# On odd parameters in geometry

Dimitry Leitesa,b aNew York University Abu Dhabi, Division of Science and
Mathematics, P.O. Box 129188, United Arab Emirates<EMAIL_ADDRESS>
bDepartment of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden;
<EMAIL_ADDRESS>
###### Abstract.

1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.
Bernstein and I, and independently M. Duflo, observed that certain divergence-
free vectorial Lie superalgebras have deformations with odd parameters and
conjectured that other simple Lie superalgebras have no such deformations
(unpublished). Here, I prove this conjecture and overview the known
classification of simple finite-dimensional complex Lie superalgebras, their
presentations, realizations, and (very sketchily) relations with simple Lie
(super)algebras over fields of positive characteristic.

2) Any supermanifold which is a ringed space of the form (a manifold $M$, the
sheaf of sections of the exterior algebra of a vector bundle over $M$) is
called split. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth
supermanifolds is split. In 1982, P. Green and Palamodov showed that a
complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a
split supermanifold. So far, researchers considered, mostly, even obstructions
to splitness. This lead them to the conclusion that any supermanifolds of
superdimension $m|1$ is split. I’ll show that there are non-split
supermanifolds of superdimension $m|1$; for example, certain $1|1$-dimensional
superstrings, the obstructions to their splitness correspond to odd
parameters.

###### Key words and phrases:

Simple Lie superalgebra, deformation, non-split supermanifold

###### 2010 Mathematics Subject Classification:

Primary 17A70; Secondary 14M30, 17B20

###### Contents

  1. 1 Introduction
    1. 1.1 Prologue: Four types of main characters, two questions
    2. 1.2 Disclamer
    3. 1.3 Simple Lie superalgebras. First definitions and examples
    4. 1.4 Realizations
    5. 1.5 Cartan matrices
    6. 1.6 Presentations
    7. 1.7 Deformations with odd parameters
    8. 1.8 Other appearances of odd parameters
  2. 2 What “Lie superalgebra” is, naively
    1. 2.1 The Sign Rule, skew and anti
  3. 3 What “Lie superalgebra” is from categorical point of view ([KLLS, KLLS1])
    1. 3.1 Morphisms of supervarieties
    2. 3.2 On automorphisms that do not preserve parity
    3. 3.3 Supervarieties and superschemes (after [L0])
    4. 3.4 The functor of points (co)represented by a supervariety or a superscheme
      1. 3.4.1 Linear supervariety$\longleftrightarrow$linear superspace.
      2. 3.4.2 Lie superalgebras ([KLLS]).
  4. 4 Deformations
    1. 4.1 Deformations and deforms. Odd parameters
  5. 5 Deformations of complex supermanifolds. Odd parameters
    1. 5.1 The complex supermanifolds
    2. 5.2 The retract of the supermanifold
    3. 5.3 On the claims of Bettadapura, see [BK1, BK2, BK3]
  6. 6 Finite-dimensional simple Lie superalgebras considered naively
    1. 6.1 Linear (matrix) Lie superalgebras
      1. 6.1.1 Supermatrices of operators
      2. 6.1.2 Supermatrices of bilinear forms
    2. 6.2 Exceptional finite-dimensional simple Lie superalgebras
    3. 6.3 Finite-dimensional simple vectorial Lie superalgebras
    4. 6.4 The list of finite-dimensional simple Lie superalgebras considered naively
      1. 6.4.1 Exceptional simple Lie superalgebras: realizations
  7. 7 Description of deformations
    1. 7.3 General facts on cohomology of Lie algebras and Lie superalgebras
    2. 7.4 Lie superalgebras ${\mathfrak{g}}(A)$ with indecomposable Cartan matrix $A$
    3. 7.5 Lie superalgebras ${\mathfrak{g}}={\mathfrak{psq}}(n)$ for $n>2$
    4. 7.6 Lie superalgebras ${\mathfrak{g}}={\mathfrak{spe}}(n)$ for $n>3$
      1. 7.6.1 Induced and coinduced modules.
      2. 7.6.2 The long exact sequence.
      3. 7.6.3 Vectorial Lie superalgebras as coinduced modules.
    5. 7.7 What was known about deformations of simple vectorial Lie superalgebras
      1. 7.7.1 Filtered deformations.
    6. 7.8 Deformations of ${\mathfrak{h}}^{\prime}(0|n)$ ([Tyut, LSh3])
  8. 8 Summary of ideas in Kac’s proof ([K2])

##  1\. Introduction

In this paper, I consider two types of occurrences of odd parameters:

(i) in deformations of several simple Lie superalgebras;

(ii) in obstructions to splitness of complex-analytic supermanifolds.

All Lie superalgebras considered are finite-dimensional over ${\mathbb{C}}$,
unless otherwise specified; the characteristic $p$ of the field ${\mathbb{K}}$
can not be confused with parity also denoted by $p$.

###  1.1. Prologue: Four types of main characters, two questions

In the late 1960s, many mathematicians (first, in Moscow, and later in the
whole Universe, until the vogue changed) were discussing two topics:

(1) Kac’s classification of simple ${\mathbb{Z}}$-graded Lie algebras of
polynomial growth over ${\mathbb{C}}$ (under a technical assumption later
dismissed by O. Mathieu in a difficult paper [M]) consisting of finite-
dimensional examples (classified by Cartan and Killing), vectorial Lie
algebras with polynomial coefficients (classified by Cartan) and two new
ingredients: loop algebras with values in simple finite-dimensional algebras
and twisted versions of these loops, and the complexified Lie algebra of
vector fields on the circle, see [K];

(2) The Kostrikin-Shafarevich conjecture on classification of simple finite-
dimensional modular Lie algebras over algebraically closed fields of
characteristic $p>7$ (initially formulated only for restricted algebras); for
a review and update, see [BGLLS].

In both classifications, the ${\mathbb{Z}}$-graded infinite-dimensional simple
Lie algebras of vector fields over ${\mathbb{C}}$ play a prominent role.

It became more and more obvious with time that two digressions from the main
goal (classification of simple objects) are important: classification of (a)
deformations of ${\mathbb{Z}}$-graded Lie algebras and (b) Lie algebras of
derivations and central extensions of simple Lie algebras.

The deformation of the Lie algebra of Hamiltonian vector fields induced by
quantization of the Poisson Lie algebra leads out of the set of
${\mathbb{Z}}$-graded Lie algebras considered in topic (1), and was ignored by
mathematicians interested in classifications (1) and (2), whereas all other
examples of simple ${\mathbb{Z}}$-graded Lie algebras are rigid (from a
certain point of view; I will not digress into this mysterious area here; let
me just mention an ostensible contradiction between, on one hand, the correct,
under the definitions considered, proof of the fact that
$H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$ for the loop algebras
${\mathfrak{g}}$, see [LR], and, on the other hand, the explicit examples of
such deformations in [Go1, Ho2] later evolved into a rich theory of Krichever-
Novikov algebras, see a review [Shei]. There is no contradiction, actually,
because $H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$ does not guarantee rigidity,
which is lack of deformations. Observe also that
$H^{2}({\mathfrak{g}};{\mathfrak{g}})\neq 0$ does not imply that the deformed
algebra is not isomorphic to the initial algebra; this phenomenon is called
semi-trivial deformation, see [BGL1, BLLS] with many examples.)

Although in topic (2), i.e., in the non-restricted KSh-conjecture, the
deformations come to the foreground, they still did not draw as much attention
as I think they should have, except for the filtered deformations. (A
plausible, very probable, reason for this negligence: of the two types of
simple Lie algebras, the non-filtered deforms — results of deformations — of
the vectorial Lie algebras are isomorphic to the known simple Lie algebras,
whereas the Lie algebras of the other type — the ones with Cartan matrix (or
their simple relatives) are rigid — at least for $p>3$. The situation is
totally different if $p=2$, see Shchepochkina’s exceptional example — the
deform of the divergence-free algebra — in [BGLLS].)

Observe two more types of characters that came to the foreground in the 1980s:

(3) Lie superalgebras were known in topology and deformation theory under the
self-contradictory name “graded Lie algebras” since at least 1940, mainly over
finite fields or ${\mathbb{Z}}$ (in which case they are super Lie rings). From
the mid-1970s, Lie superalgebras started to draw attention of researchers:
first, in characteristic 0 thanks to applications to high energy physics and
solid body physics; next, in characteristic $p>0$ prompted by relation with
quantum groups $U_{q}({\mathfrak{g}})$ when $q$ is a $p$th root of unity and
${\mathfrak{g}}$ is a simple finite-dimensional Lie algebra over
${\mathbb{C}}$. (The old name is self-contradictory because Lie superalgebras
and Lie algebras (graded or not) satisfy different identities.)

Some physicists, more clairvoyant than other, began to perceive the usefulness
— in physics — of what is nowadays called “Lie superalgebras” even before the
notion “Lie superalgebra” was explicitly distinguished and separated from
“${\mathbb{Z}}/2$-graded Lie algebra”, see prefaces in [DSB]. Encouraged by
one such physicist, Kac gave examples of three series of simple vectorial Lie
superalgebras, see [K0] and more accessible Kac’s preface in [DSB].

(4) For decades the mathematicians interested in simple Lie (super)algebras
mostly ignored the examples of filtered simple Lie algebras which can not be
obtained as a result of deformation of a ${\mathbb{Z}}$-graded simple Lie
algebra. In physics, such Lie (super)algebras appeared in higher spin
theories, see [Vas, Vas1]; in mathemtics, they were used to interpret known
orthogonal polynomials, see [LSer, Ser]. Among such simple Lie
(super)algebras, the simplest to describe is ${\mathfrak{sl}}(\lambda)$, where
$\lambda\in{\mathbb{C}}{\mathbb{P}}^{1}$, the Lie algebra of traceless
“matrices of complex size $\lambda\in{\mathbb{C}}{\mathbb{P}}^{1}$”.

Let $\mathfrak{g}=\mathfrak{sl}(2)$, and $c_{1}=\lambda^{2}-1$, where
$\lambda=\mu+1\in(\mathbb{C}\setminus\mathbb{Z})\cup\\{0\\}$ for the highest
weight $\mu$ of an irreducible $\mathfrak{sl}(2)$-module. Let $C_{1}$ be the
quadratic Casimir operator. The associative algebra
$U_{\lambda}:=U(\mathfrak{g})/(C_{1}-c_{1})$ has an ideal

$I_{\lambda}=\begin{cases}0&\text{if
$\lambda\not\in\mathbb{Z}\setminus\\{0\\}$}\\\ \text{of finite
codimension}&\text{otherwise}.\\\ \end{cases}$

Let $A$ be any associative algebra, let $A^{L}$ be the Lie algebra whose space
is $A$ but multiplication is given by the commutator $[a,b]:=ab-ba$. If
$\lambda=n\in\mathbb{Z}\setminus\\{0\\}$, then
$U_{n}/I_{n}\simeq\text{Mat}(|n|)$, hence the name
${\mathfrak{gl}}(\lambda):=(U_{\lambda}/I_{\lambda})^{L}$ for any
$\lambda\in\mathbb{C}$, see [Fei].

While the trace can be normalized at will, J. Bernstein suggested a definition
of the trace on ${\mathfrak{gl}}(\lambda)$ such that
$\mathop{\mathrm{tr}}\nolimits(1)=\lambda$ for
$\lambda\in(\mathbb{C}\setminus\mathbb{Z})\cup\\{0\\}$, see [KhM], resembling
$\mathop{\mathrm{tr}}\nolimits(1_{|n|})=|n|$ for
$n\in{\mathbb{Z}}\setminus\\{0\\}$. The traceless elements of
${\mathfrak{gl}}(\lambda)$ constitute ${\mathfrak{sl}}(\lambda)$.

If $\lambda\not\in\mathbb{Z}\setminus\\{0\\}$, then $U_{\lambda}$ and its
limit as $\lambda\to\infty\in{\mathbb{C}}{\mathbb{P}}^{1}$, see [GL1], are
central simple as well as $U_{n}/I_{n}$ for $n\in\mathbb{Z}\setminus\\{0\\}$.
According to a Herstein–Montgomery theorem, see [Mon], for any central simple
associative algebra $A$ with center $Z$, the Lie algebra
$L(A):=(A^{L})^{(1)}/((A^{L})^{(1)}\cap Z)$ is simple.

The Lie algebras $(U_{\lambda})^{L}$ are “quantized” versions of the Lie
algebras of functions on the orbits of the co-adjoint representation of simple
finite-dimensional Lie algebras over $\mathbb{C}$, see [Kon], considered with
the Poisson bracket. The natural filtration of $U({\mathfrak{sl}}(2))$ induces
a filtration in $(U_{\lambda})^{L}$; the associated graded Lie algebra is
isomorphic for $\lambda\neq 0$ to the Lie algebra
${\mathfrak{po}}(2)_{\bar{0}}$ realized on the space of even degree
polynomials in 2 indeterminates with respect to the Poisson bracket, see
[HKP]; the elements of non-negative degree in the Lie grading of
${\mathfrak{po}}(2)_{\bar{0}}$ form a Lie algebra with codimension-3 radical
complementing ${\mathfrak{sl}}(2)$.

The generalizations of ${\mathfrak{sl}}(\lambda)$ are defined as follows. Let
$\mathfrak{g}$ be any simple finite-dimensional Lie algebra over $\mathbb{C}$,
let $\rho$ be the half-sum of positive roots of $\mathfrak{g}$, let $\mu$ be
the highest weight of an irreducible $\mathfrak{g}$-module with highest weight
vector $v^{\mu}$, let the $C_{i}$ be the Casimirs, i.e., the generators of the
center of $U(\mathfrak{g})$. Set (the shift by $\rho$ is need to get the
$c_{i}$ symmetric relative the Weyl group)

(1)
$U_{c}:=U(\mathfrak{g})/(C_{i}-c_{i})_{i=1}^{\text{rk}\mathfrak{g}},\text{~{}~{}where
$c_{i}=C_{i}|_{v^{\mu+\rho}}$}.$

For almost all values of $c=(c_{1},\dots,c_{\text{rk}\mathfrak{g}})$, the
algebra $U_{c}$ is central simple, so $L(U_{c})$ is a simple Lie algebra, see
[Lq], where Lie superalgebras ${\mathfrak{g}}$ are also considered.

Question 1: What are the deformations of the simple ${\mathbb{Z}}$-graded Lie
superalgebras (finite-dimensional and of polynomial growth) that can not be
captured by classification of equivalence classes of the tensors the algebra
preserves, cf. [BGLLS, Subsection 24.2]? In § 7, I will show that no simple
finite-dimensional vectorial simple Lie superalgebra over ${\mathbb{C}}$ has
such deformations.

Contrariwise, the deforms of ${\mathfrak{osp}}(4|2)$ provide us with examples
of Lie algebras that can not be described by equivalence classes of the
bilinear forms preserved by certain members of the parametric family of these
algebras.

Question 2: Over ${\mathbb{C}}$, is there a simple finite-dimensional Lie
(super)algebra whose associated ${\mathbb{Z}}$-graded algebra has no negative
part, like ${\mathfrak{sl}}(\lambda)$? For the answer, see § 8 (no such Lie
superalgebras). The answer for the simple finite-dimensional Lie algebras over
fields of characteristic $>3$ is the same. The same question for Lie
superalgebras over fields of positive characteristic and Lie algebras in
characteristics 3 and 2 is open.

###  1.2. Disclamer

It was clear from the very beginning that certain “relatives” of some of the
simple Lie (super)algebras ${\mathfrak{g}}$, such as Lie (super)algebras of
derivations and central extensions of ${\mathfrak{g}}$, are more (at least, no
less) interesting in applications than ${\mathfrak{g}}$ itself: e.g., affine
Kac–Moody (super)algebras vs. loop (super)algebras and their twisted versions,
Poisson Lie (super)algebras vs. Hamiltonian Lie (super)algebras. (Observe that
the affine Kac–Moody algebras are most known examples of recently
distinguished notion of double extensions, this notion simultaneously requires
a derivation, a central extension, and a third ingredient — a non-degenerate
${\mathfrak{g}}$-invariant symmetric bilinear form on ${\mathfrak{g}}$, see
[BLS] and references therein.)

Until now, the vectorial Lie superalgebras (same as vectorial Lie algebras
over fields of positive characteristic) and several other types of Lie
(super)algebras are often described as if the author of the description is
obsessed with simple Lie (super)algebras. This approach leads to the notation
with a wrong emphasis that manifestly has to be improved: the natural and
basic object is the Cartan prolong (the result of the Cartan prolongation)
rediscovered by Tanaka and generalized by Shchepochkina (see [Shch, BGLLS]),
whereas the simple object is the derived algebra or a quotient of the prolong.
Similar arguments identify ${\mathfrak{g}}(A)$, ${\mathfrak{pe}}(n)$,
${\mathfrak{q}}(n)$, ${\mathfrak{po}}(0|n)$ as the protagonists, more natural
than their simple subquotients.

However, the simple Lie (super)algebras are the first to be classified, of
course.

###  1.3. Simple Lie superalgebras. First definitions and examples

First examples of Lie superalgebras, actually Lie super rings over
${\mathbb{Z}}$, appeared in 1941, in topology as the sets of homotopy groups
with the Whitehead product, see [Wh]. Associated with these examples were
modular (i.e., over ${\mathbb{K}}$ of characteristic $p>0$) Lie superalgebras
over finite fields. Similarity of Lie superalgebras and modular Lie algebras
is so striking that sometimes one hears and reads that “when $p=2$, there is
no difference between Lie algebras and Lie superalgebras”, which is not true
as is lucidly explained in [BGLLS].

