_{{\bar{0}}}=2k$ and
$n_{{\bar{1}}}=2l+1$},\end{cases}$

where $t=x_{0}$, and $p_{i}=x_{i}$, $q_{i}=x_{k+i}$ for $1\leq i\leq k$ are
the even indeterminates; $\xi_{i}=x_{n_{{\bar{0}}}+i}$,
$\eta_{i}=x_{n_{{\bar{0}}}+l+i}$ for $1\leq i\leq l$, and $\theta=x_{n}$ for
$n_{{\bar{1}}}=2l+1$ are the odd indeterminates.

####  3.15.2. Generating functions over ${\mathbb{K}}$.

Recall that the contact Lie superalgebra consists of the vector fields $D$
that preserve the contact structure (nonintegrable distribution given by a
contact form $\alpha_{1}$, cf. (19)) on the supervariety ${\mathcal{M}}$
associated with the superspace ${\mathbb{K}}^{n_{\bar{0}}+1|n_{\bar{1}}}$:

(56) $L_{D}(\alpha_{1})=F_{D}\alpha_{1}\;\text{ for some
$F_{D}\in{\mathcal{F}}$, where ${\mathcal{F}}$ is the space of functions on
${\mathcal{M}}$.}$

Consider the form (here $n_{\bar{0}}=2k_{\bar{0}}+1$; if $n_{\bar{0}}$ is
even, no contact form exists)

(57) $\alpha_{1}=dx_{0}+\mathop{\sum}\limits_{1\leq i\leq
k}x_{i}dx_{k+i}\begin{cases}&\text{if $n=n_{\bar{0}}+n_{\bar{1}}=2k$}\\\
+x_{2k+1}dx_{2k+1}&\text{if $n=n_{\bar{0}}+n_{\bar{1}}=2k+1$,}\end{cases}$

such that

$\begin{array}[]{l}\Pi(x_{0})=\Pi(x_{1})=\dots=\Pi(x_{k_{\bar{0}}})=\Pi(x_{k+1})=\dots=\Pi(x_{k+k_{\bar{0}}})={\bar{0}},\\\
\Pi(x_{k_{\bar{0}}+1})=\dots=\Pi(x_{k})=\Pi(x_{k+k_{\bar{0}}+1})=\dots=\Pi(x_{2k+1})={\bar{1}}.\end{array}$

The vector fields $D$ satisfying (56) for some function $F_{D}$ look
differently for different characteristics:

For $p\neq 2$, and also if $p=2$ and $n=2k$, the fields $D$ satisfying (56)
have, for any $f\in{\mathcal{F}}$, the following form (compare with (20)):

(58)
$\begin{array}[]{c}K_{f}=(1-E^{\prime})(f)\displaystyle{\frac{\partial}{\partial{x_{0}}}}+{\frac{\partial{f}}{\partial{x_{0}}}}E^{\prime}+\mathop{\sum}\limits_{1\leq
i\leq k_{\bar{0}}}\left(\frac{\partial f}{\partial
x_{k+i}}{\frac{\partial}{\partial{x_{i}}}}-\frac{\partial f}{\partial
x_{i}}{\frac{\partial}{\partial{x_{k+i}}}}\right)-\\\
(-1)^{\Pi(f)}\displaystyle\left(\mathop{\sum}\limits_{k_{\bar{0}}+1\leq i\leq
k}\left(\frac{\partial f}{\partial
x_{i}}{\frac{\partial}{\partial{x_{k+i}}}}+\frac{\partial f}{\partial
x_{k+i}}{\frac{\partial}{\partial{x_{i}}}}\right)\begin{cases}&\text{if
$n=2k$}\\\ +\frac{1}{2}\frac{\partial f}{\partial
x_{2k+1}}{\frac{\partial}{\partial{x_{2k+1}}}}&\text{if
$n=2k+1$}\end{cases}\right),\end{array}$

where $E^{\prime}=\mathop{\sum}\limits_{1\leq i\leq
k}x_{i}\partial_{x_{i}}\begin{cases}&\text{if $n=2k$}\\\
+\frac{1}{2}x_{2k+1}\partial_{x_{2k+1}}&\text{if $n=2k+1$}.\end{cases}$

For $p=2$ and $n=2k+1$, we can not use formula (58) anymore (at least, not for
arbitrary $f$) since it contains $\frac{1}{2}$. In this case, the elements of
the contact algebra are of the following two types:

a) For any $f$ such that
${\frac{\partial{f}}{\partial{x_{0}}}}={\frac{\partial{f}}{\partial{x_{2k+1}}}}=0$,
i.e., for any $f\in\mathcal{O}[x_{1},\dots,x_{2k};\underline{N}]$, we have

(59)
$\begin{array}[]{ll}K_{f}=&(1+E^{\prime})(f)\displaystyle{\frac{\partial}{\partial{x_{0}}}}+\mathop{\sum}\limits_{1\leq
i\leq k_{\bar{0}}}\left(\frac{\partial f}{\partial
x_{k+i}}{\frac{\partial}{\partial{x_{i}}}}+\frac{\partial f}{\partial
x_{i}}{\frac{\partial}{\partial{x_{k+i}}}}\right)+\\\
&\displaystyle\mathop{\sum}\limits_{k_{\bar{0}}+1\leq i\leq
k}\left(\frac{\partial f}{\partial
x_{i}}{\frac{\partial}{\partial{x_{k+i}}}}+\frac{\partial f}{\partial
x_{k+i}}{\frac{\partial}{\partial{x_{i}}}}\right),\text{~{}~{}where
$E^{\prime}=\mathop{\sum}\limits_{1\leq i\leq
k}x_{i}{\frac{\partial}{\partial{x_{i}}}}$.}\end{array}$

b) For any $g\in{\mathcal{O}}[x_{0},\dots,x_{2k};\underline{N}]$, we set:

(60)
$\begin{array}[]{ll}\text{b1)}&A_{g}:=g\displaystyle\left(x_{2k+1}\partial_{x_{0}}+\partial_{x_{2k+1}}\right)\\\
\text{b2)}&B_{g}:=gx_{2k+1}\partial_{x_{2k+1}}.\end{array}$

The brackets and squarings of contact vector fields, and the corresponding
contact brackets of generating functions, are as follows:

(61)
$\begin{array}[]{ll}{}[K_{f},K_{f^{\prime}}]=K_{[f,f^{\prime}]_{k.b.}};&{}[M_{f},M_{f^{\prime}}]=M_{[f,f^{\prime}]_{m.b.}};\\\
{}(K_{f})^{2}=K_{\mathop{\sum}\limits_{1\leq i\leq
k}{\frac{\partial{f}}{\partial{x_{i}}}}{\frac{\partial{f}}{\partial{x_{k+i}}}}};&{}(M_{f})^{2}=M_{\mathop{\sum}\limits_{1\leq
i\leq
k}{\frac{\partial{f}}{\partial{x_{i}}}}{\frac{\partial{f}}{\partial{x_{k+i}}}}};\\\
{}[K_{f},A_{g}]=A_{[f,g]_{k.b.}};&(A_{g})^{2}=B_{g{\frac{\partial{g}}{\partial{x_{0}}}}};\\\
{}[K_{f},B_{g}]=B_{[f,g]_{k.b.}};&[A_{g},B_{g^{\prime}}]=A_{gg^{\prime}};\\\
{}[A_{g},A_{g^{\prime}}]=B_{{\frac{\partial{(gg^{\prime})}}{\partial{x_{0}}}}};&[B_{g},B_{g^{\prime}}]=(B_{g})^{2}=0,\\\
\end{array}$

where the contact bracket (for either ${\mathfrak{k}}$ or ${\mathfrak{m}}$) is
of the form