Weibel writes “Homological algebra had its origins in the 19-th century, via
the work of Riemann (1857) and Betti (1871) on “homology numbers”, and the
rigorous development of the notion of homology numbers by Poincaré in 1895. An
observation of Emmy Noether in 1925 shifted the attention to the “homology
groups” of a space, and algebraic techniques were developed for computational
purposes in the 1930’s. Yet, homology remained a part of the realm of topology
until about 1945.”, see [We, pp. 797–836].

The Lie superalgebras or super rings that thus appeared in topology over
${\mathbb{Z}}$ or finite fields were nilpotent (hence, “not very interesting”
and nobody tried to find out any simple (without ideals) examples), and did
not draw much attention until it was discovered that the representations of
the quantum groups $U_{q}({\mathfrak{g}})$ when $q$ is the $p$th root of unit
is related with the representation of the characteristic $p$ version of the
simple complex Lie algebra ${\mathfrak{g}}$ and a certain Lie superalgebra,
see, e.g., [Vi, ChCh, A] and references therein. By that time simple finite-
dimensional Lie superalgebras over ${\mathbb{C}}$ were already widely known.

However, even now, the study of modular Lie (super)algebras, even simple ones,
is not a “trendy” topic, despite very interesting recent advances, see, e.g.,
[Kan] and references therein.

For the steps of classification of simple Lie superalgebras considered
naively, see the first works with examples, starting with [K0], and
classification in particular cases [Kapp, Kap], [SNR] and [Ho, Dj1, Dj2, Dj3,
DjH]. For a classification of simple ${\mathbb{Z}}$-graded finite-dimensional
Lie superalgebras together with rounding up of all (in particular, vectorial)
finite-dimensional cases, see [K1, K1C, K1.5] culminated in [K2]. In § 8, I
give a summary of the ideas in the proof given in [K2]; I hope this summary is
more helpful than [Sch]. In these first papers, in the book [Sch], and in many
works published later, Lie superalgebras ${\mathfrak{g}}$ were, and often
still are, considered as pairs
$({\mathfrak{g}}_{\bar{0}},{\mathfrak{g}}_{\bar{1}})$ consisting of the Lie
algebra ${\mathfrak{g}}_{\bar{0}}$ and ${\mathfrak{g}}_{\bar{0}}$-module
${\mathfrak{g}}_{\bar{1}}$ such that the multiplication
$S^{2}({\mathfrak{g}}_{\bar{1}})\longrightarrow{\mathfrak{g}}_{\bar{0}}$
satisfies the super Jacobi identity. Interpretations of each Lie superalgebra
as preserving “something” (realizations) are more insightful; they help to
understand both the structure of the superalgebra, its representations, and
this “something”. (Observe that such an interpretation of several important
Lie (super)algebras — e.g., of affine Kac-Moody (super)algebras and of non-
trivial central extensions of simple stringy Lie (super)algebras — is lacking
to this day.)

###  1.4. Realizations

Several of simple Lie superalgebras are realized as linear algebras preserving
a tensor, see [BGL, BGLL1], or a distribution. That is precisely how — as
preserving a distribution — Cartan and Killing used to describe not only all 5
exceptional simple Lie algebras, but also ${\mathfrak{sl}}(n)$,
${\mathfrak{sp}}(2n)$, and ${\mathfrak{o}}(n)$, at the time when description
in terms of roots and Dynkin diagrams was not yet discovered. For usefulness
of such a realization of certain simple Lie algebras in characteristic 3, see
[GL3].

Each of the serial finite-dimensional simple Lie algebras has two super
analogs: a direct (simple-minded) one and one of a purely super nature. None
of the exceptional simple Lie algebras over ${\mathbb{C}}$ has a super analog,
at least over ${\mathbb{C}}$. For various interesting interpretations of the
exceptional complex simple Lie superalgebras, see [Sud]; by way of considering
their versions in characteristic 3, see [Eld, Eldi]. I particularly like
Elduque’s Super Magic Super Square (rectangle, actually): Elduque’s approach
indicates a road not fully explored yet (cf. [Kan]) and shows the important
role the “incarnations” of exceptional simple Lie superalgebras over fields of
positive characteristic play in interpretation of their namesakes over
${\mathbb{C}}$. This Super Square also describes new examples indigenous to
positive characteristic, see [Eldi], where several examples from [BGL] not
directly entering the Super Square are further interpreted, whereas [BGL]
contains classification of finite-dimensional Lie superalgebras with
indecomposable Cartan matrix.

The infinite-dimensional Lie algebras of vector fields with polynomial (or
formal power series) coefficients have finite-dimensional analogs both over
fields of positive characteristic and among Lie superalgebras (over any
fields), see [BGLLS]. For the steps of classification of the former, see [K2]
with numerous corrections, the latest few being [CaKa1, CaKa2, CaKa4]; for
another approach, see [Sh5, Sh14, LSh1, LSh2, LSh3, LSh5] and references
therein.

###  1.5. Cartan matrices

Kac observed, see [K1.5, K2], that several of simple Lie superalgebras
${\mathfrak{g}}$ can be realized as ${\mathfrak{g}}={\mathfrak{g}}(A)$, where
$A$ is an analog of Cartan matrix he defined. Kac mistakenly claimed that
${\mathfrak{psl}}(n|n)$ also had Cartan matrix, whereas it is its double
extension ${\mathfrak{gl}}(n|n)$ that has Cartan matrix. For correct
definitions of Cartan matrices of a Lie superalgebra, analogs of the Dynkin
diagrams, and roots, see [BGL, CCLL, BLLoS]; for a review of double
extensions, see [BLS], especially its two last sections.

###  1.6. Presentations

For defining relations (replacing Serre relations) between analogs of the
Chevalley generators of ${\mathfrak{g}}(A)$, see [GL2] and — for comparison —
[Zh] where, same as in [GL2], non-Serre relations are also given, despite the
title of [Zh].

There are also other presentations of ${\mathfrak{g}}(A)$, e.g., in terms of a
pair (triple, actually) of Jacobson generators, see [GL1], where all simple
finite-dimensional Lie algebras are considered. Not every simple finite-
dimensional Lie superalgebra with Cartan matrix can be similarly described,
see [LSS, GL5].

For presentations of simple vectorial Lie superalgebras needed in this paper,
see [GLP]. Sixteen years after [GLP], presentations of 2 of the 4 series of
simple finite-dimensional vectorial Lie superalgebras were given in [CCP]; the
form of these latter presentations, though interesting, is not useful for this
paper.

###  1.7. Deformations with odd parameters

Some of simple Lie superalgebras — over fields of any characteristic — have
deformations thus resembling simple modular Lie algebras.

Here, I give a complete classification of deformations of simple finite-
dimensional Lie superalgebras over ${\mathbb{C}}$, with proofs for the first
time. In order to do this, I need to introduce the adequate definition of Lie
superalgebra and deformations with odd parameters. Help of A. Lebedev in
clarifying these notions was invaluable.

###  1.8. Other appearances of odd parameters

Two instances of odd parameters were observed earlier, see [L3]. More
accessible sources in English where these instances are described are the
following ones:

$\bullet$ A. A. Kirillov observed a supersymmetry between solutions of the
Korteweg–de Vries and Schrödinger equations, but missed the odd parameters;
for these parameters, a generalization of Kirillov’s construction to some of
the $N$-extended stringy superalgebras and the restriction on $N$, see [L9].

$\bullet$ In [MaG], Manin defined the connection in terms of the differential
forms, which is dual to the classical definition. This definition of
connections on supermanifolds naturally leads to odd parameters, see [L10].

$\bullet$ One more instance of odd parameters was discovered by Shander who
expounded few pages on integration in [L3] to 80 pages in [L5, no. 31/1988-14,
45–131 pp], see his Chapter “Integration” [LSoS, Ch.6]. In particular, in
1986, he defined cochains dual to chains and hence bigraded by pairs of
numbers; here the odd parameters are needed. Shander also hinted at a way to
define an analog of the Stokes’ formula with “over-supermanifold” of
codimension $(0|-1)$, see [LSoS, Ch.6]. A similar formula (with the
$(0|1)$-dimensional superline as the boundary of the point) was suggested by
Palamodov, see [PalI].

$\bullet$ One more instance of odd parameter is given on the group
$\mathop{\mathrm{{GQ}}}\nolimits(n;{\mathcal{C}})$ of invertible matrices of
the form $\begin{pmatrix}A&B\\\ B&A\end{pmatrix}$, where
$A\in{\mathfrak{gl}}(n;{\mathcal{C}}_{\bar{0}})$ and
$B\in{\mathfrak{gl}}(n;{\mathcal{C}}_{\bar{1}})$, and where ${\mathcal{C}}$ is
a supercommutative superalgebra, by the queer determinant, a group
homomorphism
$\mathop{\mathrm{{GQ}}}\nolimits(n;{\mathcal{C}})\longrightarrow\mathop{\mathrm{{GL}}}\nolimits(1;{\mathcal{C}}[\tau])$,
defined by the formula

$\mathop{\mathrm{qet}}\nolimits(A,B):=1+\tau\sum_{i\geq
1}\frac{\mathop{\mathrm{tr}}\nolimits((A^{-1}B)^{2i-1})}{2i-1}\in{\mathcal{C}}[\tau],\text{where
$p(\tau)={\bar{1}}$ and $\tau^{2}=0$}.$

In [Sha1], Shander described, using odd parameters, invariant functions on
supermatrices of the type never considered before (and, regrettably, never
after — to this day).

$\bullet$ The papers of Bettadapura, see [BK1, BK2] also address odd
parameters of deformations. The claims of his texts are, however, misleading,
as I’ll show in Subsection 5.3.

##  2\. What “Lie superalgebra” is, naively

The elements of ${\mathbb{Z}}/2$ will be denoted by ${\bar{0}}$ and
${\bar{1}}$ to distinguish them from elements of ${\mathbb{Z}}$. The
${\mathbb{Z}}/2$-graded space $V=V_{\bar{0}}\oplus V_{\bar{1}}$ is called a
superspace. The parity of any non-zero element $v\in V_{i}$ is denoted by
$p(v)=i\in{\mathbb{Z}}/2$. Superizing Milnor’s $K_{0}$-functor, we say that
the superdimension of $V$ is $\mathop{\mathrm{sdim}}\nolimits V=\dim
V_{\bar{0}}+\varepsilon\dim V_{\bar{1}}$, where $\varepsilon^{2}=1$. This is
briefly denoted $a|b$, where $a=\dim V_{\bar{0}}$ and $b=\dim V_{\bar{1}}$.

A superalgebra is any ${\mathbb{Z}}/2$-graded algebra. In the conventional
theory, the morphisms of superalgebras must preserve parity, i.e., only even
homomorphisms are morphisms of superalgebras; however, see Subsection 3.2.

###  2.1. The Sign Rule, skew and anti

The definition of Lie superalgebra for any characteristic $p\neq 2$ or 3 is
obtained from the definition of the Lie algebra using the Sign Rule “if
something of parity $a$ is moved past something of parity $b$, the sign
$(-1)^{ab}$ accrues; formulas defined on homogeneous elements are extended to
all elements via linearity”.

In this note, all supercommutative superalgebras are supposed to be
associative with $1$; their morphisms should send $1$ to $1$; the morphisms of
all superalgebras (supercommutative and Lie) must preserve parity. (Observe
that the requirement to preserve parity, currently universally accepted, is
not required if the superalgebras are considered as “abstract”, i.e., as given
by generators and relations that do not involve parity, cf. Subsection 3.2.)

Since commutative algebra and supercommutative superalgebra can appear in one
sentence, I never drop the prefix “super” as is often done in futile attempt
to save space, but successfully causing confusion.

Observe that sometimes applying the Sign Rule requires some dexterity. For
example, we have to distinguish between two versions both of which turn in the
nonsuper case into one, called either skew- or anti-commutativity, see [Gr]:

$\begin{array}[]{ll}ba=(-1)^{p(b)p(a)}ab&(\text{super commutativity})\\\
ba=-(-1)^{p(b)p(a)}ab&(\text{super anti-commutativity})\\\
ba=(-1)^{(p(b)+1)(p(a)+1)}ab&(\text{super skew-commutativity})\\\
ba=-(-1)^{(p(b)+1)(p(a)+1)}ab&(\text{super antiskew-
commutativity})\end{array}$

The skew formulas are those that can be “straightened” by the change of parity
of the space on which the structure is considered, whereas the prefix anti
requires an overall minus sign regardless of parity. The super commutator of
two elements is denoted by the same symbol as the commutator; the first
derived superalgebra of the Lie superalgebra ${\mathfrak{g}}$ is denoted by
${\mathfrak{g}}^{\prime}$.

An example not immediately seen as a result of application of the Sign Rule is
the trace, or supertrace for emphasis, which is any linear functional on
${\mathfrak{g}}$ that vanishes on ${\mathfrak{g}}^{\prime}$. For its correct
formula on odd supermatrices in
${\mathfrak{gl}}(m|n;\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n))$ with elements in the Grassmann superalgebra
$\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(n)$, see (11), (12).

A Lie superalgebra understood naively is any superalgebra satisfying the anti-
commutativity and Jacobi identities amended by the Sign Rule. (NB: any
${\mathbb{Z}}/2$-graded Lie algebra is a superalgebra according to this
definition, but it is not a “Lie” superalgebra because it satisfies the anti-
commutativity and Jacobi identities NOT amended by the Sign Rule and squaring
of odd elements is not defined.)

The supercommutative superalgebra of “functions” on the $0|n$-dimensional
supermanifold (which is just a superpoint in this situation) is the Grassmann
algebra $\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(n)$ on $n$
anticommuting generators, call them $\xi_{i}$. We consider
$\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n):={\mathbb{C}}[\xi]$, where
$\xi=(\xi_{1},\dots,\xi_{n})$, as a superalgebra by setting
$p(\xi_{i})={\bar{1}}$ for all $i$; clearly, $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$ is supercommutative.

##  3\. What “Lie superalgebra” is from categorical point of view ([KLLS,
KLLS1])

###  3.1. Morphisms of supervarieties

Over ${\mathbb{R}}$. Recall that the smooth supermanifolds are ringed
spaces,111I recommend [MaAG] as the most clear and shortest source from which
to understand the definition of ringed spaces. i.e., pairs
${\mathcal{M}}:=(M,{\mathcal{O}}_{\mathcal{M}})$, where $M$ is an
$m$-dimensional manifold called the underlying manifold of ${\mathcal{M}}$,
and ${\mathcal{O}}_{\mathcal{M}}$ is the structure sheaf of ${\mathcal{M}}$.
In the same way as the manifold $M$ is glued from coordinate patches locally
diffeomorphic to a ball in ${\mathbb{R}}^{m}$, the space of sections of
${\mathcal{O}}_{\mathcal{M}}$ over the domain $U$ in $M$ is isomorphic to
${C^{\infty}(U)\otimes\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)}$. Pairs
${\mathcal{U}}:=(U,{\mathcal{O}}_{U}\otimes\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n))$, where ${\mathcal{O}}_{U}(U):=C^{\infty}(U)$, are
called smooth superdomains, see, e.g., [Del, p.65].

If the ground field is ${\mathbb{C}}$ and
${\mathcal{O}}_{U}(U):={\mathcal{A}}(U)$ is the algebra of complex-analytic
functions, then pairs
${\mathcal{U}}:=(U,{\mathcal{O}}_{U}\otimes\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n))$ are called analytic superdomains that glue into
an almost complex supermanifold, see Subsection 5.1.

We say that the superdimension of ${\mathcal{M}}$ is
$\mathop{\mathrm{sdim}}\nolimits{\mathcal{M}}=m|n$. A morphism of
supermanifolds ${{\mathcal{M}}\longrightarrow{\mathcal{N}}}$ is any pair
$(\varphi,\varphi^{*})$, where $\varphi:M\longrightarrow N$ is a
diffeomorphism and
$\varphi^{*}:{\mathcal{O}}_{\mathcal{N}}\longrightarrow{\mathcal{O}}_{\mathcal{M}}$
is a morphism of sheaves of superalgebras (conventionally, preserving the
natural parity of the structure sheaves, cf. Subsection 3.2). Observe that,
unlike the case of manifolds, the morphism $\varphi^{*}$ is not recovered from
$\varphi$ and this is precisely what makes the supermanifold theory rich.

Over any ground field ${\mathbb{K}}$. Consider a superdomain ${\mathcal{U}}$
of superdimension $0|n$. Unlike superdomains of superdimension $a|b$ with
$a\neq 0$, we can consider ${\mathcal{U}}$ over any ground field
${\mathbb{K}}$ and call it superpoint. The underlying domain of
${\mathcal{U}}$ is a single point $\ast$. Since
${\mathcal{O}}_{\mathcal{U}}(\ast)=\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$, the superpoint ${\mathcal{U}}$ has a lot of
nontrivial automorphisms, namely the group
$\mathop{\mathrm{{Aut\/}}}\nolimits^{{\bar{0}}}\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$ of parity preserving automorphisms of
$\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(n)$ (or — in the
category to be studied — even larger group
$\mathop{\mathrm{{Aut\/}}}\nolimits\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$ of all automorphisms, cf. Subsection 3.2). Let the
$\xi_{j}$ for $1\leq j\leq n$ be generators of $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$. Any automorphism in
$\mathop{\mathrm{{Aut\/}}}\nolimits^{{\bar{0}}}\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$ is of the form

(2) $\xi_{j}\;\mapsto\;\sum_{r}\varphi_{j}^{r}\xi_{r}+\sum_{s\geq
1}\;\sum_{j_{1}<\dots<j_{2s+1}}\varphi_{j}^{j_{1}\dots
j_{2s+1}}\xi_{j_{1}}\dots\xi_{j_{2s+1}},$

where the matrix $(\varphi_{j}^{r})$ with elements in ${\mathbb{K}}$ is
invertible. We see that such automorphisms constitute an algebraic group (or a
Lie group if $\mathop{\text{\rm char}}\nolimits{\mathbb{K}}=0$) whose Lie
algebra consists of the even elements of the Lie superalgebra
${\mathfrak{vect}}(0|n):={\mathfrak{der}}\,\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$.

What corresponds to the odd vector fields from
${\mathfrak{der}}\,\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(n)$?
Let me give the answer in the following more general setting, the one
involving both even and odd indeterminates, but only for
${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{C}}$. In the absence of even
indeterminates, i.e., for superpoints, the formulas (3) and (4) are meaningful
for any ${\mathbb{K}}$.

Let $E$ be the trivial vector bundle over a domain $U$ of dimension $m$ with
fiber $V$ of dimension $n$; let $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(E)$ be the exterior algebra of $E$. To the bundle $E$,
we assign the superdomain ${\mathcal{U}}=(U,C^{\infty}({\mathcal{U}}))$, where
$C^{\infty}({\mathcal{U}})$ is the superalgebra of smooth sections of
$\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(E)$. Clearly, each
automorphism of the pair $(U,\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(E))$, i.e., of the vector bundle
$\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(E)$, induces an
automorphism of the superdomain ${\mathcal{U}}$.

However, not all automorphisms of the superdomain ${\mathcal{U}}$ are obtained
in this way. By definition, every morphism of superdomains
$(\varphi,\varphi^{*})\colon{\mathcal{U}}\longrightarrow{\mathcal{V}}$ is in
one-to-one correspondence with a homomorphism of the superalgebras of
functions $\varphi^{*}\colon C^{\infty}({\mathcal{V}})\longrightarrow
C^{\infty}({\mathcal{U}})$ since $\varphi$ is determined by $\varphi^{*}$.

Every homomorphism $\varphi^{*}$ is defined on the (topological222A
topological algebra $A$ over a topological field ${\mathbb{K}}$ is a
topological vector space together with a bilinear multiplication $A\times
A\longrightarrow A$, continuous in a certain sense, and such that $A$ is an
algebra over ${\mathbb{K}}$. Usually the continuity of the multiplication
means that the multiplication is continuous as a map between topological
spaces ${A\times A\longrightarrow A}$. A set $S$ is a generating set of a
topological algebra $A$ if the smallest closed subalgebra of $A$ containing
$S$ is $A$.) generators of the superalgebra, in other words: on “coordinates”
(the even ones $u:=(u_{1},\dots,u_{m})$ and the ones
$\xi:=(\xi_{1},\dots,\xi_{n})$). Consider the corresponding formulas

(3) $\left\\{\begin{aligned}
\varphi^{*}(u_{i})\;&=\;\varphi^{0}_{i}(u)+\framebox{$\sum\limits_{r\geq
1}\;\sum\limits_{i_{1}<\dots<i_{2r}}\varphi_{i}^{i_{1}\dots
i_{2r}}(u)\xi_{i_{1}}\dots\xi_{i_{2r}}$}\text{~{}~{}for all $i$},\\\
\varphi^{*}(\xi_{j})\;&=\;\sum_{r\geq
0}\;\sum_{j_{1}<\dots<j_{2r+1}}\psi_{j}^{j_{1}\dots
j_{2r+1}}(u)\xi_{j_{1}}\dots\xi_{j_{2r+1}}\text{~{}~{}for all
$j$}.\end{aligned}\right.$

A) The terms $\varphi^{*}(u_{i})=\varphi^{0}_{i}(u)$ for all $i$ determine an
endomorphism of the underlying domain $U$.

B) The linear terms $\varphi^{*}(\xi_{j})=\sum_{i}\psi_{j}^{i}(u)\xi_{i}$ for
all $j$ determine endomorphisms of the fiber $V$ (over each point its own
fiber, as the dependence on $u$ shows).

C) The terms of higher degree in $\xi$ in the right-hand side of the
expression of $\varphi^{*}(\xi_{j})$ in eq. (3) determine an endomorphism of
the larger fiber — the Grassmann superalgebra $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(V)$.

The endomorphisms A)–C) existed in Differential Geometry, and no need to
introduce a flashy prefix “super” was felt.

The difference between the vector bundle $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(E)$ and the superdomain ${\mathcal{U}}$ is most easily
understood when the reader looks at the boxed terms in (3). These terms,
meaningless in the conventional Differential Geometry (on manifolds), make
sense in the new paradigm:

In the category of superdomains there are more morphisms than in the category
of vector bundles: morphisms with non-vanishing boxed terms in (3) are exactly
the additional ones.

However, even the boxed terms in eq. (3) is not all we get in the new setting:
we still did not describe any of odd parameters of endomorphisms. To account
for the odd parameters, we have to consider the functor from the category of
supercommutative superalgebras to the category of groups
$C\longmapsto\mathop{\mathrm{{Aut\/}}}\nolimits_{C}^{{\bar{0}}}(C^{\infty}({\mathcal{U}})\otimes
C)$, i.e., the parity preserving $C$-linear automorphisms of the form

(4)
$\left\\{\begin{array}[]{ll}\varphi^{*}(u_{i})&=\;\varphi_{i}(u,\xi):=\varphi^{0}_{i}(u)+\framebox{$\sum\limits_{r\geq
1}\;\sum\limits_{i_{1}<\dots<i_{r}}\varphi_{i}^{i_{1}\dots
i_{r}}(u)\xi_{i_{1}}\dots\xi_{i_{r}}$}\text{~{}~{}for all $i$},\\\
\varphi^{*}(\xi_{j})&=\;\psi_{j}(u,\xi):=\sum\limits_{r\geq
0}\;\sum\limits_{j_{1}<\dots<j_{2r+1}}\psi_{j}^{j_{1}\dots
j_{2r+1}}(u)\xi_{j_{1}}\dots\xi_{j_{2r+1}}\\\
&+\framebox{$\psi^{0}_{j}(u)+\sum\limits_{r\geq
1}\;\sum\limits_{j_{1}<\dots<j_{2r}}\psi_{j}^{j_{1}\dots
j_{2r}}(u)\xi_{j_{1}}\dots\xi_{j_{2r}}$}\text{~{}~{}for all $j$},\\\
\end{array}\right.$

where for all $r$ the even and the odd parameters are, respectively,

$\begin{array}[]{l}\varphi^{0}_{i}(u),\varphi_{i}^{i_{1}\dots i_{2r}}(u),\
\psi_{j}^{j_{1}\dots j_{2r+1}}(u)\in C_{\bar{0}},\\\
\psi^{0}_{j}(u),\psi_{j}^{j_{1}\dots j_{2r}}(u),\ \varphi_{i}^{i_{1}\dots
i_{2r+1}}(u)\in C_{\bar{1}}.\end{array}$

These are parameters of the infinite-dimensional supergroup of automorphisms
of $C^{\infty}({\mathcal{U}})$ or, equivalently, of diffeomorphisms of
${\mathcal{U}}$; infinitesimally: of Lie superalgebra
${\mathfrak{vect}}(m|n)$.

###  3.2. On automorphisms that do not preserve parity

It took a while to acknowledge the fact that there are automorphisms of the
Grassmann algebra ${\mathbb{C}}[\xi]$, considered as an associative algebra
(and more generally, of ${\mathbb{C}}[x,\xi]$, where $x=(x_{1},\dots,x_{m})$
and $\xi=(\xi_{1},\dots,\xi_{n})$) not preserving parity, compare [B, Pa] with
[Dj4, LSe1, LSe2, Ba]. The meaning of such more general than (4) automorphisms
is unclear at the moment; most of researchers ignore them as “lacking physical
meaning”. The situation with arbitrary automorphisms and inhomogeneous
subalgebras reminds me the first cautious greetings of supersymmetries
(through 1970s till the end of 1980s, some researchers, more outspoken than
other, even branded them “useless to physics”, e.g., see a reference in [BM]).
The inhomogeneous with respect to parity subalgebras of supercommutative
superalgebras are metabelean, i.e., satisfying the identity

$[a,[b,c]]=0\text{~{}~{}for any its elements $a,b,c$, where $[a,b]:=ab-ba$}.$

A theorem by Volichenko proves that any metabelean algebra has a universal
enveloping supercommutative superalgebra. Recently, U. Iyer proved that
Volichenko algebras, defined as the inhomogeneous (relative parity)
subalgebras of Lie superalgebras, play the role of Lie algebras for the groups
of $C$-points of such general automorphisms, see [I]. Both metabelean and
Volichenko algebras are not ${\mathbb{Z}}/2$-graded, but $2$-step filtered
which is a very natural generalization of ${\mathbb{Z}}/2$-gradedness. Observe
that Volichenko’s attempt to describe what we now call Volichenko algebras by
a set of polynomial identities failed; to find the full set of such identities
is an Open problem to this day, see [BrP].

###  3.3. Supervarieties and superschemes (after [L0])

Over the ground field ${\mathbb{R}}$ or ${\mathbb{C}}$, let $E$ be a vector
bundle over $M$ with fiber $V$. Let $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(V)$ be the Grassmann superalgebra on $V$, let
$U\subset M$ be an open domain.

A supermanifold is a ringed space
${\mathcal{M}}=(M,{\mathcal{O}}_{\mathcal{M}})$, where $M$ is a manifold, and
the sheaf ${\mathcal{O}}_{\mathcal{M}}$ is locally isomorphic to
${\mathcal{O}}_{U}\otimes\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(V)$. A supermanifold isomorphic to the ringed space
whose structure sheaf is the sheaf of sections of the vector bundle
$\Lambda^{\mathbin{\hbox{\raise 1.20552pt\hbox{\bf.}}}}(E)$ with fiber $V$
over $M$ is called split.

A supervariety is a ringed space
${\mathcal{M}}=(M,{\mathcal{O}}_{\mathcal{M}})$, where $M$ is a variety (i.e.,
not necessarily smooth), and the sheaf ${\mathcal{O}}_{\mathcal{M}}$ is
locally isomorphic to a quotient of
${\mathcal{O}}_{U}\otimes\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(V)$.

Observe that every object in the category of smooth supermanifolds is split
(for a clear proof, more instructive than the first publications [Ga, Bat],
see [MaG, Subsection 4.1.3]), so there is a one–to–one correspondence between
the set of objects in the category of vector bundles over supermanifolds and
the set of objects in the category of smooth supermanifolds. The latter
category has, however, many more morphisms than the former, see eq. (4), to
say nothing about even more general automorphisms in the category considered
in Subsection 3.2.

A purely algebraic version of the supermanifold over any field (or any
commutative ring) of any characteristic is an affine superscheme
$\mathop{\mathrm{Spec}}\nolimits C$, where $C$ is a supercommutative
superalgebra or a superring. The affine superscheme is defined literally as
the affine scheme: its points are prime ideals defined literally as in the
commutative case, i.e., ${\mathfrak{p}}\subsetneq C$ is prime333K. Coulembier
pointed out to me that the so far conventional definitions in the non-
commutative case are at variance with the commutative case and common sense:
at the moment, if (5) holds, ${\mathfrak{p}}$ is called (say, in Wikipedia)
completely prime while it would be natural to retain the term prime, as is
done in [L0] and by J. Bernstein, P. Deligne et al in [Del], since the
definition is the same as in the commutative case despite the fact that
supercommutative rings are not commutative, whereas the term prime is
(currently) applied to any ideal $P\subsetneq R$ of the non-commutative ring
$R$ which for any two ideals $A$ and $B$ in $R$ satisfies the following
version of condition (5):  $\text{if $AB\subset P$, then either $A\subset P$
or $B\subset P$}.$

(5) $\text{if $a,b\in C$ and $ab\in{\mathfrak{p}}$, then either
$a\in{\mathfrak{p}}$ or $b\in{\mathfrak{p}}$}.$

The space of any affine scheme is endowed with Zariski topology and the
structure sheaf, defined as in the commutative case, see [MaAG], whose 1968
edition was the source of inspiration for [L0].

There is just one subtlety: the localization of the superalgebra (or
superring) $C$ at the prime ideal ${\mathfrak{p}}$ should be performed with
respect to the multiplicative system
$S_{\mathfrak{p}}:=C\setminus{\mathfrak{p}}$ and it is not obvious what should
we consider — in order to have well-defined fractions with non-homogeneous
denominators $b$ — only left fractions $b^{-1}a$ or only right fractions
$ab^{-1}$ or the equivalence (equality) of fractions does not depend on the
choice (left or right). Mathematically, a 1-line-proof of the boldfaced
statement is the only non-trivial place in the superization of Grothendieck’s
schemes (succinctly described in [MaAG]) to superschemes.

###  3.4. The functor of points (co)represented by a supervariety or a
superscheme

Smooth manifolds can be described as sets of points with a topology. For
manifolds-with-boundary (which, strictly speaking, are not manifolds, hence
the suggestive notation “in-one-word”) or over fields of characteristic $p>0$,
the set of points does not define the variety or scheme; the same is true for
supervarieties and superschemes. To determine one of such objects
${\mathcal{M}}$, we consider it parametrized by a superscheme
$\mathop{\mathrm{Spec}}\nolimits C$ for various $C$. In other words, we
consider collections
$\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Spec}}\nolimits
C,{\mathcal{M}})$ of $(\mathop{\mathrm{Spec}}\nolimits C)$-points of
${\mathcal{M}}$, usually called $C$-points of ${\mathcal{M}}$. If
${\mathcal{M}}$ can be recovered from its algebra of functions
${\mathcal{F}}$, as is the case, e.g., with affine (super)schemes
${\mathcal{M}}$, we can consider
$\mathop{\mathrm{Hom}}\nolimits({\mathcal{F}},C)$ instead of
$\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Spec}}\nolimits
C,{\mathcal{M}})$.

####  3.4.1. Linear supervariety$\longleftrightarrow$linear superspace.

The set ${\mathbb{K}}^{n}$ can be considered as a linear space and as a
manifold. Linear superspaces are just ${\mathbb{Z}}/2$-graded linear spaces,
they are not supermanifolds. So, first of all, let me introduce linear
supervariety or linear supermanifold

${\mathcal{V}}=(V_{\bar{0}},{\mathcal{O}}_{V_{\bar{0}}}\otimes\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(V_{\bar{1}}^{*})).$

corresponding to the linear superspace $V=V_{\bar{0}}\oplus V_{\bar{1}}$.

The morphisms of linear superspaces constitute the space
$\mathop{\mathrm{Hom}}\nolimits_{\bar{0}}(V,W)$, whereas the supervariety of
linear homomorphisms $V\longrightarrow W$ is the linear supervariety
corresponding to the superspace
$\underline{\mathop{\mathrm{Hom}}\nolimits}(V,W):=\mathop{\mathrm{Hom}}\nolimits(V,W)$.

In various instances, e.g., dealing with actions of supergroups, it is more
convenient in computations444This convenience is not just a matter of taste or
experience or habit. More precisely, one has to work with either commuting
diagrams, like in [MaAG, Section 1.15.4], or with matrix realizations, as one
does when working with Lie group actions. The language of points allows one to
actually compute something, like when turning from the invariant language of
operators to matrices. to consider, instead of the vector superspace $V$ or
the linear supermanifold ${\mathcal{V}}$, the functor $\mathop{\mathrm{{\sf
ScommSalgs}}}\nolimits_{\mathbb{K}}\leadsto{\sf Mod}_{\mathbb{K}}$ from the
category of supercommutative ${\mathbb{K}}$-superalgebras $C$ to the category
of $C$-modules represented by $V$ or, equivalently, ${\mathcal{V}}$:

$C\longmapsto{\mathcal{V}}(C)=V(C):=V\otimes C\text{~{}~{} for any $C$},$

where “any” is understood inside a suitable category (e.g., of superalgebras
finitely generated over ${\mathbb{K}}$).

####  3.4.2. Lie superalgebras ([KLLS]).

In the above terms, a Lie superalgebra in the category of supervarieties is a
vector superspace ${\mathfrak{g}}$, or a linear supervariety (supermanifold)
${\mathcal{G}}$ corresponding to it, corepresenting the functor from the
category of supercommutative ${\mathbb{K}}$-superalgebras $C$ to the category
of Lie superalgebras understood “naively”. In other words, considering
corepresenting functor instead of a representing one, we replace
${\mathfrak{g}}$, or the linear supervariety corresponding to it, by the
algebra $P({\mathfrak{g}})$ of polynomial functions on ${\mathfrak{g}}$. (Even
over ${\mathbb{R}}$ we have to replace the spaces by the algebras of
polynomial functions on these spaces.)

Clearly, $P({\mathfrak{g}})$ is a free supercommutative superalgebra generated
by ${\mathfrak{g}}^{*}$, i.e., there is a natural isomorphism of functors

$C\mapsto\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathrm{{\sf
ScommSalgs}}}\nolimits_{{\mathbb{K}}}}(P({\mathfrak{g}}),C)$ and
$C\mapsto\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}\text{-Vect}}({\mathfrak{g}}^{*},C)$,

whereas the second functor is naturally isomorphic to the forgetful function

${\mathbb{K}}\text{-Vect}\to\mathop{\mathrm{{\sf
Sets}}}\nolimits:C\mapsto{\mathfrak{g}}\otimes C.$

To the Lie superalgebra homomorphisms (in particular, to representations) a
morphism of the respective functors should correspond.