(62)
$\\{f,g\\}={\frac{\partial{f}}{\partial{x_{0}}}}(1+E^{\prime})(g)+(1+E^{\prime})(f){\frac{\partial{g}}{\partial{x_{0}}}}+\mathop{\sum}\limits_{1\leq
i\leq
k}\left({\frac{\partial{f}}{\partial{x_{i}}}}{\frac{\partial{g}}{\partial{x_{k+i}}}}+{\frac{\partial{f}}{\partial{x_{k+i}}}}{\frac{\partial{g}}{\partial{x_{i}}}}\right)$

and where

$E^{\prime}=\mathop{\sum}\limits_{1\leq i\leq
k}x_{i}{\frac{\partial}{\partial{x_{i}}}}\text{~{}~{}and~{}~{}}x_{0}:=\begin{cases}\tau&\text{for
${\mathfrak{m}}(k;\underline{N}|k+1)$}\\\ t&\text{for
${\mathfrak{k}}(2k_{\bar{0}}+1;\underline{N}|2k_{\bar{1}})$ and
$k=k_{\bar{0}}+k_{\bar{1}}$}.\end{cases}$

For $p\neq 2$ and the bracket (62), the action of $x_{0}$ gives the grading in
${\mathfrak{g}}$. For $p=2$, the element $x_{0}$ annihilates a subspace of
${\mathfrak{g}}_{-1}$ and acts as multiplication by 1 on the complementary
subspace of ${\mathfrak{g}}_{-1}$, and ${\mathfrak{g}}_{-2}$. Instead of
${\mathbb{K}}[i]$ we denote the corresponding ${\mathbb{K}}x_{0}$-module by

(63) ${\mathbb{K}}[*]$

Set $\Phi:=\mathop{\sum}\limits_{1\leq i\leq k}x_{i}x_{k+i}$. For $p=2$, the
elements $\Phi$ and $x_{0}$ interchange their roles: now $\Phi$ commutes with
${\mathfrak{g}}_{0}$.

####  3.15.3. The two types of superizations of the contact series over
${\mathbb{C}}$.

The type ${\mathfrak{k}}$: Define the Lie superalgebra
${\mathfrak{hei}}(2n|m)$ on the direct sum of a $(2n,m)$-dimensional
superspace $W$ endowed with a nondegenerate antisymmetric bilinear form $B$
and a $(1,0)$-dimensional space spanned by $z$. Clearly, we have

(64)
${\mathfrak{k}}(2n+1|m)=({\mathfrak{hei}}(2n|m),{\mathfrak{c}}{\mathfrak{osp}}(m|2n))_{*}$

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

The type ${\mathfrak{m}}$: 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 (${\mathfrak{ba}}$ is
Anti-Bracket read backwards). Its space is $W\oplus{\mathbb{C}}\cdot z$, where
$W$ is an $n|n$-dimensional superspace endowed with a nondegenerate
antisymmetric odd bilinear form $B$; the bracket in ${\mathfrak{ba}}(n)$ is
given by the following relations:

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

###  3.16. Exceptional vectorial Lie superalgebras over ${\mathbb{K}}$
analogous to their namesakes over ${\mathbb{C}}$

We describe the exceptional vectorial Lie superalgebras, and their
desuperizations, as complete Cartan prolongs of their nonpositive parts, see
Tables 23.2, 23.4; the corresponding gradings are explicitly given in Table
(181).

Every Lie superalgebras of series ${\mathfrak{vect}}$, ${\mathfrak{k}}$ and
${\mathfrak{m}}$ is the complete prolong of its negative part in the standard
grading, see (18). The simple exceptional Lie superalgebra ${\mathfrak{g}}$ of
depth $>1$, whose negative part in its W-grading different from the negative
part of ${\mathfrak{k}}$ or ${\mathfrak{m}}$ in their standard grading, and
not isomorphic to ${\mathfrak{kas}}(5;\underline{N}|5)$ is the complete
prolong of its negative part.

Desuperizations of two nonisomorphic Lie superalgebras realized by vector
fields on supervarieties of different superdimension might become vectorial
Lie algebras realized on varieties of the same dimension. We distinguish these
cases by their depths (${\mathfrak{m}}{\mathfrak{b}}_{2}(11;\underline{N})$
and ${\mathfrak{m}}{\mathfrak{b}}_{3}(11;\underline{N})$ and also
${\mathfrak{k}}{\mathfrak{le}}_{2}(20;\underline{N})$ and
${\mathfrak{k}}{\mathfrak{le}}_{3}(20;\underline{N})$); for the case of equal
depths, we denote ${\mathfrak{v}}{\mathfrak{le}}(9;\underline{N})$ and — for
the time-being — $\widetilde{{\mathfrak{v}}{\mathfrak{le}}}(9;\underline{N})$.

##  4\. A description of
$\widetilde{{\mathfrak{sb}}}(2^{n}-1;\widetilde{}\underline{N})$ for $p=2$

###  4.1. Recapitulation: $p=0$, $n$ even

Let $q=(q_{1},\dots,q_{n})$ and $\xi=(\xi_{1},\dots,\xi_{n})$. We consider the
subsuperspace of functions ${\mathbb{C}}[q,\xi]$ of the form

$\\{(1+\Xi)f(q,\xi)\mid\Delta(f)=0\text{~{}~{}and~{}~{}}\int_{\xi}f\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}=0\\},\text{~{}~{}where
$\Xi:=\xi_{1}\cdot\dots\cdot\xi_{n}$},$

with the Buttin bracket. In this section we use only this bracket and omit
index “B.b”.

Let us compute the bracket in $(1+\Xi){\mathfrak{sb}}^{(1)}(n;n)$ realized by
elements of ${\mathfrak{sb}}^{(1)}(n)$. We have

(65) $\begin{array}[]{l}\\{(1+\Xi)f,\ (1+\Xi)g\\}=\\\ \phantom{.}\hskip
56.9055pt\begin{cases}\\{f,\ g\\}=(1+\Xi)\\{f,g\\}&\text{if
$\deg_{\xi}(f),\deg_{\xi}(g)>0$};\\\ \\{(1+\Xi)f,\
g\\}=(1+\Xi)\\{f,g\\}&\text{if $\deg_{\xi}(f)=0$, $\deg_{\xi}(g)>1$};\\\
\\{(1+\Xi)f,\ g\\}=&\\\ (1+\Xi)\\{f,\ g\\}+\sum\partial_{\xi_{i}}\Xi
f\partial_{q_{i}}(g_{i})\xi_{i}&\text{if $\deg_{\xi}(f)=0$,
$\deg_{\xi}(g)=1$,}\\\ \stackrel{{\scriptstyle\text{Since
$\sum\partial_{\xi_{i}}\Xi f\partial_{q_{i}}(g_{i})\xi_{i}=\Xi
f\sum\partial_{q_{i}}(g_{i})=0$.}}}{{=}}&\\\ (1+\Xi)\\{f,\ g\\}&\text{i.e.,
$g=\sum g_{i}(q)\xi_{i}$ and $\Delta(g)=0$};\\\
\sum\partial_{\xi_{i}}\Xi(\partial_{q_{i}}(f)g-\partial_{q_{i}}(g)f)&\text{if
$\deg_{\xi}(f)=\deg_{\xi}(g)=0$}.\end{cases}\end{array}$

In the ${\mathbb{Z}}$-grading of
${\mathfrak{g}}=\widetilde{{\mathfrak{sb}}}(n;n)$ by degrees of the $q$
shifted by $-1$, we have:

${\mathfrak{g}}_{-1}$ is spanned by monomials in $\xi$ of degrees 1 through
$n-1$, and by $1+\Xi$;

${\mathfrak{g}}_{0}$ is spanned by functions of the form $g=(1+\Xi)\sum
g_{i}(\xi)q_{i}$, where $\sum\partial_{\xi_{i}}g_{i}=0$.

The ${\mathfrak{g}}_{0}$-action on ${\mathfrak{g}}_{-1}$ is as follows. If
$\deg_{\xi}(g_{i})>0$, then we can ignore $\Xi$ in the factor $1+\Xi$ since
$\Xi$ annihilates $g_{i}$, and hence $\mathop{\mathrm{ad}}\nolimits_{g}$ acts
on ${\mathfrak{g}}_{-1}$ as the vector field $\sum g_{i}\partial_{\xi_{i}}$
acts on the space of functions in $\xi$.

If $g=(1+\Xi)q_{i}$, then the $\mathop{\mathrm{ad}}\nolimits_{g}$-acts on
${\mathfrak{g}}_{-1}$ precisely an element of
$\widetilde{{\mathfrak{svect}}}(0|n)$ acts on the space
$\mathop{\mathrm{{Vol\/}}}\nolimits[\xi]$:

(66) $\begin{array}[]{l}\\{g,\ \xi_{j}\\}=(1+\Xi)\delta_{ij},\\\ \\{g,\
1+\Xi\\}=\partial_{\xi_{i}}(\Xi),\\\
\mathop{\mathrm{ad}}\nolimits_{g}(f)=\partial_{\xi_{i}}(f)\text{~{}~{} for
monomials~{}~{}}f=f(\xi).\end{array}$

Since $(1-\Xi)\mathop{\mathrm{{vol\/}}}\nolimits$ is the invariant subspace in
$\mathop{\mathrm{{Vol\/}}}\nolimits[\xi]$, it follows that, in the quotient
space, we can take for a basis elements of the form
$f(\xi)\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}$, where monomials $f$ differ
from $1$ and $\Xi$, and either $1$ or $\Xi$. (For reasons unknown, SuperLie
selected $\Xi$.)

####  4.1.1. Recapitulation: $p=0$, $n$ odd.

Everything is as above but with an odd parameter $\tau$ which should be a
coefficient of $\Xi$. These deforms with odd parameters have interpretation in
the category of supervarieties, see [BGL4], but not in the naive setting of
superspaces over ${\mathbb{K}}$.

###  4.2.
$\textbf{F}(\widetilde{{\mathfrak{sb}}}(2^{n-1}-1;\underline{N}|2^{n-1}))$ for
$n$ odd, $p=2$

For $p=2$, it is possible to desuperize deforms with odd parameters and
consider them in the category of superspaces, see [BGL4]. We assume that
$p(\mathop{\mathrm{{vol\/}}}\nolimits_{\xi})\cong n\pmod{p}$.

Example: $n=3$. For a basis in ${\mathfrak{g}}_{-1}$, where
$\partial_{i}:=\partial_{x_{i}}$, we take:

(67)
$\small\begin{array}[]{l}\partial_{1}=\xi_{1}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi},\
\partial_{2}=\xi_{2}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi},\
\partial_{3}=\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi},\
\partial_{4}=\xi_{1}\xi_{2}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi},\
\partial_{5}=\xi_{1}\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi},\\\
\partial_{6}=\xi_{2}\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi},\
\partial_{7}=\xi_{1}\xi_{2}\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}.\end{array}$

For a basis of ${\mathfrak{g}}_{0}$, where $\delta_{i}:=\partial_{\xi_{i}}$ we
take:

(68)
$\footnotesize\begin{array}[]{ll}\begin{array}[]{l}(1+\xi_{1}\xi_{2}\xi_{3})\delta_{1}\longleftrightarrow
x_{1}\partial_{7}+x_{4}\partial_{2}+x_{5}\partial_{3}+x_{7}\partial_{6}\\\
(1+\xi_{1}\xi_{2}\xi_{3})\delta_{2}\longleftrightarrow
x_{2}\partial_{7}+x_{4}\partial_{1}+x_{6}\partial_{3}+x_{7}\partial_{5}\\\
(1+\xi_{1}\xi_{2}\xi_{3})\delta_{3}\longleftrightarrow
x_{3}\partial_{7}+x_{5}\partial_{1}+x_{6}\partial_{2}+x_{7}\partial_{4}\\\
\xi_{1}\delta_{2}\longleftrightarrow x_{2}\partial_{1}+x_{6}\partial_{5}\\\
\xi_{1}\delta_{3}\longleftrightarrow x_{3}\partial_{1}+x_{6}\partial_{4}\\\
\xi_{2}\delta_{1}\longleftrightarrow x_{1}\partial_{2}+x_{5}\partial_{6}\\\
\xi_{1}\delta_{1}+\xi_{2}\delta_{2}\longleftrightarrow
x_{1}\partial_{1}+x_{2}\partial_{2}+x_{5}\partial_{5}+x_{6}\partial_{6}\\\
\xi_{1}\delta_{1}+\xi_{3}\delta_{3}\longleftrightarrow
x_{1}\partial_{1}+x_{3}\partial_{3}+x_{4}\partial_{4}+x_{6}\partial_{6}\\\
\end{array}&\begin{array}[]{l}\xi_{2}\delta_{3}\longleftrightarrow
x_{3}\partial_{2}+x_{5}\partial_{4}\\\ \xi_{3}\delta_{1}\longleftrightarrow
x_{1}\partial_{3}+x_{4}\partial_{6}\\\ \xi_{3}\delta_{2}\longleftrightarrow
x_{2}\partial_{3}+x_{4}\partial_{5}\\\
\xi_{1}\xi_{2}\delta_{3}\longleftrightarrow x_{3}\partial_{4}\\\
\xi_{1}\xi_{3}\delta_{2}\longleftrightarrow x_{2}\partial_{5}\\\
\xi_{1}\xi_{2}\delta_{2}+\xi_{1}\xi_{3}\delta_{3}\longleftrightarrow
x_{2}\partial_{4}+x_{3}\partial_{5}\\\
\xi_{2}\xi_{3}\delta_{1}\longleftrightarrow x_{1}\partial_{6}\\\
\xi_{1}\xi_{3}\delta_{1}+\xi_{2}\xi_{3}\delta_{2}\longleftrightarrow
x_{1}\partial_{5}+x_{2}\partial_{6}\\\
\xi_{1}\xi_{2}\delta_{1}+\xi_{2}\xi_{3}\delta_{3}\longleftrightarrow
x_{1}\partial_{4}+x_{3}\partial_{6}\\\ \end{array}\end{array}$

The highest weight vectors are:

(69)
$\footnotesize\begin{array}[]{l}w_{1}=x_{2}x_{3}\partial_{7}+x_{2}x_{5}\partial_{1}+x_{2}x_{6}\partial_{2}+x_{2}x_{7}\partial_{4}+x_{3}x_{4}\partial_{1}+x_{3}x_{6}\partial_{3}+x_{3}x_{7}\partial_{5}+x_{4}x_{6}\partial_{4}+x_{5}x_{6}\partial_{5},\\\
w_{2}=x_{2}x_{3}\partial_{1}+x_{2}x_{6}\partial_{4}+x_{3}x_{6}\partial_{5},\\\
w_{3}=x_{3}^{(2)}\partial_{1}+x_{3}x_{6}\partial_{4}.\end{array}$

The lowest weight vectors are:

(70)
$\footnotesize\begin{array}[]{l}v_{1}=x_{1}^{(2)}\partial_{1}+x_{1}x_{2}\partial_{2}+x_{1}x_{5}\partial_{5}+x_{1}x_{6}\partial_{6}+x_{2}x_{5}\partial_{6}+x_{1}^{(2)}\partial_{1}+x_{1}x_{3}\partial_{3}+x_{1}x_{4}\partial_{4}+x_{1}x_{6}\partial_{6}+x_{3}x_{4}\partial_{6},\\\
v_{2}=x_{1}^{(2)}\partial_{1}+x_{1}x_{2}\partial_{2}+x_{1}x_{5}\partial_{5}+x_{1}x_{6}\partial_{6}+x_{2}x_{5}\partial_{6}+x_{1}^{(2)}\partial_{1}+x_{1}x_{3}\partial_{3}+x_{1}x_{4}\partial_{4}+x_{1}x_{6}\partial_{6}+x_{3}x_{4}\partial_{6},\\\
v_{3}=x_{1}^{(2)}\partial_{6},\end{array}$

####  4.2.1. Partial prolongs.

The elements of ${\mathfrak{g}}_{0}$ absent in
$\widetilde{\mathfrak{g}}_{0}:=[{\mathfrak{g}}_{1},{\mathfrak{g}}_{-1}]$ are
$\xi_{1}\xi_{2}\delta_{3}$, $\xi_{1}\xi_{3}\delta_{2}$,
$\xi_{2}\xi_{3}\delta_{1}$. The $\widetilde{\mathfrak{g}}_{0}$-module
${\mathfrak{g}}_{-1}$ is irreducible.