Clearly, if ${\mathfrak{g}}$ is a Lie superalgebra, then
${\mathfrak{g}}(C):={\mathfrak{g}}\otimes C$ is also a Lie superalgebra for
any $C$ functorially in $C$. The last three words mean that

(6) for any morphism of supercommutative superalgebras $C\longrightarrow
C^{\prime}$, there exists a morphism of Lie superalgebras
${\mathfrak{g}}\otimes C\longrightarrow{\mathfrak{g}}\otimes C^{\prime}$ so
that a composition of morphisms of supercommutative superalgebras
$C\longrightarrow C^{\prime}\longrightarrow C^{\prime\prime}$ goes into the
composition of Lie superalgebra morphisms
${\mathfrak{g}}(C)\longrightarrow{\mathfrak{g}}(C^{\prime})\longrightarrow{\mathfrak{g}}(C^{\prime\prime});$
the identity map goes into the identity map, etc.

An ideal ${\mathfrak{h}}\subset{\mathfrak{g}}$ represents the collection of
ideals ${\mathfrak{h}}(C)\subset{\mathfrak{g}}(C)$ for every $C$.

A Lie or algebraic (over any field) supergroup ${\mathcal{G}}$ is a group in
the category of supervariaties. The action of ${\mathcal{G}}$ in the
superspace $V$ is the collection of actions of ${\mathcal{G}}(C)$ in $V(C)$
for every $C$ such that these actions are compatible with morphisms of
supercommutative superalgebras $C\longrightarrow C^{\prime}$ in the same sense
as in (6).

However, we can consider any category ${\sf C}$; then any object
${\mathfrak{g}}\in\mathop{\mathrm{Ob}}\nolimits{\sf C}$ of this category
representing the functor
$C\mapsto{\mathfrak{g}}(C):=\mathop{\mathrm{Hom}}\nolimits_{\sf
C}(\mathop{\mathrm{Spec}}\nolimits C,{\mathfrak{g}})$, i.e., satisfying
conditions (6), is said to be a Lie superalgebra in the category ${\sf C}$.

##  4\. Deformations

###  4.1. Deformations and deforms. Odd parameters

Which of the infinitesimal deformations can be extended to a global one is a
separate much tougher question, usually solved ad hoc. Let me give two graphic
examples.

1) Deformations of representations. Consider a representation
$\rho:{\mathfrak{g}}\longrightarrow{\mathfrak{gl}}(V)$. The tangent space of
the moduli superspace of deformations of $\rho$ is isomorphic to
$H^{1}({\mathfrak{g}};V\otimes V^{*})$. For example, if ${\mathfrak{g}}$ is
the $0|n$-dimensional (i.e., purely odd) Lie superalgebra (with the only
bracket possible: identically equal to zero), its only irreducible
representations are the 1-dimensional trivial one, $\mathbbmss{1}$, and
$\Pi(\mathbbmss{1})$, where $\Pi$ is the change of parity functor, i.e.
$\Pi(V)_{i}:=V_{i+{\bar{1}}}$ for any superspace $V$. Clearly,

$\mathbbmss{1}\otimes\mathbbmss{1}^{*}\simeq\Pi(\mathbbmss{1})\otimes\Pi(\mathbbmss{1})^{*}\simeq\mathbbmss{1},$

and, because the Lie superalgebra ${\mathfrak{g}}$ is commutative (bracket is
identically zero), the differential in the cochain complex is zero. Therefore

$H^{1}({\mathfrak{g}};\mathbbmss{1})=E^{1}({\mathfrak{g}}^{*})\simeq\Pi({\mathfrak{g}}^{*}),$

so there is a $(\dim\,{\mathfrak{g}})$-dimensional space of odd parameters of
deformations of the trivial representation. If we consider ${\mathfrak{g}}$
“naively”, all of these odd parameters will be lost.

2) Deformations of the brackets. Let $C$ be a supercommutative superalgebra.

After [Ru], where the non-super case is considered, we say that a deformation
of a Lie superalgebra ${\mathfrak{g}}$ over $\mathop{\mathrm{Spec}}\nolimits
C$, is a Lie algebra ${\mathfrak{G}}$ such that
${\mathfrak{G}}\simeq{\mathfrak{g}}\otimes C$, as spaces. The deformation is
trivial if ${\mathfrak{G}}\simeq{\mathfrak{g}}\otimes C$, as Lie superalgebras
over $C$, not just as $C$-modules, and non-trivial otherwise.

Generally, the deforms — the results of deformations — of a given Lie
superalgebra ${\mathfrak{g}}$ over ${\mathbb{K}}$ are Lie superalgebras
${\mathfrak{G}}\otimes_{I}{\mathbb{K}}$, where $I$ is any closed point in
$\mathop{\mathrm{Spec}}\nolimits C$.

In particular, consider a deformation with an odd parameter $\tau$. This is a
Lie superalgebra ${\mathfrak{G}}$ isomorphic to
${\mathfrak{g}}\otimes{\mathbb{K}}[\tau]$ as a superspace. If, moreover,
${\mathfrak{G}}\simeq{\mathfrak{g}}\otimes{\mathbb{K}}[\tau]$ as a Lie
superalgebra, i.e.,

$[a\otimes f,b\otimes g]=(-1)^{p(f)p(b)}[a,b]\otimes fg\text{~{}~{}for any
$a,b\in{\mathfrak{g}}$ and $f,g\in{\mathbb{K}}[\tau]$},$

then the deformation is considered trivial (and non-trivial otherwise).
Observe that ${\mathfrak{g}}\otimes\tau$ is not an ideal of ${\mathfrak{G}}$:
the ideal should be a free ${\mathbb{K}}[\tau]$-module.

Comment. In a sense, the people who ignore odd parameters of deformations have
a point: we (rather they) consider classification of simple Lie superalgebras
(or whatever other problem) over the ground field ${\mathbb{K}}$, not over
${\mathbb{K}}[\tau]$, right? However, the odd parameters of deformations are
no less natural, actually, than the odd part of the Lie superalgebra itself.
In order to see these parameters, we have to consider whatever we are
deforming not over ${\mathbb{K}}$, but over ${\mathbb{K}}[\tau]$.

We do the same, actually, when $\tau$ is even and we consider formal
deformations over ${\mathbb{K}}[[\tau]]$. If the formal series in $\tau$
converges in a domain $U$, we can evaluate $\tau$ for any $\tau\in U$ and
consider copies ${\mathfrak{g}}_{\tau}$, where $\tau\in U$ (of the same
dimension as that of ${\mathfrak{g}}$ if $\dim{\mathfrak{g}}<\infty$). If the
parameter is formal or odd, such an evaluation is only possible trivially:
$\tau\mapsto 0$.

##  5\. Deformations of complex supermanifolds. Odd parameters

Objects in the category of smooth supermanifolds are in one-to-one
correspondence with vector bundles, see [Ga, Bat], [MaG, Subsection 4.1.3]. P.
Green, see [G], was the first to observe that in the category of complex-
analytic supermanifolds there are more objects than there are vector bundles;
Palamodov submitted the paper with the same result 2 months later than [G],
see [Pal]. This topic gained a new interest with the discovery by Donagi and
Witten of non-splitness of the moduli spaces of super Riemann surfaces, see
[DW]. For an overview of results on non-split supermanifolds obtained by
Onishchik and his students, see [Lob]. All papers on non-split supermanifolds
published so far ignored odd parameters of deformations. Let me consider the
odd parameters.

###  5.1. The complex supermanifolds

Let the ground field be ${\mathbb{C}}$, and $V$ an $m$-dimensional space. The
sheaf $\mathcal{F}_{n|m}$ of germs of $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(V)$-valued holomorphic functions on ${\mathbb{C}}^{n}$
defines the supermanifold ${\mathcal{C}}^{n|m}$. We consider the elements of
$\mathcal{F}_{n|0}$ even, the elements $\xi_{j}$ of a basis of $V$ odd. A
superdomain in ${\mathcal{C}}^{n|m}$ is a ringed space of the form
$(U,\mathcal{F}_{n|m}|_{U})$, where $U$ is an open domain in
${\mathbb{C}}^{n}$. An almost complex-analytic supermanifold of dimension
$n|m$ is a ringed space locally isomorphic to a superdomain in
${\mathcal{C}}^{n|m}$. Thus, if ${\mathcal{M}}$ is a supermanifold, then for
any point $p_{0}\in M$ there exist a neighborhood $U$ of $p_{0}$ in $M$ and an
isomorphism of the ringed space $(U,\mathcal{O}|_{U})$ onto a superdomain
$(\widetilde{U},\mathcal{F}_{n|m}|_{\widetilde{U}})$, where $\widetilde{U}$ is
an open domain in ${\mathbb{C}}^{n}$, called a chart in ${\mathcal{C}}^{n|m}$.
Let $x_{1},\dots,x_{n}$ be coordinates in ${\mathbb{C}}^{n}$. Identifying
$(U,\mathcal{O}|_{U})$ with the superdomain by means of a chart, we get the
sections $x_{i}$ for $i=1,\dots,n$, and $\xi_{j}$ for $j=1,\dots,m$ of
$\mathcal{O}|_{U}$ called local coordinates on $(U,\mathcal{O}|_{U})$.

For complex supermanifolds and recently distinguished notion of real-complex
supermanifolds, such as Minkowski supermanifolds, see [BGLS].

###  5.2. The retract of the supermanifold

Let $(M,\mathcal{O}_{\mathcal{M}})$ be a supermanifold,
$\mathcal{I}\subset\mathcal{O}_{\mathcal{M}}$ a subsheaf of ideals,
$\pi:\mathcal{O}_{\mathcal{M}}\longrightarrow\mathcal{I}$ the natural
projection. Setting

$N=\\{x\in M\mid\pi(\varphi)(x)=0\ \ \text{~{}~{}for
all~{}~{}}\varphi\in(\mathcal{O}_{\mathcal{M}})_{x}\\},\ \
\mathcal{O}_{\mathcal{N}}=(\mathcal{O}_{\mathcal{M}}/\mathcal{I})|_{N},$

we get the ringed space $(N,\mathcal{O}_{\mathcal{N}})$ called a
subsupermanifold of $(M,\mathcal{O}_{\mathcal{M}})$. If the sheaf
$\mathcal{I}$ is generated, over an open set $U\subset M$, by its homogeneous
sections $\varphi_{1},\ldots,\varphi_{s}$, then it is usual to say that the
subsupermanifold is determined in $U$ by the system of equations
$\varphi_{i}=0$, where $i=1,\ldots,s$.

To any supermanifold ${\mathcal{M}}=(M,{\mathcal{O}}_{\mathcal{M}})$ a split
supermanifold corresponds. Consider the filtration

(7)
${\mathcal{O}}_{\mathcal{M}}:={\mathcal{J}}^{0}\supset{\mathcal{J}}^{1}\supset{\mathcal{J}}^{2}\supset\ldots$

of $\mathcal{O}_{\mathcal{M}}$ by the powers of the subsheaf of ideals
${\mathcal{J}}={\mathcal{J}}^{1}$ generated by the odd elements. The
associated graded sheaf

$\mathop{\mathrm{gr}}\nolimits{\mathcal{O}}_{\mathcal{M}}=\bigoplus_{k\geq
0}\mathop{\mathrm{gr}}\nolimits^{k}\mathcal{O}_{\mathcal{M}},$

where
$\mathop{\mathrm{gr}}\nolimits^{k}{\mathcal{O}}_{\mathcal{M}}={\mathcal{J}}^{k}/{\mathcal{J}}^{k+1}$,
gives rise to the split supermanifold
$(M,\mathop{\mathrm{gr}}\nolimits{\mathcal{O}}_{\mathcal{M}})$ called the
retract of the supermanifold ${\mathcal{M}}$.

The quotient modulo the subsheaf $\mathcal{N}$ generated by all nilpotents of
$\mathcal{O}_{\mathcal{M}}$ determines the reduction whose result is
conventionally denoted ${\mathcal{M}}_{\text{red}}$. The quotient modulo the
subsheaf $\mathcal{J}^{1}$ generated by only odd nilpotents determines the odd
reduction, whose result Manin denoted
${\mathcal{M}}_{\mathop{\mathrm{rd}}\nolimits}$, of
$(M,\mathcal{O}_{\mathcal{M}})$; thus,
${\mathcal{M}}_{\mathop{\mathrm{rd}}\nolimits}$ is a submanifold of
$(M,\mathcal{O}_{\mathcal{M}})$.

Thus, we see that with any supermanifold ${\mathcal{M}}$ two objects of the
classical complex analytic geometry are associated: the complex manifold
$(M,\mathcal{O}_{M})$ and the holomorphic vector bundle $\mathbf{E}$ over
$(M,\mathcal{O}_{M})$ with fiber $V$ corresponding to the sheaf
$\mathcal{E}={\mathcal{J}}^{1}/{\mathcal{J}}^{2}$. It turns out that the
complex supermanifold ${\mathcal{M}}$ is not, in general, determined by these
two objects up to an isomorphism: there exist non-split supermanifolds.

The obstructions to splitness were considered and calculated, so far, for
transition functions of the form (3); e.g., see [BO, Bash] and references
therein. In this — restricted — approach (considering only transition
functions of the form (3)), the first obstructions to splitness are given by
the terms of degree $>1$ in the $\xi_{i}$ in the expansion of
$\varphi^{*}(x_{i})$ and $\varphi^{*}(\xi_{j})$. If the $x_{i}$ and $\xi_{j}$
are arbitrary local coordinates of ${\mathcal{M}}$ in a neighborhood $U\subset
M$, then $X_{i}=x_{i}+\mathcal{J}^{2}$ and $\Xi_{j}=\xi_{j}+\mathcal{J}^{3}$
are splitting local coordinates of
$(M,\mathop{\mathrm{gr}}\nolimits\mathcal{O}_{\mathcal{M}})$ in $U$, and one
gets the transition functions between these splitting coordinates, if one
takes the terms of degree 0 (respectively 1) in $\xi_{j}$ in the transition
functions $\varphi_{i}$ (respectively $\psi_{j}$) for ${\mathcal{M}}$, see eq.
(3).

In the restricted approach, the first obstructions to splitness are
represented by cocycles in the Čech cohomology
$H^{1}(M;{\mathcal{T}}\otimes{\mathcal{E}}^{2}(\mathbf{E}))$, where
${\mathcal{E}}^{i}(\mathbf{E})$ is the sheaf corresponding to the $i$th
exterior power $E^{i}(\mathbf{E})$ of the bundle $\mathbf{E}$, and
${\mathcal{T}}$ is the tangent bundle, see [MaG, Ch.4, §2, Prop. 9, p. 191]
and Palamodov’s chapter in [Ber1, Ch.4, §4, Sections 6–9] reproduced in [Ber2,
Ch.3, Th.2, p.126]. These Palamodov’s chapters expound a short note [Pal]
submitted 2 months after [G] was submitted, but published much later than [G].

The higher obstructions to splitness in the restricted approach are
represented by cocycles in the Čech cohomology
$H^{1}(M;{\mathcal{T}}\otimes{\mathcal{E}}^{2i}(\mathbf{E}))$ for $i>1$, see
[Ber2, Ch.3, §7, Th.2, p.135]; these obstructions describe even obstructions
to splitness in the non-restricted approach as well, but miss odd
obstructions.

In particular, this restricted approach (ignoring odd parameters, as in eq.
(3)) immediately leads us to the conclusion that “any complex supermanifold of
superdimension $m|1$ is split” since $E^{2}(\mathbf{E})=0$, see [MaG, Ch.4,§2,
Prop. 8 (a), p. 190] and [Va2, Example 3.3.1 (1)]. In [MaG, Va1, Va2], various
deformation problems were considered over a supervariety of parameters, but —
inexplicably — not in such generality in this particular case: in the study of
non-splitness.

To incorporate odd parameters, recall the following well known facts, see,
e.g., [OSS, § 1.1]. Let us cover ${\mathbb{CP}}^{1}$ by two affine charts
$U_{0}$ and $U_{1}$ with local coordinates $x$ and $y=\frac{1}{x}$,
respectively. Let $\xi$ and $\eta$ be basis sections of line bundle
${\mathbf{L}}_{k}$ over $U_{0}$ and $U_{1}$, respectively, such that in
$U_{0}\cap U_{1}$ we have (up to a non-zero constant factor)

$\begin{array}[]{l}\eta={x}^{k}\xi.\end{array}$

The bundle ${\mathbf{L}}_{k}$ is said to be of degree $k\in{\mathbb{Z}}$. The
sheaf of sections of this bundle is designated ${\mathcal{O}}(k)$. In
particular, since $dy=-x^{-2}dx$, then the cotangent sheaf of 1-forms
$\Omega^{1}$ is ${\mathcal{O}}(-2)$, and hence the dual to $\Omega^{1}$
tangent sheaf ${\mathcal{T}}$ is ${\mathcal{O}}(2)$; clearly,
${\mathcal{O}}:={\mathcal{O}}(0)$ is the structure sheaf (of functions). We
also have the following formulas, see “Bott’s formulas” in [OSS, p.4]:

(8)
$\begin{array}[]{l}{\mathcal{O}}(a)\otimes{\mathcal{O}}(b)\simeq{\mathcal{O}}(a+b);\\\
\dim
H^{0}({\mathbb{C}}{\mathbb{P}}^{1};{\mathcal{O}}(a))=\begin{cases}a+1&\text{if
$a\geq 0$}\\\ 0&\text{otherwise};\end{cases}\\\
\text{$\xi,x\xi,\dots,x^{a}\xi$ is a~{}basis of the space of sections over
$U_{0}$};\\\ \end{array}$ (9) $\begin{array}[]{l}\dim
H^{1}({\mathbb{C}}{\mathbb{P}}^{1};{\mathcal{O}}(a))=\begin{cases}|a|-1&\text{if
$a\leq-2$}\\\ 0&\text{otherwise};\end{cases}\\\ \text{for a~{}basis of the
space of sections over $U_{0}$ we can take}\\\
\begin{cases}\xi,x^{-1}\xi,\dots,x^{2-|a|}\xi&\text{if $a<-2$}\\\ \xi&\text{if
$a=-2$}.\end{cases}\\\ \end{array}$

Now, if we consider transition functions taking odd parameters into account,
then eq. (4) shows that there is a linear in $\xi$ term in the new coordinates
$\varphi(x_{i})$. The odd obstructions to splitness are represented by
cocycles in $H^{1}(M;{\mathcal{T}}\otimes{\mathcal{E}}^{2i+1}(\mathbf{E}))$
for $i\geq 0$.