The weights are considered with respect to
${\mathfrak{sl}}(3)\subset\textbf{F}(\widetilde{\mathfrak{svect}}(0|3))$,
i.e., $w(\xi_{1})=(1,0)$, $w(\xi_{2})=(-1,1)$, $w(\xi_{3})=(0,-1)$. The
raising are the elements for which either $w_{1}+w_{2}>0$, or
$w_{1}=-w_{2}>0$, the lowering are the elements for which either
$w_{1}+w_{2}<0$, or $w_{1}=-w_{2}<0$. (To find lowering and raising operators,
we could have considered a ${\mathbb{Z}}$-grading of
$\widetilde{\mathfrak{svect}}(0|n)$ by setting
$\deg\xi_{1}=\ldots=\deg\xi_{n-1}=1$ and $\deg\xi_{n}=-n+1$ with ensuing
natural division into “positive” and “negative” parts.)

Let $V_{i}$ and $W_{i}$ denote the $\widetilde{\mathfrak{g}}_{0}$-modules
generated by $v_{i}$ and $w_{i}$, respectively. We have

(71) $\footnotesize\begin{array}[]{l}\dim{\mathfrak{g}}_{1}=31,\
\dim{\mathfrak{g}}_{2}=49,\ \dim{\mathfrak{g}}_{3}=71;\\\ \dim V_{1}=\dim
W_{1}=7,\ \dim V_{2}=\dim W_{2}=8,\ \dim V_{3}=\dim W_{3}=16;\\\ V_{1}=W_{1},\
V_{1}\subset V_{2}\subset V_{3},\ W_{1}\subset W_{2}\subset W_{3},\\\
\dim(V_{2}+W_{2})=9,\ \dim(V_{2}+W_{3})=\dim(V_{3}+W_{2})=17,\
\dim(V_{3}+W_{3})=24.\end{array}$

The brackets with ${\mathfrak{g}}_{-1}$:

(72)
$\footnotesize\begin{array}[]{l}\dim([{\mathfrak{g}}_{-1},V_{1}])=\dim[{\mathfrak{g}}_{-1},V_{2}+W_{2}]=14,\\\
\dim([{\mathfrak{g}}_{-1},V_{3}])=15\text{~{}~{}(absent are
$\xi_{1}\xi_{2}\delta_{3}$, $\xi_{1}\xi_{3}\delta_{2}$)},\\\
\dim([{\mathfrak{g}}_{-1},W_{3}])=15\text{~{}~{}(absent are
$\xi_{1}\xi_{3}\delta_{2}$, $\xi_{2}\xi_{3}\delta_{1}$)},\\\
\dim([{\mathfrak{g}}_{-1},V_{3}+W_{3}])=16\text{~{}~{}(absent is
$\xi_{1}\xi_{3}\delta_{2}$)}.\end{array}$

Therefore

(73) Partial prolongs in the direction of | dimensions
---|---
$V_{1}\text{~{}~{}or~{}~{}}V_{2}\text{~{}~{}or~{}~{}}V_{2}+W_{2}$ | $\dim{\mathfrak{g}}_{2}=1;\ {\mathfrak{g}}_{3}=0$
$V_{3}\text{~{}~{}or~{}~{}}V_{3}+W_{2}$ | $\dim{\mathfrak{g}}_{2}=\dim{\mathfrak{g}}_{3}=16$
$V_{3}+W_{3}$ | $\dim{\mathfrak{g}}_{2}=32,\dim{\mathfrak{g}}_{3}=40$
$[{\mathfrak{g}}_{-1},V_{1}]\text{~{}~{}or~{}~{}}[{\mathfrak{g}}_{-1},V_{2}+W_{2}]$ |  $\dim{\mathfrak{g}}_{1}=10\text{~{}~{}absent are $v_{3}$ and $w_{3}$};\ \dim{\mathfrak{g}}_{2}=1;\ {\mathfrak{g}}_{3}=0$.

The noncritical coordinates of
$\widetilde{{\mathfrak{sb}}}(7;\widetilde{}\underline{N})=\textbf{F}(\widetilde{{\mathfrak{sb}}}(4;\underline{N}|3))$
are $N_{1}$, $N_{2}$, and $N_{3}$.

###  4.3.
$\textbf{F}(\widetilde{{\mathfrak{sb}}}(2^{n-1}-1;\underline{N}|2^{n-1}))$ for
$n$ even, $p=2$

For the unconstrained shearing vector $\underline{N}_{\infty}$, see (10), the
dimensions of homogeneous components of
$\widetilde{{\mathfrak{sb}}}(2^{n}-1;\underline{N}_{\infty})$ are the same as
those of ${\mathfrak{sb}}^{(1)}(n)$ in the nonstandard grading
${\mathfrak{sb}}^{(1)}(n;n)$ for $p=0$. The main idea:
${\mathfrak{sb}}^{(1)}(n)=\mathop{\text{\rm
Im}}\nolimits\Delta|_{{\mathfrak{b}}(n)}$, where
$\Delta=\sum\frac{\partial^{2}}{\partial q_{i}\partial\xi_{i}}$. The
dimensions of homogeneous components of ${\mathfrak{b}}(n;n)$ for $n$ even
are:

(74) $i$ | $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{i}$ | $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{sb}}^{(1)}(n;n)_{i}$
---|---|---
$-1$ | $2^{n-1}|2^{n-1}$ | $2^{n-1}|2^{n-1}-1$
$0$ | $n(2^{n-1}|2^{n-1})$ | $(n-1)(2^{n-1}|2^{n-1})+1|0$
$1$ | $\frac{1}{2}n(n+1)(2^{n-1}|2^{n-1})$ | $\frac{1}{2}(n^{2}-n+2)(2^{n-1}|2^{n-1})-0|1$

Let the weights of $\xi_{i}$ be $w(\xi_{i})=(0,...0,1,0,...,0)$ with a 1 on
the $i$th place for $i<n$ and $w(\xi_{n})=(-1,...,-1)$.

####  4.3.1. Example $\widetilde{{\mathfrak{sb}}}(8;\underline{N}|7)$.

For a basis (even $\mid$ odd) of the ${\mathfrak{g}}_{0}$-module
${\mathfrak{g}}_{-1}\simeq\frac{\Pi(\mathop{\mathrm{{Vol\/}}}\nolimits(0|n))}{{\mathbb{K}}(1+\xi_{1}\dots\xi_{n})\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}}$,
where ${\mathfrak{g}}_{0}\simeq\widetilde{{\mathfrak{svect}}}(0|4))$, we take:

(75)
$\small\begin{array}[]{l|l}\partial_{1}:=\xi_{1}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\;\partial_{2}:=\xi_{2}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};&\partial_{9}:=\xi_{1}\xi_{2}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\
\partial_{10}:=\xi_{1}\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\\\
\partial_{3}=\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\;\partial_{4}=\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};&\partial_{11}:=\xi_{1}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\
\partial_{12}:=\xi_{2}\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\\\
\partial_{5}=\xi_{1}\xi_{2}\xi_{3}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\partial_{6}=\xi_{1}\xi_{2}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};&\partial_{13}:=\xi_{2}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\;\partial_{14}:=\xi_{3}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\\\
\partial_{7}=\xi_{1}\xi_{3}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi};\;\partial_{8}=\xi_{2}\xi_{3}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}&\partial_{15}:=\xi_{1}\xi_{2}\xi_{3}\xi_{4}\mathop{\mathrm{{vol\/}}}\nolimits_{\xi}\end{array}$

Critical coordinates of
$\widetilde{{\mathfrak{sb}}}(15;\widetilde{}\underline{N})=\textbf{F}(\widetilde{{\mathfrak{sb}}}(8;\underline{N}|7))$
are the same as those of $\widetilde{{\mathfrak{sb}}}(8;\underline{N}|7)$:
$N_{5}=N_{6}=N_{7}=N_{8}=1$, and all those corresponding to the formerly odd
indeterminates: $N_{9}$–$N_{15}$.

####  4.3.2. Partial prolongs.

We have $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{0}=25|24$, and
${\mathfrak{g}}_{0}$ contains a simple ideal of
$\mathop{\mathrm{sdim}}\nolimits=21|24$, the quotient is commutative;
${\mathfrak{g}}_{-1}$ is irreducible over this ideal. We have
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{1}=56|55$, there are 3 highest
weight vectors and 2 lowest weight vectors in ${\mathfrak{g}}_{1}$;

(76) $\begin{array}[]{l}V_{1}=W_{1},\ V_{1}\subset V_{2},\ W_{1}\subset
W_{2}\subset W_{3};\\\ \mathop{\mathrm{sdim}}\nolimits V_{1}=24|21,\
\mathop{\mathrm{sdim}}\nolimits V_{2}=\mathop{\mathrm{sdim}}\nolimits
W_{3}=32|31,\ \mathop{\mathrm{sdim}}\nolimits W_{2}=24|22,\\\
\mathop{\mathrm{sdim}}\nolimits(V_{2}+W_{2})=32|32,\
\mathop{\mathrm{sdim}}\nolimits(V_{2}+W_{3})=40|40;\\\
\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=105|104.\end{array}$

The highest weight vector of $W_{3}$ is $w_{3}=x_{4}^{(2)}\partial_{5}$. This
answer seems strange: the algebra is symmetric with respect to the permutation
of the $\xi_{i}$ while the list of highest weight vectors is not. Performing
all possible permutations we obtain similar vectors $x_{1}^{(2)}\partial_{8}$,
$x_{2}^{(2)}\partial_{7}$, $x_{3}^{(2)}\partial_{6}$ (which are not
highest/lowest with respect to the division into positive/negative weights we
have selected first) but generate similar submodules $Y_{1}$, $Y_{2}$, $Y_{3}$
(and $Y_{4}=W_{3}$).

We have $Y_{1}+Y_{2}+Y_{3}+Y_{4}={\mathfrak{g}}_{1}$ and
$\mathop{\mathrm{sdim}}\nolimits(Y_{1}+Y_{2}+Y_{3})=48|48$.

Other highest weight vectors:

(77)
$\footnotesize\begin{array}[]{l}w_{1}=x_{3}x_{4}\partial_{10}+x_{3}x_{13}\partial_{5}+x_{4}^{(2)}\partial_{11}+x_{4}x_{12}\partial_{5}+x_{4}x_{13}\partial_{6}+x_{2}x_{4}\partial_{9}+x_{2}x_{14}\partial_{5}+x_{4}^{(2)}\partial_{11}+x_{4}x_{12}\partial_{5}+x_{4}x_{14}\partial_{7},\\\
w_{2}=x_{3}x_{4}\partial_{9}+x_{3}x_{14}\partial_{5}+x_{4}x_{14}\partial_{6}.\end{array}$

The lowest weight vectors:

(78)
$\footnotesize\begin{array}[]{l}v_{1}=x_{1}x_{2}\partial_{4}+x_{1}x_{5}\partial_{7}+x_{1}x_{9}\partial_{11}+x_{1}x_{12}\partial_{14}+x_{2}x_{5}\partial_{8}+x_{2}x_{9}\partial_{13}+x_{2}x_{10}\partial_{14}+x_{9}x_{10}\partial_{7}+x_{9}x_{12}\partial_{8},\\\
v_{2}=x_{1}^{(2)}\partial_{4}+x_{1}x_{5}\partial_{8}+x_{1}x_{9}\partial_{13}+x_{1}x_{10}\partial_{14}+x_{9}x_{10}\partial_{8}.\end{array}$

We have

(79)
$\footnotesize\begin{array}[]{ll}\mathop{\mathrm{sdim}}\nolimits([{\mathfrak{g}}_{-1},\
V_{1}])=21|24;&\mathop{\mathrm{sdim}}\nolimits([{\mathfrak{g}}_{-1},\
Y_{i}])=22|24,\\\ \mathop{\mathrm{sdim}}\nolimits([{\mathfrak{g}}_{-1},\
Y_{i}+Y_{j}])=23|24,&\mathop{\mathrm{sdim}}\nolimits([{\mathfrak{g}}_{-1},Y_{i}+Y_{j}+Y_{k}])=24|24.\end{array}$

Partial prolongs of ${\mathfrak{g}}_{0}$ and the following parts of
${\mathfrak{g}}_{1}$:

From $V_{1}=W_{1}$ and $W_{2}$:
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=11|8$,
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{3}=0|1$, no parameters;

From $V_{2}$: $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=33|32$, 1
parameter $N_{1}$ (same for $W_{3}$, parameter $N_{4}$);

From $V_{2}+W_{3}$: $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=56|56$,
2 parameters $N_{1}$ and $N_{4}$, similar for $Y_{i}+Y_{j}$;

From $Y_{1}+Y_{2}+Y_{3}$:
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=80|80$, 3 parameters
$N_{1}$, $N_{2}$ and $N_{3}$.

Partial prolongs of the following parts of ${\mathfrak{g}}_{0}$:

From $(21|24)$: $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{1}=24|27$,
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=11|8$, no parameters;

From $(22|24)$: $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{1}=32|34$,
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=33|32$, 1 parameter;

From $(23|24)$: $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{1}=40|41$,
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=56|56$, 2 parameters;

From $(24|24)$: $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{1}=48|48$,
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{2}=80|80$, 3 parameters.

##  5\. The Lie superalgebra ${\mathfrak{vle}}(4|3)$ over ${\mathbb{C}}$ and
its $p>2$ versions

###  5.1. Recapitulations, see [ShP]

In the realization of ${\mathfrak{le}}(3)$ by means of generating functions,
we identify the space of ${\mathfrak{le}}(3)$ with
$\Pi({\mathbb{C}}[\theta,q]/{\mathbb{C}}\cdot 1)$, where
$\theta=(\theta_{1},\theta_{2},\theta_{3})$, $q=(q_{1},q_{2},q_{3})$, see
(31). In the standard grading $\deg_{Lie}$ of ${\mathfrak{le}}(3)$, we assume
that $\deg q_{i}=\deg\theta_{i}=1$ for all $i$, and the grading is given by
the formula

$\deg_{Lie}(f):=\deg\mathop{\mathrm{Le}}\nolimits_{f}=\deg f-2\text{~{}~{}for
any monomial $f\in{\mathbb{C}}[\theta,q]$.}$

The nonstandard grading $\deg_{Lie;3}$ of
${\mathfrak{g}}={\mathfrak{le}}(3;3)$ is determined by the formulas

$\begin{array}[]{l}\deg_{3}\theta_{i}=0\text{~{}~{}and~{}~{}}\deg_{3}q_{i}=1\text{~{}~{}for~{}~{}}i=1,2,3;\\\
\deg_{Lie;3}(f)=\deg_{3}f-1\text{~{}~{} for any monomial
$f\in{\mathbb{C}}[\theta,q]$}.\end{array}$

This grading of ${\mathfrak{g}}={\mathfrak{le}}(3;3)$ is of depth 1, and its
homogeneous components are of the form:

${\mathfrak{g}}_{-1}=\Pi({\mathbb{C}}[\theta_{1},\theta_{2},\theta_{3}]/{\mathbb{C}}\cdot
1);\quad{\mathfrak{g}}_{k}=\Pi({\mathbb{C}}[\theta_{1},\theta_{2},\theta_{3}])\otimes
S^{k+1}(q_{1},q_{2},q_{3})\text{~{}~{}for $k\geq 0$}.$

In particular, ${\mathfrak{g}}_{0}\simeq{\mathfrak{vect}}(0|3)$, and
${\mathfrak{g}}_{1}$ is an irreducible ${\mathfrak{g}}_{0}$-module with the
lowest weight vector $q_{1}^{2}$. The whole Lie superalgebra
${\mathfrak{le}}(3;3)$ is the Cartan prolong of its nonpositive part and the
component ${\mathfrak{g}}_{1}$ generates the whole positive part.