In the case of the linear bundle $\mathbf{E}={\mathbf{L}}_{k}$ over
$M={\mathbb{C}}{\mathbb{P}}^{1}$, the corresponding obstructions belong to
(recall that $E^{1}(V):=\Pi(V)$)

(10)
$H^{1}(M;{\mathcal{T}}\otimes{\mathcal{E}}^{1}(\mathbf{E}))=H^{1}(M;{\mathcal{O}}(2)\otimes\Pi({\mathcal{O}}(k)))\simeq
H^{1}(M;\Pi({\mathcal{O}}(k+2)).$

Eqs. (8), (9) and (10) imply the following theorem, where
$\mathop{\mathrm{{GL}}}\nolimits(V)=\mathop{\mathrm{{GL}}}\nolimits(1)$.

######  Theorem (Odd parameters of the non-split $1|1$-dimensional superstring
whose underlying is ${\mathbb{C}}{\mathbb{P}}^{1}$).

The non-split supermanifolds ${\mathcal{M}}$ of complex superdimension $1|1$,
a.k.a. superstrings, with underlying ${\mathbb{C}}{\mathbb{P}}^{1}$ whose
retract corresponds to the line bundle $\mathbf{E}={\mathbf{L}}_{k}$ with
fiber $V$ are described by $\mathop{\mathrm{{GL}}}\nolimits(V)$-orbits in the
space $H^{1}(M;\Pi({\mathcal{O}}(k+2))$ — its projectivization, the number of
orbits being equal to $|k+2|-1$ for $k\leq-4$, whereas ${\mathcal{M}}$ is
split for $k>-4$.

###  5.3. On the claims of Bettadapura, see [BK1, BK2, BK3]

Although Bettadapura sometimes uses standard notations in nonstandard ways
(e.g., his $\Pi E$ is an ad hoc notation, which in the text on super geometry
causes confusion), writes that something “will” whereas it already “is”,
Palamodov’s technique published in the chapter Palamodov wrote ([Ber1, Ch.4,
§4]) is misattributed to Berezin, some symbols are left undefined (e.g.,
$E^{\text{v}}$), etc., etc., and claims are formulated assuming the change of
coordinates is performed via formulas of the form (3), not (4), it is possible
to decipher that some statements are wrong if we take into account odd
parameters (e.g., [BK1, Lemma 2.4] — a copy of the corresponding claims in
[MaG] and [Va2], although these references are not given). These claims are at
variance with Theorem 5.2.1. Theorem), the other statements do take the odd
parameters into account and contain ostensibly new examples of non-split super
Riemann surfaces ${\mathcal{S}}$, but in reality they do not describe odd
parameters of deformations of ${\mathcal{S}}$, but rather an incomplete study
of deformations of ${\mathcal{S}}\times{\mathcal{C}}^{0|n}$.

Observe that, like many, but fortunately not everybody, Bettadapura applies
the term “super Riemann surface” to $1|1$-dimensional supermanifolds with a
contact structure, whereas it seems natural to consider several types of
“super Riemann surfaces” of dimension $1|N$ either with and without contact
structure but, perhaps, preserving a volume element, see [Lsrs] where
restrictions on $N$ are justified (explained) from a certain point of view.

##  6\. Finite-dimensional simple Lie superalgebras considered naively

Certain simple Lie superalgebras (or their relatives, like
${\mathfrak{gl}}(n|n)$ and ${\mathfrak{sl}}(n|n)$) were easy to describe
meaningfully, by some properties of these algebras. Even before a matrix
realization of ${\mathfrak{osp}}(m|2n)$ and its interpretation as a Lie
superalgebra preserving a non-degenerate even bilinear form were discovered it
was natural to call it “ortho-symplectic”, just by looking at its even part.
Two series of simple Lie superalgebras were considered “strange” until the
interpretations of their non-simple relatives as preserving a tensor was
discovered, hence their first notation and certain awkward names (e.g., “di-
spin algebra of Mitchell—Gell-Mann—Radicati”) were soon abandoned. The
notation “${\mathfrak{psl}}$” was also being avoided for a long time, although
it was already usual in the modular setting.

###  6.1. Linear (matrix) Lie superalgebras

Let $\text{Size}=(p_{1},\dots,p_{|\text{Size}|})$ be an ordered collection of
parities of the basis vectors of $V$ for which we take only vectors
homogeneous with respect to parity and $|\text{Size}|:=\dim V$. The general
linear Lie superalgebra of all supermatrices of size Size corresponding to
linear operators in the superspace $V=V_{\bar{0}}\oplus V_{\bar{1}}$ over the
ground field ${\mathbb{K}}$ is denoted by ${\mathfrak{gl}}(\text{Size})$.
Usually, for the standard (a simplest from a certain point of view) format,
${\mathfrak{gl}}({\bar{0}},\dots,{\bar{0}},{\bar{1}},\dots,{\bar{1}})$ is
abbreviated to ${\mathfrak{gl}}(\dim V_{\bar{0}}|\dim V_{\bar{1}})$. Any non-
zero supermatrix in ${\mathfrak{gl}}(\text{Size})$ can be uniquely expressed
as the sum of its even and odd parts; in the standard format this is the
following block expression; on non-zero summands the parity is defined:

$\text{\small$\begin{pmatrix}A&B\\\
C&D\end{pmatrix}$}=\text{\small$\begin{pmatrix}A&0\\\
0&D\end{pmatrix}$}+\text{\small$\begin{pmatrix}0&B\\\ C&0\end{pmatrix}$},\quad
p\left(\text{\small$\begin{pmatrix}A&0\\\
0&D\end{pmatrix}$}\right)={\bar{0}},\;p\left(\text{\small$\begin{pmatrix}0&B\\\
C&0\end{pmatrix}$}\right)={\bar{1}}.$

The trace or supertrace, for emphasis, is any map
$\mathop{\mathrm{tr}}\nolimits:{\mathfrak{g}}\longrightarrow{\mathbb{K}}$ that
vanishes on the first derived Lie (super)algebra ${\mathfrak{g}}^{\prime}$. In
particular, on ${\mathfrak{gl}}(\text{Size})$, there is just one (up to a non-
zero scalar) such map; it is given by the formula

(11) $\mathop{\mathrm{str}}\nolimits(X):=\sum(-1)^{p_{i}}X_{ii}.$

Thus, in the standard format,

$\mathop{\mathrm{str}}\nolimits\begin{pmatrix}A&B\\\
C&D\end{pmatrix}=\mathop{\mathrm{tr}}\nolimits A-\mathop{\mathrm{tr}}\nolimits
D.$

Observe that for the Lie superalgebra ${\mathfrak{gl}}_{\mathcal{C}}(a|b)$
over a supercommutative superalgebra ${\mathcal{C}}$, i.e., for supermatrices
with homogeneous (with respect to parity) elements in ${\mathcal{C}}$, we have

(12) $\begin{array}[]{l}\mathop{\mathrm{str}}\nolimits
X=\mathop{\mathrm{tr}}\nolimits A-(-1)^{p(X)}\mathop{\mathrm{tr}}\nolimits
D\text{~{}~{}for any $X=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$,}\\\
\text{where
$p(X)=p(A_{ij})=p(D_{kl})=p(B_{il})+{\bar{1}}=p(C_{kj})+{\bar{1}}$},\\\
\end{array}$

so on odd supermatrices with entries in ${\mathcal{C}}$ such that
${\mathcal{C}}_{\bar{1}}\neq 0$, the expression of supertrace coincides with
that of the conventional trace of a matrix.

Since $\mathop{\mathrm{str}}\nolimits[x,y]=0$, the supertraceless
supermatrices span a Lie subsuperalgebra called special linear and denoted
${\mathfrak{sl}}(\text{Size})$.

There are, however, at least two super versions of ${\mathfrak{gl}}(n)$, not
one; for reasons, see [LSoS, Ch1, Ch.7], where the super version of the Schur
lemma and classification of central simple (finite-dimensional) associative
superalgebras are considered. The natural (from the Schur lemma’s point of
view) other version — ${\mathfrak{q}}(n)$ — is called the queer Lie
superalgebra and is defined as the one that preserves — if $p\neq 2$ — the
complex structure given by an odd operator $J$, i.e., ${\mathfrak{q}}(n)$ is
the supercentralizer $\mathrm{C}(J)$ of $J$:

${\mathfrak{q}}(n):=\mathrm{C}(J)=\\{X\in{\mathfrak{gl}}(n|n)\mid[X,J]=0\\},\text{
where }J^{2}=-\mathop{\mathrm{id}}\nolimits.$

It is clear that over an algebraically closed field, we can by a change of
basis reduce $J$ to the normal form (shape) $J_{2n}$ in the standard format,
see eq. (13), and then the elements of ${\mathfrak{q}}(n)$ take the form

(13)
${\mathfrak{q}}(n):=\left\\{(A,B):=\text{\footnotesize$\begin{pmatrix}A&B\\\
B&A\end{pmatrix}$},\text{~{}~{}where $A,B\in{\mathfrak{gl}}(n)$ and
$J_{2n}:=\text{\footnotesize$\begin{pmatrix}0&1_{n}\\\
-1_{n}&0\end{pmatrix}$}$}\right\\}.$

(Over any field ${\mathbb{K}}$, instead of $J$ we can take any odd operator
$K$ such that $K^{2}=a\mathop{\mathrm{id}}\nolimits_{n|n}$, the multiple of
the identity operator, where $a\in{\mathbb{K}}^{\times}$. If ${\mathbb{K}}$ is
algebraically closed, the Lie superalgebras $\mathrm{C}(K)$ are isomorphic for
distinct $a$; if $p=2$, it is natural to select
$K^{2}=\mathop{\mathrm{id}}\nolimits$.)

On ${\mathfrak{q}}(n)$, the odd trace, nowadays called queertrace, is defined:

$\mathop{\mathrm{qtr}}\nolimits\colon(A,B)\longmapsto\mathop{\mathrm{tr}}\nolimits
B.$

Denote by ${\mathfrak{sq}}(n)$ the Lie superalgebra of queertraceless
matrices, first described in [BL1] together with the odd (queer) determinant
of the $C$-points of the corresponding supergroup, see a more accessible
[LSoS, Ch.1].

Clearly, ${\mathfrak{gl}}$ and ${\mathfrak{q}}$ correspond to the two cases of
the super version of Schur’s lemma over an algebraically closed field: the
${\mathfrak{gl}}$ case: an irreducible module $M$ over a collection $S$ of
homogeneous operators can be absolutely irreducible, i.e., have no proper
invariant subspaces, then the only operator commuting with $S$ is a scalar,
and the ${\mathfrak{q}}$ case: $M$ can have in invariant subspace, which is
not a subsuperspace, then the superdimension of the module is of the form
$n|n$ and an odd operator $K$ interchanges the homogeneous components of the
module.

####  6.1.1. Supermatrices of operators

To the linear map of superspaces $F:V\longrightarrow W$ there corresponds the
dual map $F^{*}:W^{*}\longrightarrow V^{*}$ between the dual superspaces. In
bases consisting of homogeneous vectors $v_{i}\in V$ of parity $p(v_{i})$, and
$w_{j}\in W$ of parity $p(w_{j})$, the formula
$F(v_{j})=\mathop{\sum}_{i}w_{i}X_{ij}$ assigns to $F$ the supermatrix $X$. In
the dual bases, the supertransposed matrix $X^{st}$ corresponds to $F^{*}$:

$(X^{st})_{ij}:=(-1)^{(p(v_{i})+p(w_{j}))p(w_{j})}X_{ji}.$

In the standard supermatrix format, we have

$X=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\longmapsto
X^{st}:=\begin{pmatrix}A^{t}&(-1)^{p(X)}C^{t}\\\
-(-1)^{p(X)}B^{t}&D^{t}\end{pmatrix}=\begin{cases}\begin{pmatrix}A^{t}&C^{t}\\\
-B^{t}&D^{t}\end{pmatrix}&\text{if $p(X)={\bar{0}}$},\\\
\begin{pmatrix}A^{t}&-C^{t}\\\ B^{t}&D^{t}\end{pmatrix}&\text{if
$p(X)={\bar{1}}$}.\end{cases}$

####  6.1.2. Supermatrices of bilinear forms

Having selected a basis (by definition consisting of vectors homogeneous with
respect to parity) of the superspace $V$, we define the Gram matrix
$B=(B_{ij})$ of the bilinear form $B^{f}$ on $V$ by the formula

(14) $B_{ij}:=(-1)^{p(B)p(v_{i})}B^{f}(v_{i},v_{j})\text{~{}~{} for the basis
vectors $v_{i}\in V$.}$

This formula for the Gram matrix of $B^{f}$ allows us to identify any bilinear
form $B(V,W)$ with an operator, an element of
$\mathop{\mathrm{Hom}}\nolimits(V,W^{*})$, see [LSoS, Ch.1].

Recall that the upsetting of bilinear forms
$u\colon\mathop{\mathrm{Bil}}\nolimits(V,W)\longrightarrow\mathop{\mathrm{Bil}}\nolimits(W,V)$
is given by the formula

(15) $u(B^{f})(w,v):=(-1)^{p(v)p(w)}B^{f}(v,w)\text{~{}~{} for any $v\in V$
and $w\in W$.}$

Let now $W=V$, and
$\mathop{\mathrm{Bil}}\nolimits(V):=\mathop{\mathrm{Bil}}\nolimits(V,V)$. The
shape of the Gram matrix $B^{u}$ of a homogeneous form $u(B^{f})$ in the
standard format of $V$ is as follows

(16) $B^{u}:=\text{\small$\begin{pmatrix}R^{t}&(-1)^{p(B)}T^{t}\\\
(-1)^{p(B)}S^{t}&-U^{t}\end{pmatrix}$}\text{~{}~{} for
$B=\text{\small$\begin{pmatrix}R&S\\\ T&U\end{pmatrix}$}$,}$

The form $B^{f}$ is said to be symmetric if $B^{u}=B$, and anti-symmetric if
$B^{u}=-B$. In particular, the form on ${\mathfrak{gl}}(V)$ (resp.
${\mathfrak{q}}(V)$) given by the respective trace is symmetric:

$(X,Y):=\mathop{\mathrm{str}}\nolimits(XY)\text{~{}~{}for any
$X,Y\in{\mathfrak{gl}}(V)$ (resp. $(X,Y):=\mathop{\mathrm{qtr}}\nolimits(XY)$
for any $X,Y\in{\mathfrak{q}}(V)$)}.$

Clearly, the upsetting $B\mapsto B^{u}$ of Gram matrices of bilinear forms is
not supertransposition.

Observe that the passage from $V$ to $\Pi(V)$ turns every symmetric form
${\mathcal{B}}$ on $V$ into an anti-symmetric one ${\mathcal{B}}^{\Pi}$ on
$\Pi(V)$ and every anti-symmetric ${\mathcal{B}}$ into a symmetric
${\mathcal{B}}^{\Pi}$ by setting

$\text{${\mathcal{B}}^{\Pi}(\Pi(x),\Pi(y)):=(-1)^{p(B)+p(x)+p(x)p(y)}{\mathcal{B}}(x,y)$
for any $x,y\in V$}.$

Observe that there are no “supersymmetric” bilinear forms; the property

(17) $B^{f}(w,v):=(-1)^{p(v)p(w)}B^{f}(v,w)\text{~{}~{} for any $v,w\in V$}$

which is sometimes given as their definition just reflects braiding in the
category of superspaces and has nothing to do with the symmetry of bilinear
forms. (Therefore, the title of [BKLS] should have been corrected. The same
confusing term is used in [BS], where inhomogeneous with respect to parity
bilinear forms are classified on superspaces of dimension $\leq 7$ in an
attempt to define a generalized mixture of the Heisenberg superalgebra and
what in [BGLLS] was called the antibracket superalgebra. The mainstream
supersymmetry theory ignores the inhomogeneous bilinear forms since they are
not preserved by Lie superalgebras; but it is interesting, nevertheless, to
explore their hidden depths, cf. Subsection 3.2; e.g., what is the “something”
that preserves them? What linear transformations of the superspace $V$ (only
even or any) is reasonable in some (which?) sense to consider in the
classification of inhomogeneous bilinear forms on $V$?)

Most popular normal shapes of the (Gram matrices of the) even non-degenerate
symmetric form are the ones which in the standard format are as follows:

$\begin{array}[]{l}B_{ev}(m|2n)=\mathop{\mathrm{diag}}\nolimits(1_{m},J_{2n}):=\text{\small$\begin{pmatrix}1_{m}&0\\\
0&J_{2n}\end{pmatrix}$}\text{~{}or~{}}\mathop{\mathrm{diag}}\nolimits(A_{m},J_{2n}):=\text{\small$\begin{pmatrix}A_{m}&0\\\
0&J_{2n}\end{pmatrix}$},\\\ \text{where
$J_{2n}=\mathop{\mathrm{antidiag}}\nolimits(1_{n},-1_{n}):=\text{\small$\begin{pmatrix}0&1_{n}\\\
-1_{n}&0\end{pmatrix}$}$ and
$A_{m}=\mathop{\mathrm{antidiag}}\nolimits(1,\dots,1)$}.\\\ \end{array}$

The Lie superalgebra ${\mathfrak{aut}}(B)\subset{\mathfrak{gl}}(\text{Size})$
that preserves the Gram matrix $B$ of the form
${B^{f}\in\mathop{\mathrm{Bil}}\nolimits(V)}$ consists of the supermatrices
$X\in{\mathfrak{gl}}(\text{Size})$ such that

$X^{st}B+(-1)^{p(X)p(B)}BX=0\quad\text{for an homogeneous matrix
$B\in{\mathfrak{gl}}(\text{Size})$}.$

The usual notation for ${\mathfrak{aut}}(B_{ev}(m|2n))$ is
${\mathfrak{osp}}(m|2n)$; sometimes one writes more explicitly,
${\mathfrak{osp}}^{sy}(m|2n)$. Observe that the antisymmetric non-degenerate
bilinear form is preserved by the “symplectico-orthogonal” Lie superalgebra
${\mathfrak{osp}}^{a}(m|2n)$ isomorphic to ${\mathfrak{osp}}^{sy}(m|2n)$.

A non-degenerate symmetric odd bilinear form $B_{odd}(n|n)$ can be reduced to
a normal shape whose matrix in the standard format is $J_{2n}$, see (16), not
$\Pi_{2n}:=\mathop{\mathrm{antidiag}}\nolimits(1_{n},1_{n})$ which is
antisymmetric, see [LSoS], contrary to a hasty conclusion induced by the
“even” experience. The usual notation for
${\mathfrak{aut}}(B_{odd}(\text{Size}))$ is ${\mathfrak{pe}}(\text{Size})$.
The passage from $V$ to $\Pi(V)$ establishes an isomorphism
${\mathfrak{pe}}^{sy}(\text{Size})\cong{\mathfrak{pe}}^{a}(\text{Size})$.
These isomorphic Lie superalgebras are nowadays called, as A. Weil suggested,
periplectic.

The traceless (special) subalgebra of ${\mathfrak{pe}}(n)$ is denoted
${\mathfrak{spe}}(n)$; it is simple for $n>2$.

For a large class of Lie superalgebras either simple, or relatives of simple,
Kac introduced the notion of a Cartan matrix, see [K2] with improvements of
the definition in [CCLL, BGL, BLS, BLLoS]. If $p\neq 2$, then neither
periplectic superalgebras nor their simple relatives have Cartan matrices;
this is not so for $p=2$, see [BGL] containing the classification of all
finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix
over any algebraically closed field of positive characteristic.

If ${\mathfrak{c}}$ is a Lie algebra of scalar matrices, and
${\mathfrak{g}}\subset{\mathfrak{gl}}(n|n)$ is a Lie subsuperalgebra
containing ${\mathfrak{c}}$, then the projective Lie superalgebra of type
${\mathfrak{g}}$ is ${\mathfrak{pg}}:={\mathfrak{g}}/{\mathfrak{c}}$, e.g.,
${\mathfrak{psl}}(n|n)$ and ${\mathfrak{psq}}(n)$.

###  6.2. Exceptional finite-dimensional simple Lie superalgebras

There are 3 of them: ${\mathfrak{osp}}_{a}(4|2)$, ${\mathfrak{ag}}(2)$ and
${\mathfrak{ab}}(3)$. I denote them following meaningful Serganova’s notation
from the paper [Se2], where she proved that the notion of roots generalized to
non-degenerate pseudo-euclidean space instead of euclidean, leads —
unexpectedly — to Lie superalgebras ${\mathfrak{g}}(A)$ with indecomposable
matrix $A$. A tempting Open problem: superise the result of [P] by considering
non-degenerate but not sign-definite sesquilinear form — an analog of
Serganova’s result for complex reflections.

Kaplansky was the first to discover all three exceptional simple Lie
superalgebras ([Kapp]). They are of the form ${\mathfrak{g}}(A)$, i.e., have a
Cartan matrix $A$, and thus are described in [BGL] over fields of any
characteristic $p$ (non-existing for some values of $p$), where the
description as pairs $({\mathfrak{g}}_{\bar{0}},{\mathfrak{g}}_{\bar{1}})$,
although snubbed at in Subsection 1.3 above as “clumsy”, is also given because
it is useful in certain calculations, see also [CCLL]. For interpretations of
the exceptional simple Lie algebras, see [Eld, Eldi] and [Sud], see also
Subsection 6.4.1.

###  6.3. Finite-dimensional simple vectorial Lie superalgebras

Define partial derivatives in $\Lambda^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}(n)$ by setting
$\partial_{\xi_{i}}(\xi_{j}):=\delta_{ij}$ and considering only left action of
the derivations on functions. Let $\partial_{i}:=\partial_{\xi_{i}}$. Then,
setting $\deg\xi_{i}=1$ for all $i$, we get the standard
${\mathbb{Z}}$-grading of

$\bullet$ the general vectorial Lie superalgebra

${\mathfrak{vect}}(0|n):=\left\\{\sum f_{i}\partial_{i}\mid
f_{i}\in\Lambda(n)\text{~{}~{}for all
$i$}\right\\}=\mathop{\oplus}\limits_{-1\leq i\leq n-1}{\mathfrak{g}}_{i}.$

$\bullet$ The divergence (with respect to the volume element
$\mathop{\mathrm{{vol\/}}}\nolimits(\xi)$ in “coordinates” $\xi$ which means
that the Lie derivative $L_{D}$ along the vector field $D$ multiplies
$\mathop{\mathrm{{vol\/}}}\nolimits(\xi)$ by $\mathop{\text{\rm
div}}\nolimits(D)$) is defined by the formula

$\mathop{\text{\rm div}}\nolimits\left(\sum
f_{i}\partial_{i}\right):=\sum(-1)^{p(f_{i})p(\partial_{i})}\partial_{i}f_{i}.$

The special or divergence–free vectorial Lie superalgebra is

${\mathfrak{svect}}(0|n):=\\{D\in{\mathfrak{vect}}(0|n)\mid\mathop{\text{\rm
div}}\nolimits(D)=0\\}=\\{D\in{\mathfrak{vect}}(0|n)\mid
L_{D}(\mathop{\mathrm{{vol\/}}}\nolimits(\xi))=0\\}.$

Kac was the first to observe (in analogy with characteristic $>0$, cf. [W])
that for $n=2k$ any volume form $f\mathop{\mathrm{{vol\/}}}\nolimits(\xi)$,
where $f\in\Lambda(\xi)$, can be reduced by a change of indeterminates (an
automorphism of ${\mathbb{C}}[\xi]$) to one of the two normal shapes: with
either $f=1$ or with $f=(1+\prod\xi_{i})$, see [K1.5], [K2, Section 3.3.1],
where the volume forms (a particular case of integrable forms, see [BL]) are
erroneously considered as a particular case of differential forms. Kac
formulated this even before the notion of the volume form was correctly
defined. The same argument works for functions in $n=2k+1$ indeterminates with
values in ${\mathbb{C}}[\tau]$, the normal shapes being with either $f=1$ or
with $f=(1+\tau\prod\xi_{i})$. Accordingly, the deformed special or
divergence–free vectorial Lie superalgebra is

$\widetilde{{\mathfrak{svect}}}(0|2k):=\\{D\in{\mathfrak{vect}}(0|2k)\mid
L_{D}((1+t\prod\xi_{i})\mathop{\mathrm{{vol\/}}}\nolimits(\xi))=0\\}.$

Any two such Lie superalgebras for $t,t^{\prime}\in{\mathbb{C}}^{\times}$ are
clearly isomorphic, so we assume $t=1$.

Considering the supergroup of automorphisms
${\mathbb{C}}[\xi_{1},\dots,\xi_{n}]$ in terms of the functor

${\mathcal{C}}\longmapsto\mathop{\mathrm{{Aut\/}}}\nolimits^{\bar{0}}_{\mathcal{C}}({\mathcal{C}}\otimes{\mathbb{C}}[\xi]),$

where ${\mathcal{C}}$ is a supercommutative superalgebra (cf. with the
description of a group scheme in [MaAG]), we arrive at the normal shapes of
the volume forms $f\mathop{\mathrm{{vol\/}}}\nolimits(\xi)$:

$f=\begin{cases}(1+t\prod\xi_{i}),\text{~{}~{}where
$t\in{\mathcal{C}}_{\bar{0}}^{\times}$}&\text{for $n$ even},\\\
(1+\tau\prod\xi_{i}),\text{~{}~{}where
$\tau\in{\mathcal{C}}_{\bar{1}}$}&\text{for $n$ odd},\end{cases}$

and hence to an odd parameter $\tau$ and the deformed special or
divergence–free vectorial Lie superalgebra

$\widetilde{{\mathfrak{svect}}}(0|2k+1):=\\{D\in{\mathbb{C}}[\tau]\otimes{\mathfrak{vect}}(0|2k)\mid
L_{D}((1+\tau\prod\xi_{i})\mathop{\mathrm{{vol\/}}}\nolimits(\xi))=0\\}.$

Any two such Lie superalgebras for odd parameters $\tau,\tau^{\prime}$ are,
clearly, isomorphic.

$\bullet$ Define the space of differential (a.k.a. exterior) forms
${\mathbb{C}}[\xi,d\xi]$ by setting $p(d\xi_{i})={\bar{0}}$ and letting the
$d\xi_{i}$ commute with the $\xi_{j}$ and $d\xi_{j}$; define the exterior
differential by the formula (together with the Leibniz and Sign Rules)

$d(f):=\sum d\xi_{i}\frac{\partial f}{\partial\xi_{i}}.$

Define the action of Lie derivative along the vector field $D$ on the space of
differential forms ${\mathbb{C}}[\xi,d\xi]$ by setting

$L_{D}(df):=(-1)^{p(D)}d(D(f)).$

For a normal shape of the symplectic, i.e., a non-degenerate and closed,
2-form on ${\mathcal{C}}^{0|n}$ we can take (as was claimed in [Ln], for the
first proof, see [Sha]; for a conceptual proof, see [GLr, BGLS])

$\omega:=\sum(d\xi_{i})^{2}.$

Let the Lie superalgebra of Hamiltonian vector fields on ${\mathcal{C}}^{0|n}$
be

${\mathfrak{h}}(0|n):=\left\\{D\in{\mathfrak{vect}}(0|n)\mid
L_{D}(\omega)=0\right\\}.$

Let the space of the Poisson Lie superalgebra ${\mathfrak{po}}(0|n)$ on
${\mathcal{C}}^{0|n}$ be $\Lambda(n)$ with the Poisson bracket

$\\{f,g\\}_{P.b.}:=(-1)^{p(f)}\sum\frac{\partial
f}{\partial\xi_{i}}\frac{\partial g}{\partial\xi_{i}}.$

Let $H_{-}:f\longmapsto H_{f}$ be the projection
${\mathfrak{po}}(0|n)\longrightarrow{\mathfrak{h}}(0|n)$ given by the formula

$H_{f}:=(-1)^{p(f)}\sum\frac{\partial f}{\partial\xi_{i}}\partial_{\xi_{i}}.$

Clearly, the following sequence is exact

(18)
$0\longrightarrow{\mathbb{C}}\longrightarrow{\mathfrak{po}}(0|n)\stackrel{{\scriptstyle
H_{-}}}{{\longrightarrow}}{\mathfrak{h}}(0|n)\longrightarrow 0.$

###  6.4. The list of finite-dimensional simple Lie superalgebras considered
naively

Recall that the Lie superalgebra ${\mathfrak{g}}$ without proper ideals and of
dimension $>1$ is said to be simple. Eq. (19) lists all non-isomorphic
(minding isomorphisms (21), (22)) simple finite-dimensional Lie superalgebras
over ${\mathbb{C}}$; for first examples and subcases of classification, see
[K0, K1, Kapp, K1C, SNR, Ho, Dj1, Dj2, Dj3, DjH]; for a summary (ignoring odd
parameters), see [K2, Sch]:

(19) $\begin{array}[]{l}\text{${\mathfrak{sl}}(m|n)$ for $m>n\geq 1$,
${\mathfrak{psl}}(n|n)$ for $n>1$, ${\mathfrak{osp}}(m|2n)$ for $mn\neq 0$}\\\
\text{${\mathfrak{psq}}(n)$ and ${\mathfrak{spe}}(n)$ for $n>2$;}\\\
\text{${\mathfrak{osp}}_{\alpha}(4|2)$ for $\alpha\neq 0,-1$ is given by any
of its four Cartan matrices (\cite[cite]{[\@@bibref{}{BGL}{}{}]})}\\\
\begin{pmatrix}0&1&-1-\alpha\\\ -1&0&-\alpha\\\ -1-\alpha&\alpha&0\\\
\end{pmatrix}\text{~{}~{}}\begin{pmatrix}2&-1&0\\\ \alpha&0&-1-\alpha\\\
0&-1&2\\\ \end{pmatrix}\text{~{}~{}}\begin{pmatrix}2&-1&0\\\ -1&0&-\alpha\\\
0&-1&2\\\ \end{pmatrix}\text{~{}~{}}\begin{pmatrix}2&-1&0\\\ -1-\alpha&0&1\\\
0&-1&2\\\ \end{pmatrix}\\\ \text{${\mathfrak{a}}{\mathfrak{g}}(2)$ and
${\mathfrak{a}}{\mathfrak{b}}(3)$ are given by their Cartan matrices
(\cite[cite]{[\@@bibref{}{BGL}{}{}]}},\\\ \text{see also
\cite[cite]{[\@@bibref{}{Kapp, DWN,Sud}{}{}]} and very informative
\cite[cite]{[\@@bibref{}{Eld, Eldi}{}{}]});}\\\ \text{${\mathfrak{vect}}(0|n)$
for $n>1$,}\\\ \text{${\mathfrak{svect}}(0|n)$ and
$\widetilde{\mathfrak{svect}}(0|2n)$ for $n>2$,}\\\
\text{${\mathfrak{h}}^{\prime}(0|n):=[{\mathfrak{h}}(0|n),{\mathfrak{h}}(0|n)]$
for $n>3$.}\\\ \end{array}$

Occasional isomorphisms:

(20) $\begin{array}[]{l}{\mathfrak{sl}}(m|n)\simeq{\mathfrak{sl}}(n|m),\ \
{\mathfrak{vect}}(0|2)\simeq{\mathfrak{sl}}(1|2)\simeq{\mathfrak{osp}}(2|2),\\\
{\mathfrak{spe}}(3)\simeq{\mathfrak{svect}}(0|3),\ \
{\mathfrak{psl}}(2|2)\simeq{\mathfrak{h}}^{\prime}(0|4).\\\ \end{array}$

The isomorphisms
${\mathfrak{osp}}_{\alpha}(4|2)\simeq{\mathfrak{osp}}_{\alpha^{\prime}}(4|2)$
are generated by the transformations ([BGL]):

(21) $\alpha\longmapsto\alpha^{\prime}:=-1-\alpha\
,\qquad\alpha\longmapsto\alpha^{\prime}:=\mathchoice{\raisebox{0.25pt}{$\dfrac{1}{\alpha}$}}{\raisebox{0.25pt}{$\dfrac{1}{\alpha}$}}{\raisebox{-0.5pt}{$\tfrac{1}{\alpha}$}}{\raisebox{-0.5pt}{$\tfrac{1}{\alpha}$}}\
,$

so the other isomorphisms are

(22)
$\alpha\longmapsto\alpha^{\prime}:=-\mathchoice{\raisebox{0.25pt}{$\dfrac{1+\alpha}{\alpha}$}}{\raisebox{0.25pt}{$\dfrac{1+\alpha}{\alpha}$}}{\raisebox{-0.5pt}{$\tfrac{1+\alpha}{\alpha}$}}{\raisebox{-0.5pt}{$\tfrac{1+\alpha}{\alpha}$}}\
,\quad\alpha\longmapsto\alpha^{\prime}:=-\mathchoice{\raisebox{0.25pt}{$\dfrac{1}{\alpha+1}$}}{\raisebox{0.25pt}{$\dfrac{1}{\alpha+1}$}}{\raisebox{-0.5pt}{$\tfrac{1}{\alpha+1}$}}{\raisebox{-0.5pt}{$\tfrac{1}{\alpha+1}$}}\
,\quad\alpha\longmapsto\alpha^{\prime}:=-\mathchoice{\raisebox{0.25pt}{$\dfrac{\alpha}{\alpha+1}$}}{\raisebox{0.25pt}{$\dfrac{\alpha}{\alpha+1}$}}{\raisebox{-0.5pt}{$\tfrac{\alpha}{\alpha+1}$}}{\raisebox{-0.5pt}{$\tfrac{\alpha}{\alpha+1}$}}.$

In what follows, I’ll prove the old conjecture The Lie superalgebras
$\widetilde{\mathfrak{svect}}(0|2n+1)$ for $n\geq 1$ are the only (bar
${\mathfrak{spe}}(3)$, see eqs. (20)) deformations of simple Lie superalgebras
with an odd parameter.

####  6.4.1. Exceptional simple Lie superalgebras: realizations

The irreducible modules of least dimension over ${\mathfrak{ag}}(2)$,
${\mathfrak{ab}}(3)$, and ${\mathfrak{osp}}_{a}(4|2)$ for $a$ generic are the
adjoint ones. Therefore, for the best interpretation of these algebras, see
[Eldi].