To obtain ${\mathfrak{vle}}(4|3)$, we add the central element $d$ to the
zeroth component of ${\mathfrak{le}}(3;3)$; so that
$\mathop{\mathrm{ad}}\nolimits_{d}$ is the grading operator on the Cartan
prolong of its nonpositive part; this prolong is strictly bigger than
${\mathfrak{le}}(3;3)\ltimes{\mathbb{C}}\cdot d$.

This Cartan prolong is the exceptional simple Lie superalgebra
${\mathfrak{vle}}(4|3)$. Its component ${\mathfrak{vle}}_{1}$ is reducible but
indecomposable ${\mathfrak{vle}}_{0}$-module, and
${\mathfrak{vle}}_{1}/{\mathfrak{le}}(3;3)_{1}\simeq({\mathfrak{le}}(3;3)_{-1})^{*}$.
The other lowest weight vector in ${\mathfrak{vle}}_{1}$ is the element dual
to the highest weight vector in ${\mathfrak{g}}_{-1}$, i.e., to
$\Pi(\theta_{1}\theta_{2}\theta_{3})$.

Under the identification

$\Pi(\theta_{1}\theta_{2}\theta_{3})\longmapsto-\partial_{y};\quad\Pi(\theta_{i})\longmapsto-\partial_{x_{i}};\quad\Pi\left(\frac{\partial(\theta_{1}\theta_{2}\theta_{3})}{\partial\theta_{i}}\right)\longmapsto-\partial_{\xi_{i}}$

each vector field $D\in{\mathfrak{vle}}(4|3)$ is of the form

(80)
$D_{f,\;g}=\mathop{\mathrm{Le}}\nolimits_{f}+yB_{f}-(-1)^{p(f)}\left(y\Delta(f)+y^{2}\frac{\partial^{3}f}{\partial\xi_{1}\partial\xi_{2}\partial\xi_{3}}\right)\partial_{y}+\\\
+B_{g}-(-1)^{p(g)}\left(\Delta(g)+2y\frac{\partial^{3}g}{\partial\xi_{1}\partial\xi_{2}\partial\xi_{3}}\right)\partial_{y},$

where $f,g\in{\mathbb{C}}[x,\xi]$, and the operators $B_{F}$ and $\Delta$ are
given by the formulas

(81)
$B_{F}=\frac{\partial^{2}F}{\partial\xi_{2}\partial\xi_{3}}\frac{\partial}{\partial\xi_{1}}+\frac{\partial^{2}F}{\partial\xi_{3}\partial\xi_{1}}\frac{\partial}{\partial\xi_{2}}+\frac{\partial^{2}F}{\partial\xi_{1}\partial\xi_{2}}\frac{\partial}{\partial\xi_{3}},\quad\Delta=\mathop{\sum}\limits_{1\leq
i\leq 3}\frac{\partial^{2}}{\partial x_{i}\partial\xi_{i}}.$

There are two embeddings of ${\mathfrak{le}}(3)$ into ${\mathfrak{vle}}(4|3)$.
The embedding $i_{1}:{\mathfrak{le}}(3)\longrightarrow{\mathfrak{vle}}(4|3)$
corresponds to the grading ${\mathfrak{le}}(3;3)$. For explicit formulas, see
[ShP].

The embedding $i_{2}:{\mathfrak{le}}(3)\longrightarrow{\mathfrak{vle}}(4|3)$
corresponds to the standard grading of ${\mathfrak{le}}(3)$. In terms of
generating functions this embedding is of the form

(82) $i_{2}(f(q,\theta))\longrightarrow D_{f(i(q,\theta)),\;0},\;\text{ where
}i(q_{1},q_{2},q_{3},\theta_{1},\theta_{2},\theta_{2})=(x_{1},x_{2},x_{3},\xi_{1},\xi_{2},\xi_{3}).$

As vector spaces, we have

(83)
${\mathfrak{vle}}(4|3)=i_{1}({\mathfrak{le}}(3;3))+i_{2}({\mathfrak{le}}(3))\text{~{}~{}
while ~{}~{}}i_{1}({\mathfrak{le}}(3;3))\cap
i_{2}({\mathfrak{le}}(3))\simeq{\mathfrak{sle}}^{(1)}(3).$

Besides,

(84)
$\begin{array}[]{l}i_{1}(f(q))=i_{2}(\Delta(f\theta_{1}\theta_{2}\theta_{3})),\\\
i_{2}(f(q))=-i_{1}(\Delta(f\theta_{1}\theta_{2}\theta_{3})),\end{array}$

and

(85) $i_{1}(f)=i_{2}(f)\text{ if }f=\mathop{\sum}\limits_{1\leq i\leq
3}f_{i}(q)\theta_{i}\text{ and }\Delta f=0.$

The lowest weight vectors in ${\mathfrak{vle}}_{1}$ are $i_{1}(q_{1}^{2})$ and
$i_{2}(\theta_{1}\theta_{2}\theta_{3})$. We have

(86) $\mathop{\mathrm{sdim}}\nolimits{\mathfrak{vle}}_{1}=28|27.$

###  5.2. Passage from the ground field ${\mathbb{C}}$ to ${\mathbb{K}}$ for
$p>2$

The situation described in the previous subsection does not change under
passage from the ground field ${\mathbb{C}}$ to any field of characteristic 0
and also to ${\mathbb{K}}$ if $\mathop{\text{\rm
char}}\nolimits{\mathbb{K}}=p>2$ provided $\underline{M}_{i}=\infty$ for each
even indeterminate $x_{i}$. As is clear from Eq. (80), even over
${\mathbb{C}}$

(87) the degree with respect to $y$ of the coefficients of the vector fields
in ${\mathfrak{vle}}$ is $\leq 2$.

In all these cases, the Lie superalgebra ${\mathfrak{g}}$ — the Cartan prolong
of the nonpositive part of ${\mathfrak{vle}}$ — remains simple and of infinite
dimension. The component ${\mathfrak{g}}_{1}$ also retains its structure as
${\mathfrak{g}}_{0}$-module but it does not generate the whole
${\mathfrak{g}}$ (since the $u_{i}$ do not generate
${\mathcal{O}}(u;\underline{M})$ if $\underline{M}\neq\underline{M}_{s}$).

If $p>2$, the component ${\mathfrak{g}}_{1}$ also retains its structure as
${\mathfrak{g}}_{0}$-module even if any (or all) of the coordinates of the
shearing vector
$\underline{M}=(\underline{M}_{1},\underline{M}_{2},\underline{M}_{3})$ become
finite. If $\underline{M}_{i}<\infty$ for all $i$, the Cartan prolong is
finite dimensional. It can by described by means of Eq. (80) but we should
investigate when
$D_{f,\;g}\in{\mathfrak{vect}}(4;\widetilde{}\underline{M}|3)$, where
$\widetilde{}\underline{M}=(\underline{M},M_{y})$. The direct observation
gives the answer:

$f\in{\mathcal{O}}(x;\underline{M}|\xi)\oplus\mathop{\mathrm{{Span}}}\nolimits(x_{i}^{(s_{i})})_{i=1}^{3},\text{~{}~{}
where $s_{i}=p^{M_{i}}$,}$

i.e., we should add “virtual” (nonexisting for the given $\underline{M}$)
elements $f=x_{i}^{(s_{i})}$ for $i=1,2,3$. Since due to (84)

$D_{x_{i}^{(s_{i})},\;0}=i_{2}(q_{i}^{(s_{i})})=i_{1}(\Delta(q_{i}^{(s_{i})}\theta_{1}\theta_{2}\theta_{3})),$

we see that
$i_{2}(q_{i}^{(s_{i})})\in{\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)^{(1)}$
though $q_{i}^{(s_{i})}\notin{\mathfrak{le}}(3;\underline{M})^{(1)}$.

Now, let us investigate generating functions $g$. The kernel of the map
$g\longmapsto D_{0,\ g}$ is:

(88) $\mathop{\mathrm{{Span}}}\nolimits(D_{0,\;g}=0\mid\text{$g=g(x)$ or
$g=\mathop{\sum}\limits_{i}g_{i}(x)\xi_{i}$ with $\Delta g=0$.})$

If $g=\mathop{\sum}\limits_{i}g_{i}(x)\xi_{i}$ but $\Delta g=h(x)\neq 0$, then
$D_{0,\;g}=h(x)\partial_{y}$ depends on $h$ only. It is clear that any
function $h\in{\mathcal{O}}(x;\underline{M})$ can be expressed as $h=\Delta g$
for some $g\in{\mathcal{O}}(x;\underline{M}|\xi)$, except for

$h_{s}=x_{1}^{(s_{1}-1)}x_{2}^{(s_{2}-1)}x_{3}^{(s_{3}-1)},\text{~{}~{}where
$s=(s_{1},s_{2},s_{3})$.}$

To obtain $D={h_{s}}\partial_{y}$, we should add to the space of generating
functions any of the “virtual” functions

$g_{s,i}=\xi_{i}\partial_{j}\partial_{k}\left(x_{1}^{(s_{1})}x_{2}^{(s_{2})}x_{3}^{(s_{3})}\right)\text{~{}~{}
for $i=1,2,3$, and $j,k\neq i$, $j\neq k$.}$

Modulo the kernel (88) of $\Delta$ only one “extra” generator suffices; we
select $x_{1}^{(s_{1})}x_{2}^{(s_{2}-1)}x_{3}^{(s_{3}-1)}\xi_{1}$ for
definiteness.

As was shown in [ShP] for $p=0$; but the formulas remain true for any $p>0$,

$D_{0,\;g_{s}}=i_{1}(-h_{s}\xi_{1}\xi_{2}\xi_{3})\text{, and hence, for $p=2$,
}D_{0,\;g_{s}}\in i_{1}({\mathfrak{le}}(3;\underline{M})^{(1)}).$

As a result, if $p>2$ and $\underline{M}_{i}<\infty$ for $i=1,2,3$, then

$\begin{array}[]{rl}{\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)&=\mathop{\mathrm{{Span}}}\nolimits(D_{f,\;g}),\\\
\text{where}&f\in{\mathcal{O}}(x;\underline{M}|\xi)\oplus\mathop{\mathrm{{Span}}}\nolimits(x_{i}^{(s_{i})})_{i=1}^{3}\text{~{}and~{}}g\in{\mathcal{O}}(x;\underline{M}|\xi)\oplus{\mathbb{K}}\cdot
x_{1}^{(s_{1})}x_{2}^{(s_{2}-1)}x_{3}^{(s_{3}-1)}\xi_{1}.\end{array}$

The Lie superalgebra ${\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)$ is
finite dimensional and simple.

In this case, however, the fact (83) does not hold, even as spaces. We have

$D_{0,\;h_{s}\xi_{1}\xi_{2}\xi_{3}}\in{\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)\text{~{}~{}but~{}~{}}D_{0,\;h_{s}\xi_{1}\xi_{2}\xi_{3}}=i_{1}(f)-i_{2}(f),$

where

$f=q_{1}^{(s_{1})}q_{2}^{(s_{2}-1)}q_{3}^{(s_{3}-1)}\theta_{1}+q_{1}^{(s_{1}-1)}q_{2}^{(s_{2})}q_{3}^{(s_{3}-1)}\theta_{2}+q_{1}^{(s_{1}-1)}q_{2}^{(s_{2}-1)}q_{3}^{(s_{3})}\theta_{3}\notin{\mathcal{O}}(q;\underline{M}|\theta).$

##  6\. A description of
${\mathfrak{vle}}(7;\widetilde{}\underline{M}):=\textbf{F}({\mathfrak{vle}}(4;\underline{M}|3))$
for $p=2$

The Lie superalgebra ${\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)$ for
$p=2$ is a direct reduction modulo 2 of the integer form with divided powers
as coefficients of the complex vectorial Lie superalgebra
${\mathfrak{vle}}(4|3)$.

First of all, let us define squares of odd elements for Lie superalgebra
${\mathfrak{le}}(n;\underline{M})$, cf. eq. (61):

(89) $f^{2}:=\mathop{\sum}\limits_{1\leq i\leq n}\frac{\partial f}{\partial
q_{i}}\frac{\partial f}{\partial\theta_{i}}$

and have in mind that for $p=2$ and $\underline{M}_{i}<\infty$ for all $i$,
the Lie superalgebra ${\mathfrak{g}}={\mathfrak{le}}(n;\underline{M})$ is not
simple: the generating function of the maximal degree
$q_{1}^{(s_{1}-1)}q_{2}^{(s_{2}-1)}\ldots
q_{n}^{(s_{3}-1)}\theta_{1}\theta_{2}\ldots\theta_{n}$ does not belong to
${\mathfrak{g}}^{(1)}$, which is simple.

For $p>2$, we just reduce the expression (80) modulo $p$.

For $p=2$, we can not just reduce the expression (80) modulo 2; we should
modify it. Indeed, the system of equations on the coefficients of the field
$D\in{\mathfrak{vle}}(4|3)$ whose solution is given by the formula (80)
contain coefficients $\frac{1}{2}$, see [ShP]. The vector field
$D\in{\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)$ is of the form:

(90)
$D_{f,\;g}=\mathop{\mathrm{Le}}\nolimits_{f}+yB_{f}+y\Delta(f)\partial_{y}+B_{g}+\Delta(g)\partial_{y},\text{~{}~{}where
$\widetilde{}\underline{M}=(\underline{M},M_{y})$ and $M_{y}=1$},$

where $f,g\in{\mathcal{O}}[x;\underline{M}|\xi]$, and the operators $B_{F}$
and $\Delta$ are (recall eq. (81))

(91)
$B_{F}=\frac{\partial^{2}F}{\partial\xi_{2}\partial\xi_{3}}\frac{\partial}{\partial\xi_{1}}+\frac{\partial^{2}F}{\partial\xi_{3}\partial\xi_{1}}\frac{\partial}{\partial\xi_{2}}+\frac{\partial^{2}F}{\partial\xi_{1}\partial\xi_{2}}\frac{\partial}{\partial\xi_{3}},\quad\Delta=\mathop{\sum}\limits_{1\leq
i\leq 3}\frac{\partial^{2}}{\partial x_{i}\partial\xi_{i}}.$

The formulas (82), (84), and (85) are valid for any $p$.

For any $p>0$, the complete prolong can be described in terms of the
generating functions
$f,g\in{\mathcal{O}}(x_{1},x_{2},x_{3};\underline{M}_{\infty}|\xi_{1},\xi_{2},\xi_{3})$
and the Lie superalgebra obtained is infinite dimensional and simple.