Consider ${\mathfrak{osp}}_{a}(4|2)$ for the exceptional values of $a$. Kac
was, probably, the first to establish their irreducibles of least dimension in
an unpublished preprint [K?].

For $a=1$ (and for $a=-2$ and $a=-\frac{1}{2}$), the Lie superalgebra
${\mathfrak{osp}}_{a}(4|2)$ is isomorphic to ${\mathfrak{osp}}(4|2)$ and its
irreducible module of least dimension is (up to the change of parity) the
$(4|2)$-dimensional tautological module in which the non-degenerate even
symmetric bilinear form is preserved.

For ${\mathfrak{osp}}_{2}(4|2)$ and ${\mathfrak{osp}}_{3}(4|2)$ taken in
realization with respective Cartan matrices

$\begin{pmatrix}0&1&-3\\\ -1&0&-2\\\ -3&2&0\\\
\end{pmatrix}\text{~{}~{}and~{}~{}}\begin{pmatrix}0&1&0\\\ -1&2&-3\\\
0&-1&2\\\ \end{pmatrix}$

consider the weights of their irreducibles relative their Chevalley generators
$H_{i}:=[X_{i}^{+},X_{i}^{-}]$.

For $a=2$, as well as for $a=-3$, $\frac{1}{2}$, $-\frac{3}{2}$,
$-\frac{1}{3}$ and $-\frac{2}{3}$, the irreducible module $V$ of least
dimension is (up to the change of parity) the $(6|4)$-dimensional module with
an even highest weight vector of weight $(0,0,2)$.

For $a=3$, as well as for $a=-4$, $\frac{1}{3}$, $-\frac{4}{3}$,
$-\frac{1}{4}$ and $-\frac{3}{4}$, the irreducible module $W$ of least
dimension is (up to the change of parity) the $(6|8)$-dimensional module with
an even highest weight vector of weight $(2,3,0)$.

Interestingly, the exceptional algebras are somehow related, see [GL4]:

${\mathfrak{ag}}(2)={\mathfrak{osp}}_{3}(4|2)\oplus\Pi(W);\ \
{\mathfrak{ab}}(3)=({\mathfrak{osp}}_{2}(4|2)\oplus{\mathfrak{sl}}(2))\bigoplus(V\boxtimes
R(\omega_{1})),$

where $R(\omega_{1})$ is the tautological ${\mathfrak{sl}}(2)$-module.

##  7\. Description of deformations

######  Theorem (Classification of deformations with odd parameters).

The only (minding isomorphism
${\mathfrak{svect}}(0|3)\simeq{\mathfrak{spe}}(3)$) simple finite-dimensional
Lie superalgebras over ${\mathbb{C}}$ having deformations with odd parameters
are ${\mathfrak{g}}={\mathfrak{svect}}(0|2n+1)$ for $n\geq 1$ and
$\mathop{\mathrm{sdim}}\nolimits H^{2}({\mathfrak{g}};{\mathfrak{g}})=0|1$.

This theorem was conjectured in 1976 by Bernstein and me, and M. Duflo as he
told me in 1986; this conjecture was never published. In this section, I’ll
prove it. For the vectorial Lie superalgebras, I know only one method to prove
this conjecture; it requires an ingredient published in the paper [GLP]. This
section contains also the details of the proof and description of deformations
with even parameters; almost all cases were partly described earlier by other
researchers.

######  Theorem (Classification of deformations with even parameters).

The only (minding isomorphisms (20)) simple finite-dimensional Lie
superalgebras over ${\mathbb{C}}$ having deformations with even parameters are
${\mathfrak{g}}={\mathfrak{svect}}(0|2n)$ for $n\geq 2$,
${\mathfrak{h}}^{\prime}(0|n)$ for $n\geq 5$, and ${\mathfrak{osp}}(4|2)$. In
all these cases, $\dim H^{2}({\mathfrak{g}};{\mathfrak{g}})=1$. The respective
deforms are ${\mathfrak{g}}=\widetilde{{\mathfrak{svect}}}(0|2n)$,
${\mathfrak{psl}}(2^{n-1}|2^{n-1})$ for $n=2k$ and ${\mathfrak{psq}}(2^{n-1})$
for $n=2k-1$, the deforms being isomorphic for any non-zero value of
parameter; for isomorphisms of ${\mathfrak{osp}}_{\alpha}(4|2)$ for distinct
$\alpha$, see eq. (21).

###  7.3. General facts on cohomology of Lie algebras and Lie superalgebras

See [Kn, pp.288–289] and [Fu, Ch.1].

###  7.4. Lie superalgebras ${\mathfrak{g}}(A)$ with indecomposable Cartan
matrix $A$

Kac was the first to observe that some of simple Lie superalgebras have Cartan
matrices, and that one Lie superalgebra can have several Cartan matrices, see
[K2]. For a correct definition of Cartan matrices, and how to list all Cartan
matrices of a given Lie superalgebra, see [CCLL, BLS, BLLoS]. For the Lie
superalgebra ${\mathfrak{g}}={\mathfrak{g}}(A)$, if $A$ is invertible, the
proof of the fact “${H^{2}({\mathfrak{g}};{\mathfrak{g}})=0}$” is the same as
for simple Lie algebras (see [Ru]) with the help of quadratic Casimir element
corresponding to the even non-degenerate invariant symmetric bilinear form
(even NIS) on ${\mathfrak{g}}(A)$; for the description of such forms on Lie
superalgebras ${\mathfrak{g}}(A)$, see [BKLS].

However, ${\mathfrak{psl}}(n|n)$ has no Cartan matrix, while any of several
Cartan matrices of ${\mathfrak{gl}}(n|n)$ is not invertible.

If $A$ is not invertible, we have to consider the simple subquotient
${\mathfrak{g}}^{(1)}(A)/{\mathfrak{c}}$ of ${\mathfrak{g}}(A)$, and the proof
is still the same with the help of the quadratic Casimir element corresponding
to the NIS induced from the NIS on ${\mathfrak{g}}(A)$.

###  7.5. Lie superalgebras ${\mathfrak{g}}={\mathfrak{psq}}(n)$ for $n>2$

To prove that $H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$, we use the Hochshild-
Serre spectral sequence (see [Fu, Ch.1, no. 5]) with
$E_{2}^{i,j}=H^{j}({\mathfrak{g}}_{\bar{0}};S^{i}({\mathfrak{g}}_{\bar{1}}))=0$
for $i,j\geq 0$ and ${i+j=2}$, see [Fu]. Since $H^{j}({\mathfrak{sl}}(n);M)=0$
for any $j$ and any non-trivial (recall: finite-dimensional)
${\mathfrak{sl}}(n)$-module $M$, we only have one non-zero component:
$H^{0}({\mathfrak{g}}_{\bar{0}};S^{2}({\mathfrak{g}}_{\bar{1}}))=H^{0}({\mathfrak{sl}}(n))={\mathbb{C}}$,
because $S^{2}({\mathfrak{sl}}(n))$ contains the trivial
${\mathfrak{sl}}(n)$-module, see [OV, Table 5].

###  7.6. Lie superalgebras ${\mathfrak{g}}={\mathfrak{spe}}(n)$ for $n>3$

Observe, that the title of the paper [FSS] is misleading: the paper considers
$q$-quantized algebra $U_{q}({\mathfrak{pe}}(n))$, not deformations of
${\mathfrak{pe}}(n)$ or ${\mathfrak{spe}}(n)$. I’ll consider the case
${\mathfrak{spe}}(3)\simeq{\mathfrak{svect}}(0|3)$ while studying
${\mathfrak{svect}}(0|2n-1)$.

####  7.6.1. Induced and coinduced modules.

Let ${\mathfrak{h}}\subset{\mathfrak{g}}$ be a Lie superalgebra and its
subsuperalgebra, $M$ an ${\mathfrak{h}}$-module. The induced and coinduced
${\mathfrak{g}}$-modules are constructed from $M$ by setting

(23)
$\mathop{\mathrm{Ind}}\nolimits^{\mathfrak{g}}_{\mathfrak{h}}(M):=U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}M,\quad\quad\mathop{\mathrm{Coind}}\nolimits^{\mathfrak{g}}_{\mathfrak{h}}(M):=\mathop{\mathrm{Hom}}\nolimits_{U({\mathfrak{h}})}(U({\mathfrak{g}}),M).$

Clearly, $U({\mathfrak{g}})$ is both left and right ${\mathfrak{h}}$-module;
hence, $\mathop{\mathrm{Ind}}\nolimits^{\mathfrak{g}}_{\mathfrak{h}}(M)$ and
$\mathop{\mathrm{Coind}}\nolimits^{\mathfrak{g}}_{\mathfrak{h}}(M)$ are
${\mathfrak{g}}$-modules.

For any ${\mathfrak{h}}$-module $M$, we have ([Fu, Th. 1.5.4])

(24)
$H^{q}({\mathfrak{g}};\mathop{\mathrm{Coind}}\nolimits^{\mathfrak{g}}_{\mathfrak{h}}(M))\simeq
H^{q}({\mathfrak{h}};M);\quad\quad
H_{q}({\mathfrak{g}};\mathop{\mathrm{Ind}}\nolimits^{\mathfrak{g}}_{\mathfrak{h}}(M))\simeq
H_{q}({\mathfrak{h}};M).$

Note that for the Lie superalgebra ${\mathfrak{g}}={\mathfrak{vect}}(0|a)$,
where $a\geq 2$, or ${\mathfrak{svect}}(0|b)$, where $b\geq 3$, or
${\mathfrak{h}}(0|c)$, where $c\geq 4$, the subalgebra
${\mathfrak{g}}_{>0}:=\oplus_{i\geq 0}{\mathfrak{g}}_{i}$ is generated by
${\mathfrak{g}}_{1}$, except for ${\mathfrak{h}}(0|n)$; it is
$({\mathfrak{h}}(0|c)^{\prime})_{>0}$ which is generated by
${\mathfrak{g}}_{1}$.

For any ${\mathbb{Z}}$-graded simple vectorial Lie superalgebras
${\mathfrak{g}}$ on ${\mathcal{C}}^{0|n}$, let the ${\mathfrak{g}}_{0}$-module
$M$ be such that ${\mathfrak{g}}_{>}M=0$, so $M$ can be considered as
${\mathfrak{g}}_{\geq 0}$-module, where ${\mathfrak{g}}_{\geq
0}:=\oplus_{i\geq 0}{\mathfrak{g}}_{i}$. The ${\mathfrak{g}}$-module
$T(M):=\mathop{\mathrm{Coind}}\nolimits^{\mathfrak{g}}_{{\mathfrak{g}}\geq
0}(M)$ is interpreted as the space of tensor fields on ${\mathcal{C}}^{0|n}$ —
sections of the vector bundle with fiber $M$. For any ${\mathfrak{g}}$-module
$V$, let $V^{\mathfrak{g}}$ denote the submodule of
${\mathfrak{g}}$-invariants. We have

(25) $H^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}};T(M))\stackrel{{\scriptstyle\eqref{h2}}}{{\simeq}}H^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}}_{\geq
0};M)\stackrel{{\scriptstyle\text{[Fu]}}}{{\simeq}}H^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}}_{0})\otimes(H^{\mathbin{\hbox{\raise
1.20552pt\hbox{\bf.}}}}({\mathfrak{g}}_{>0})\otimes M)^{{\mathfrak{g}}_{0}}.$

For the proof of the second isomorphism in (25), see Th. 2.2.8 (due to Losik)
in Fuchs’ book [Fu]. It is given there for the Lie algebra
${\mathfrak{g}}={\mathfrak{vect}}(n|0)={\mathfrak{der}}\,{\mathbb{C}}[x]$
only, where $x=(x_{1},\dots,x_{n})$ are even indeterminates, but the proof can
be directly translated to any Cartan prolongs
${\mathfrak{g}}:=({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})_{*}$ and
conjecturally — we do not need this in this paper — to the Lie (super)algebras
${\mathfrak{g}}:=\oplus_{i\geq-d}{\mathfrak{g}}_{i}$ of depth $d>1$; for the
most clear definition of prolong, see [Shch, BGLLS]. The proof of (25) uses
the Hochshield-Serre spectral sequence for the algebra ${\mathfrak{g}}_{\geq
0}$ and its ideal ${\mathfrak{g}}_{>0}$; it is proved that $E_{\infty}=E_{2}$.

####  7.6.2. The long exact sequence.

Let ${\mathfrak{g}}$ be a Lie superalgebra and let

(26) $0\longrightarrow
A\buildrel\partial_{0}\over{\longrightarrow}C\buildrel\partial_{1}\over{\longrightarrow}B\longrightarrow
0,\;\;\text{~{}~{}where~{}~{}}\;\;\;p(\partial_{0})=\bar{0}\;\;\text{ and
}\;\;\;\partial_{1}\;\;\text{ is either even or odd},$

be a short exact sequence of ${\mathfrak{g}}$-modules. Let $d$ be the
differential in the standard cochain complex of ${\mathfrak{g}}$, cf. [Fu,
Subsection 1.3.6].

Consider the long cohomology sequence:

(27)
$\dots\buildrel\partial\over{\longrightarrow}H^{i}({\mathfrak{g}};A)\buildrel\partial_{0}\over{\longrightarrow}H^{i}({\mathfrak{g}};C)\buildrel\partial_{1}\over{\longrightarrow}H^{i}({\mathfrak{g}};B)\buildrel\partial\over{\longrightarrow}H^{i+1}({\mathfrak{g}};A)\buildrel\partial_{0}\over{\longrightarrow}\dots$

where the $\partial_{i}$ are the differentials induced by their namesakes in
(26), and $\partial=d\circ\partial_{1}^{-1}$. Since $\partial_{0}$ and
$\partial_{1}$ commute with $d$, the sequence (26) is well-defined and the
same arguments as for Lie algebras ([Fu]) demonstrate that the long cohomology
sequence (27) induced by (26) is exact, see [LPS].

####  7.6.3. Vectorial Lie superalgebras as coinduced modules.

Let ${\mathfrak{g}}$ be a vectorial Lie superalgebra with the standard
${\mathbb{Z}}$-grading, $\mathbbmss{1}$ be the 1-dimensional trivial
${\mathfrak{g}}_{0}$-module and $\mathbbmss{1}[k]$ the
${\mathfrak{g}}_{0}$-module trivial on the simple part and with value $k$ on
the distinguished central element ($1_{n}\in{\mathfrak{gl}}(n)$). Let the
space of functions — module over any vectorial Lie superalgebra (subalgebra of
${\mathfrak{vect}}(0|n)$) be denoted

${\mathcal{F}}:=T(\mathbbmss{1})=\Lambda(n).$

Let $\mathop{\mathrm{id}}\nolimits_{\mathfrak{h}}:=V$ be the restriction of
the tautological ${\mathfrak{gl}}(V)$-module to
${\mathfrak{h}}\subset{\mathfrak{gl}}(V)$. Then:

(28)
$\begin{array}[]{ll}{\mathfrak{vect}}(0|n)=&\begin{cases}T(\mathop{\mathrm{id}}\nolimits_{{\mathfrak{gl}}(n)})&\text{
as ${\mathfrak{vect}}(0|n)$-module};\\\
T(\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(n)})&\text{ as
${\mathfrak{svect}}(0|n)$-module};\end{cases}\\\
{\mathfrak{po}}(0|n)=&T(\mathbbmss{1})={\mathcal{F}}\text{ as an
${\mathfrak{h}}(0|n)$- and ${\mathfrak{h}}(0|n)^{\prime}$-modules.}\\\
\end{array}$

Looking at the explicit relations in ${\mathfrak{g}}_{>}$, see [GLP], we get
$H^{2}({\mathfrak{g}}_{>})$. Clearly,
$H^{1}({\mathfrak{g}}_{>})={\mathfrak{g}}_{1}$.

$\bullet$ For ${\mathfrak{v}}:={\mathfrak{vect}}(0|n)$.

(29)
$H^{2}({\mathfrak{v}};{\mathfrak{v}})\simeq\oplus_{a+b=2}H^{a}({\mathfrak{gl}}(n))\otimes(H^{b}({\mathfrak{v}}_{>})\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{gl}}(n)})^{{\mathfrak{gl}}(n)}.$

Since $H^{a}({\mathfrak{gl}}(n))\neq 0$ for $a=0,1$, we have to consider
$(H^{b}({\mathfrak{v}}_{>})\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{gl}}(n)})^{{\mathfrak{gl}}(n)}$
for $b=2$ and $1$, respectively. Looking at the explicit weights of generators
($b=1$) and relations ($b=2$) in ${\mathfrak{v}}_{>}$ given in [GLP], we
conclude that
$(H^{b}({\mathfrak{v}}_{>})\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{gl}}(n)})^{{\mathfrak{gl}}(n)}=0$,
so ${\mathfrak{vect}}(0|n)$ is rigid.