Should we wish to obtain a finite dimensional Lie superalgebra, we have to set
$\underline{M}_{i}<\infty$ for all $i$, and then (as for any $p>0$)

(92)
$\begin{array}[]{l}{\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)=\mathop{\mathrm{{Span}}}\nolimits(D_{f,\;g}),\\\
\text{for
}f\in{\mathcal{O}}(x;\underline{M}|\xi)\oplus\mathop{\mathrm{{Span}}}\nolimits(x_{i}^{(s_{i})})_{i=1}^{3}\text{~{}
and ~{} }g\in{\mathcal{O}}(x;\underline{M}|\xi)\oplus{\mathbb{K}}\cdot
x_{1}^{(s_{1})}x_{2}^{(s_{2}-1)}x_{3}^{(s_{3}-1)}\xi_{1}.\end{array}$

This Lie superalgebra
${\mathfrak{g}}={\mathfrak{vle}}(4;\widetilde{}\underline{M}|3)$ is not simple
but ${\mathfrak{g}}^{(1)}$ is simple, its codimension in ${\mathfrak{g}}$ is
equal to 2: for
$f=q_{1}^{(s_{1}-1)}q_{2}^{(s_{2}-1)}q_{3}^{(s_{3}-1)}\theta_{1}\theta_{2}\theta_{3}$,
we have

$D_{f,\;0}=i_{2}(f)\notin{\mathfrak{g}}^{(1)},\qquad
D_{0,\;x_{1}^{(s_{1})}x_{2}^{(s_{2}-1)}x_{3}^{(s_{3}-1)}\xi_{1}}=i_{1}(f)\notin{\mathfrak{g}}^{(1)}.$

If $\underline{M}_{i}>1$ for all $i$, then
$\mathop{\mathrm{sdim}}\nolimits{\mathfrak{g}}_{1}=28|27$ remains the same as
over ${\mathbb{C}}$ and any other $p>2$.

For $p=2$, the structure of the ${\mathfrak{g}}_{0}$-module
${\mathfrak{g}}_{1}$ differs drastically from that for $p\neq 2$. Instead of
two lowest weight vectors, we have three of them. Besides, these three lowest
weight vectors do not describe the whole complexity of the module.

The submodule $i_{1}({\mathfrak{le}}(3;3)_{1})$ has a complicated structure.
To describe it, observe that for any vectorial Lie (super)algebra expressed in
terms of generating functions, the shearing vector can be considered on the
level of generating functions (let us denote it $\underline{M}$ in this case)
and on the level of coefficients of vector fields they generate (let us denote
it $\widetilde{}\underline{M}$ in this case).

In the first case, we obtain the “underdeveloped” Lie superalgebra

(93)
${\mathfrak{le}}^{\underline{M}}(n):=\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{f}\mid
f\in{\mathcal{O}}(n;\underline{M}));$

in the second one we get the correct
${\mathfrak{le}}(n;\widetilde{}\underline{M})=\mathop{\mathrm{{Span}}}\nolimits(\mathop{\mathrm{Le}}\nolimits_{f}\in{\mathfrak{vect}}(n;\widetilde{}\underline{M}|n))$.
We have

(94)
${\mathfrak{le}}(n;\widetilde{}\underline{M})={\mathfrak{le}}^{\underline{M}}(n)\ltimes\mathop{\mathrm{{Span}}}\nolimits(q_{i}^{(s_{i})}\mid
i=1,\dots,n).$

Accordingly, for $\widetilde{}\underline{M}_{i}$ unconstrained, the component
${\mathfrak{le}}(3;\widetilde{}\underline{M};3)_{1}$ is of the form:

${\mathfrak{le}}(3;\widetilde{}\underline{M};3)_{1}=\Pi({\mathbb{K}}[\theta_{1}\theta_{2}\theta_{3}])\otimes
DS^{2}(q_{1},q_{2},q_{3}),\text{~{}~{}see \eqref{DS}}.$

This component contains submodules corresponding to the minimal values
$\underline{M}_{i}=1$ for some $i$. The submodule

$W_{0}:={\mathfrak{le}}^{\underline{M}_{s}}(3;3)_{1}=\mathop{\mathrm{{Span}}}\nolimits(q_{i}q_{j}\varphi(\theta)\text{~{}~{}for
any $i.j$ and $\varphi$}).$

is irreducible. It is glued to the submodules

$W_{i}:=W_{0}\ltimes{\mathbb{K}}\cdot q_{i}^{(2)}$

in which $W_{0}$ is a submodule but not a direct summand, and each $W_{i}$ can
be further enlarged to the module

$W_{i,\theta}:=W_{i}\ltimes\mathop{\mathrm{{Span}}}\nolimits(q_{i}^{(2)}\varphi(\theta))\text{~{}~{}
corresponding to $\underline{M}_{i}>1$ and $\underline{M}_{j}=1$ for $j\neq
i$}$

with shearing performed on the level of generating functions.

Let us describe the subalgebras of
${\mathfrak{vle}}(4;\widetilde{}\underline{M}_{s}|3)$, the partial prolongs.
In what follows we will often use the following convention on notation:

(95) Let $v_{i}$ be a lowest weight vector of the ${\mathfrak{g}}_{0}$-module
${\mathfrak{g}}_{1}$ and $V_{i}$ the submodule generated by $v_{i}$. Let
${\mathfrak{g}}_{k}^{(i)}$ be the $k$th prolong “in the direction of $V_{i}$”,
i.e., prolong of
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0},V_{i}\subset{\mathfrak{g}}_{1})$.

Consider the ${\mathfrak{g}}_{0}$-submodules $W\subset{\mathfrak{g}}_{1}$ not
contained in $i_{1}({\mathfrak{le}}(3;3))$. There is only one such submodule
$V_{3}\subset W$ generated by $v_{3}$, see (96). The
${\mathfrak{g}}_{0}$-module ${\mathfrak{g}}_{1,\underline{M}}$ has the
following three lowest weight vectors expressed in the form $D_{f,\;g}$, and
also as $i_{1}(\cdot)$ or $i_{2}(\cdot)$:

(96) $v_{1}$ | $x_{1}y\partial_{\xi_{2}}+x_{1}\xi_{3}\partial_{x_{1}}++x_{1}\xi_{1}\partial_{x_{3}}+x_{2}y\partial_{\xi_{1}}+$ | $i_{1}(q_{1}q_{2})$ | $D_{x_{1}\xi_{1}\xi_{3}+x_{2}\xi_{2}\xi_{3},\;0}$
---|---|---|---
| $x_{2}\xi_{3}\partial_{x_{2}}+x_{2}\xi_{2}\partial_{x_{3}}+\xi_{3}\xi_{2}\partial_{\xi_{2}}+\xi_{3}\xi_{1}\partial_{\xi_{1}}$ |  |
$v_{2}$ | $x_{1}y\partial_{\xi_{1}}+x_{1}\xi_{3}\partial_{x_{2}}+x_{1}\xi_{2}\partial_{x_{3}}+\xi_{3}\xi_{2}\partial_{\xi_{1}}$ | $i_{1}(q_{1}^{(2)})$ | $D_{x_{1}\xi_{2}\xi_{3},\;0}$
$v_{3}$ | $y\xi_{3}\partial_{\xi_{3}}+y\xi_{2}\partial_{\xi_{2}}+y\xi_{1}\partial_{\xi_{1}}+\xi_{3}\xi_{2}\partial_{x_{1}}+$ | $i_{2}(\theta_{1}\theta_{2}\theta_{3})$ | $D_{\xi_{1}\xi_{2}\xi_{3},\;0}$
| $\xi_{3}\xi_{1}\partial_{x_{2}}+\xi_{2}\xi_{1}\partial_{x_{3}}$ |  |

By increasing the value of some of the coordinates $\underline{M}_{i}$ we
enlarge
${\mathfrak{g}}_{1}^{(3)}={\mathfrak{vle}}(4;\widetilde{}\underline{M}_{s}|3)_{1}$.
As ${\mathfrak{g}}_{0}$-module, ${\mathfrak{g}}_{1}^{(3)}$ is of the following
form:

$W_{0}\subset(W_{1}+W_{2}+W_{3})\subset{\mathfrak{g}}_{1}^{(3)},$

where
${\mathfrak{g}}_{1}^{(3)}/(W_{1}+W_{2}+W_{3})\simeq({\mathfrak{g}}_{-1})^{*}$,
and $(W_{1}+W_{2}+W_{3})/W_{0}$ is the trivial $0|3$-dimensional module, and
where $\mathop{\mathrm{sdim}}\nolimits W_{0}=12|12$.

The superdimensions of the positive components of