$\bullet$ For ${\mathfrak{s}}:={\mathfrak{svect}}(0|n)$. We have an exact
sequence of ${\mathfrak{s}}$-modules:

(30)
$0\longrightarrow{\mathfrak{s}}\longrightarrow{\mathfrak{v}}\stackrel{{\scriptstyle\mathop{\text{\rm
div}}\nolimits}}{{\longrightarrow}}\mathop{\mathrm{{Vol\/}}}\nolimits^{0}\longrightarrow
0.$

In this case, the divergence is not a surjective mapping onto ${\mathcal{F}}$,
its image is the space $\mathop{\mathrm{{Vol\/}}}\nolimits^{0}$ of functions
(coefficients of $\mathop{\mathrm{{vol\/}}}\nolimits$) with integral 0. In the
associated long exact sequence

(31) $\ldots\longrightarrow
H^{1}({\mathfrak{s}};{\mathfrak{s}})\longrightarrow
H^{1}({\mathfrak{s}};{\mathfrak{v}})\longrightarrow
H^{1}({\mathfrak{s}};\mathop{\mathrm{{Vol\/}}}\nolimits^{0})\longrightarrow
H^{2}({\mathfrak{s}};{\mathfrak{s}})\longrightarrow
H^{2}({\mathfrak{s}};{\mathfrak{v}})\longrightarrow\ldots$

let us compute

(32)
$H^{i}({\mathfrak{s}};{\mathfrak{v}})\simeq\sum_{a+b=i}H^{a}({\mathfrak{sl}}(n))\otimes(H^{b}({\mathfrak{s}}_{>})\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak{sl}}(n)}))^{{\mathfrak{sl}}(n)}\text{
for $i=1,2$ and $n>1$};$

Since $H^{1}({\mathfrak{s}};{\mathfrak{v}})=0$ and
$H^{2}({\mathfrak{s}};{\mathfrak{v}})=0$ by (24), (28) and also by plugging
the weights of generators of ${\mathfrak{svect}}(0|n)_{>0}$ computed in [GLP]
in (32) for $b=1$, we see that $H^{2}({\mathfrak{s}};{\mathfrak{s}})\cong
H^{1}({\mathfrak{s}};\mathop{\mathrm{{Vol\/}}}\nolimits^{0})$. To compute it,
consider the short exact sequence

(33)
$0\longrightarrow\mathop{\mathrm{{Vol\/}}}\nolimits^{0}\longrightarrow{\mathcal{F}}\stackrel{{\scriptstyle\int}}{{\longrightarrow}}{\mathbb{C}}\longrightarrow
0.$

It gives rise to the long exact sequence

(34) $\begin{array}[]{l}0\longrightarrow
H^{0}({\mathfrak{s}};\mathop{\mathrm{{Vol\/}}}\nolimits^{0})\longrightarrow
H^{0}({\mathfrak{s}};{\mathcal{F}})\longrightarrow
H^{0}({\mathfrak{s}})\longrightarrow
H^{1}({\mathfrak{s}};\mathop{\mathrm{{Vol\/}}}\nolimits^{0})\longrightarrow
H^{1}({\mathfrak{s}};{\mathcal{F}})\longrightarrow\ldots\end{array}$

and since

$\begin{array}[]{l}H^{0}({\mathfrak{s}};\mathop{\mathrm{{Vol\/}}}\nolimits^{0})\simeq{\mathbb{C}},\quad
H^{0}({\mathfrak{s}};{\mathcal{F}})\simeq
H^{0}({\mathfrak{sl}})\simeq{\mathbb{C}};\quad
H^{1}({\mathfrak{s}};{\mathcal{F}})=H^{0}({\mathfrak{sl}})={\mathbb{C}},\end{array}$

it follows that
$H^{1}({\mathfrak{s}};\mathop{\mathrm{{Vol\/}}}\nolimits^{0})\simeq
H^{0}({\mathfrak{s}})\simeq\Pi^{n}{\mathbb{C}}$. Since we already know one
global deformation, there are no more.

###  7.7. What was known about deformations of simple vectorial Lie
superalgebras

For ${\mathfrak{h}}^{\prime}(0|n)$, and for the Lie superalgebra
${\mathfrak{svect}}(0|n)$ of divergence-free vector fields, the deformations
were computed for $n$ small by N.V.D. Hijligenberg, Yu. Kotchetkov, and G.
Post, see [HKP].

The result for ${\mathfrak{h}}^{\prime}(0|n)$ is double-checked being a
corollary of the description of deformations of the Poisson Lie superalgebras
(physicists call it quantization) obtained by I. Tyutin [Tyut]. Observe that
the coincidence of the number of non-trivial deformations of
${\mathfrak{po}}(0|n)$ and ${\mathfrak{h}}(0|n)$ is not automatic: e.g., cf.
the answers for ${\mathfrak{po}}(2a|b)$ and ${\mathfrak{h}}(2a|b)$ for
$(2a,b)=(2,2)$, see [LSh3]. I was unable to understand the idea of the proof
in [Tyut], so I have to use another method, the one described in Subsection
7.6.3, see Section 7.8.

####  7.7.1. Filtered deformations.

Let ${\mathfrak{g}}=\oplus{\mathfrak{g}}_{i}$ be a ${\mathbb{Z}}$-graded Lie
superalgebra and ${\mathfrak{G}}$ a filtered Lie superalgebra with decreasing
filtration such that
$\mathop{\mathrm{gr}}\nolimits{\mathfrak{G}}\simeq{\mathfrak{g}}$. Then, the
elements of $(H^{2}({\mathfrak{g}}_{-};{\mathfrak{g}}))^{{\mathfrak{g}}_{0}}$
of ${\mathfrak{g}}_{0}$-invariants in
$H^{2}({\mathfrak{g}}_{-};{\mathfrak{g}})$ describe the filtered deformations
of ${\mathfrak{g}}$ to ${\mathfrak{G}}$, see [CK1, CK1a]. The filtered
deformations of ${\mathfrak{svect}}(0|2n)$ and ${\mathfrak{h}}^{\prime}(0|n)$
are described in [Po2]. Observe that the filtered deformation describes only
the deformation corresponding to one ${\mathbb{Z}}$-grading and only if the
negative part in this grading is non-zero. To describe all deformations
remained an open problem which is closed in this paper.

###  7.8. Deformations of ${\mathfrak{h}}^{\prime}(0|n)$ ([Tyut, LSh3])

The answer for ${\mathfrak{h}}^{\prime}(0|n)$ can be double-checked as a
corollary of the description of the deformations of the Poisson Lie
superalgebra ${\mathfrak{po}}(0|n)$, see [Tyut], taking into account the exact
sequences (18) and

(35)
$0\longrightarrow{\mathfrak{h}}^{\prime}(0|n)\longrightarrow{\mathfrak{h}}(0|n)\longrightarrow{\mathbb{C}}H_{\theta_{1}\dots\theta_{n}}\longrightarrow
0.$

######  Theorem (Deformations of ${{\mathfrak{h}}^{\prime}}(0|n)$ ([LSh3])).

The quantizations

${\mathfrak{po}}(0|2n)\longrightarrow{\mathfrak{gl}}(\Lambda(n))$ and
${\mathfrak{po}}(0|2n-1)\longrightarrow{\mathfrak{q}}(\Lambda(n))$

induce deformations

${\mathfrak{h}}^{\prime}(0|2n)\longrightarrow{\mathfrak{psl}}(\Lambda(n))$ and
${\mathfrak{h}}^{\prime}(0|2n-1)\longrightarrow{\mathfrak{psq}}(\Lambda(n))$.

##  8\. Summary of ideas in Kac’s proof ([K2])

Notation below are same as in [K2] with translation into notation of §2 and
information on discoverers and first partial classifications from [Kapp,
Newsletters], [NRS]: For example, the simple Lie superalgebras
${\mathfrak{g}}$ with simple ${\mathfrak{g}}_{\bar{0}}$ were classified in
[Dj2, DjH]; there are 4 series of which members of the 3rd and 4th series are
actually isomorphic. In modern notation, they are: ${\mathfrak{osp}}(1|2n)$,
${\mathfrak{psq}}(n)$,
${\mathfrak{spe}}^{a}(n)\simeq{\mathfrak{spe}}^{sy}(n)$.

######  Theorem (Kac’s Main Theorem).

([K2, Theorem 5 on p. 74]) Any simple finite-dimensional Lie superalgebra
${\mathfrak{g}}$ over an algebraically closed field ${\mathbb{K}}$ of
characteristic $0$ is either isomorpic to a simple Lie algebra, or to one of
the following Lie superalgebras:

$A(m,n)$ (=${\mathfrak{sl}}(m+1|n+1)$, where $1\leq m<n$, or
${\mathfrak{p}}{\mathfrak{sl}}(n+1|n+1)$, where $n\geq 1$),

$B(m,n)$ (=${\mathfrak{osp}}(2m+1|2n)$, where $m\geq 0,n>0$),

$C(n)$ (=${\mathfrak{osp}}(2|2n-2)$, where $n\geq 2$),

$D(m,n)$ (=${\mathfrak{osp}}(2m|2n)$, where $m\geq 2,n>0$),

$D(2,1;\alpha)$ ($={\mathfrak{osp}}(4|2;\alpha)$; discovered as
$\Gamma(A,B,C)$, see [Kapp]),

$F(4)$ (=${\mathfrak{ab}}(3)$, see [Se4]; discovered as $\Gamma_{3}$, see
[Kapp]),

$G(3)$ (=${\mathfrak{ag}}(2)$, see [Se4]; discovered as $\Gamma_{2}$, see
[Kapp]),

$P(n)$ (=${\mathfrak{spe}}(n+1)$, where $n\geq 2$),

$\widetilde{Q}(n)$, $Q(n)$ (=${\mathfrak{sq}}(n+1)$, ${\mathfrak{psq}}(n+1)$,
where $n\geq 2$),

$W(n)$ (=${\mathfrak{vect}}(0|n)$, where $n\geq 2$; discovered in [K0]),

$S(n)$ (=${\mathfrak{svect}}(0|n)$, where $n\geq 3$; discovered in [K0]),

$\widetilde{S}(2n)$ (=$\widetilde{\mathfrak{svect}}(0|2n)$, where $n\geq 2$),

$\widetilde{H}(n)$, $H(n)$ (=${\mathfrak{h}}(0|n)$,
${\mathfrak{h}}^{\prime}(0|n)$; discovered in [K0]).

###### Ideas of the proof of Kac’s Main Theorem.

Let ${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}$.

First, let us look at the ${\mathfrak{g}}_{\bar{0}}$-action on
${\mathfrak{g}}_{\bar{1}}$. If this action is irreducible, then
${\mathfrak{g}}$ is “classical” (in Kac’s terms) and isomorphic to one of the
following: $B(m,n)$, $D(m,n)$, $F(4)$, $G(3)$, $Q(n)$ or $D(2,1;\alpha)$. This
is proved in Theorem 2 on p. 44. Proof is performed by means of the
representation theory of semisimple Lie algebras.

If the ${\mathfrak{g}}_{\bar{0}}$-action on ${\mathfrak{g}}_{\bar{1}}$ is
reducible, we can take a maximal proper subalgebra
$L_{0}\subset{\mathfrak{g}}$ containing ${\mathfrak{g}}_{\bar{0}}$ (there can
be several such subalgebras, take any of them). The subalgebra $L_{0}$ induces
a filtration of ${\mathfrak{g}}$:

${\mathfrak{g}}=L_{-1}\supset L_{0}\supset L_{1}\supset\dots$

Set $\mathop{\mathrm{gr}}\nolimits
L:=\oplus_{i\geq-1}\mathop{\mathrm{gr}}\nolimits_{i}L$.

Then, (Proposition 1.3.2, p. 24) $\mathop{\mathrm{gr}}\nolimits L$ has the
following properties:

1\. $\mathop{\mathrm{gr}}\nolimits L$ is transitive (this is a direct
corollary of simplicity of ${\mathfrak{g}}$);

2\. ${\mathbb{Z}}$-grading of $\mathop{\mathrm{gr}}\nolimits L$ is compatible
with ${\mathbb{Z}}/2$-grading a.k.a. parity (this is a corollary of the fact
$\mathop{\mathrm{gr}}\nolimits_{-1}L\subset\mathop{\mathrm{gr}}\nolimits
L_{\bar{1}}$ because $L_{0}\supset{\mathfrak{g}}_{\bar{0}}$).

3\. $\mathop{\mathrm{gr}}\nolimits L$ is irreducible, i.e.,
$\mathop{\mathrm{gr}}\nolimits_{0}L$ irreducibly acts on
$\mathop{\mathrm{gr}}\nolimits_{-1}L$ (since $L_{0}$ is maximal).

4\. $\mathop{\mathrm{gr}}\nolimits_{1}L\neq 0$ (since
${\mathfrak{g}}_{\bar{0}}$-action on ${\mathfrak{g}}_{\bar{1}}$ is reducible).

Theorem 4, p. 71 claims that any ${\mathbb{Z}}$-graded Lie superalgebra
satisfying the above conditions 1–4 is one of the following:

I. $A(m,n),C(n),P(n)$;

II. $W(n),S(n),\widetilde{H}(n),H(n)$;

III. $H^{\xi}=H\otimes\Lambda(\xi)+{\mathbb{K}}\cdot\partial_{\xi}$, where $H$
is a simple Lie algebra, with the grading given by setting $\deg\xi=-1$.

IV. ${\mathfrak{g}}+{\mathbb{K}}\cdot d$, where ${\mathfrak{g}}$ is of type I,
II, III, the center of ${\mathfrak{g}}_{\bar{0}}$ is trivial and $d$ is the
grading operator.

Let the ${\mathfrak{g}}_{\bar{0}}$-action on ${\mathfrak{g}}_{\bar{1}}$ be
reducible. Then, Theorem 4 implies Theorem 5 by case-by-case checking of these
four possibilities for $\mathop{\mathrm{gr}}\nolimits L$.

III. If $\mathop{\mathrm{gr}}\nolimits L=H^{\xi}$, then ${\mathfrak{g}}$ is
not simple. To prove this Kac uses Proposition 2.2.2, p. 35. In this
particular case, one can directly see why:
$L_{0}=H\oplus{\mathbb{K}}\cdot\partial_{\xi}$ (as a linear space). Hence,
${\mathfrak{g}}_{\bar{0}}=H$. Since $H$ is a simple Lie algebra, it acts on
the 1-dimensional space ${\mathbb{K}}\cdot\partial_{\xi}$ by zero, so
${\mathfrak{g}}^{\prime}\neq{\mathfrak{g}}$.

IV. This case, and the case of $\mathop{\mathrm{gr}}\nolimits L=W(n)$ or
$A(m|n)$ or $C(n)$ is dealt with by Proposition 1.3.1, p. 24, which claims

If for a finite-dimensional transitive filtered Lie superalgebra $L$ the
representation of $\mathop{\mathrm{gr}}\nolimits_{0}L$ in
$\mathop{\mathrm{gr}}\nolimits_{-1}L$ is irreducible, and the even part of the
center of $\mathop{\mathrm{gr}}\nolimits_{0}L$ is non-zero, then
$L\simeq\mathop{\mathrm{gr}}\nolimits L$.

This claim is practically obvious. Indeed, let us normalize the element $z$ of
the center of $\mathop{\mathrm{gr}}\nolimits_{0}L$ so as to act as a grading
operator on $\mathop{\mathrm{gr}}\nolimits L$. Then, its pre-image in $L$ is
diagonalizable and defines a grading on $L$ itself.

Thanks to this Proposition, the case IV does not contribute to the
classification: the Lie superalgebra ${\mathfrak{g}}$ is not simple, whereas
if $\mathop{\mathrm{gr}}\nolimits L=W(n)$ or $A(m|n)$ or $C(n)$, then $L=W(n)$
or $A(m|n)$ or $C(n)$, respectively.

The case $\mathop{\mathrm{gr}}\nolimits L=P(n)$ (as well as with many other
cases) is dealt with by Proposition 2.1.4, p. 34 which claims

If ${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}$ is
a simple finite-dimensional Lie superalgebra such that the
${\mathfrak{g}}_{\bar{0}}$ action on ${\mathfrak{g}}_{\bar{1}}$ is the same as
that of one of the Lie superalgebras $A(m,n)$, $B(m,n)$, $C(n)$, $D(m,n)$
where $(m,n)\neq(2,1)$, $F(4)$, $G(3)$, $P(n)$, $Q(n)$, then ${\mathfrak{g}}$
is isomorphic to the corresponding Lie superalgebra.

The hypothesis of this Proposition is obvious if
$\mathop{\mathrm{gr}}\nolimits L=P(n)$ (as well as if
$\mathop{\mathrm{gr}}\nolimits L=A(m|n)$ or $C(n)$): indeed,
$L_{0}=(L_{0})_{\bar{0}}\oplus(L_{0})_{\bar{1}}$, where
$(L_{0})_{\bar{0}}=\mathop{\mathrm{gr}}\nolimits_{0}L=L_{\bar{0}}$ and
$(L_{0})_{\bar{1}}=\mathop{\mathrm{gr}}\nolimits_{1}L$. Since
$\mathop{\mathrm{gr}}\nolimits_{0}L=L_{\bar{0}}$ is reductive, any its finite-
dimensional representation is completely reducible, in particular, the
$L_{\bar{0}}$-action on $L_{\bar{1}}$.

The key point in the proof of Proposition 2.1.4 is Proposition 2.1.3, which
claims that for the Lie superalgebras in question the
${\mathfrak{g}}_{\bar{0}}$-module $S^{2}({\mathfrak{g}}_{\bar{1}})$ contains
${\mathfrak{g}}_{\bar{0}}$ with multiplicity 1.

The cases of $\mathop{\mathrm{gr}}\nolimits L=\widetilde{H}(n)$ or $H(n)$ are
considered in Proposition 3.3.7, p. 66. Since ${\mathfrak{h}}(0|n)$ is not
simple, there remains only $H(n)^{\prime}$.

Finally, the case of $\mathop{\mathrm{gr}}\nolimits L=S(n)$ is dealt with by
Theorem 3.3.5, p. 64. In both these cases, the following two arguments are key
ones.

First, Proposition 3.1.3, p.59 which claims that if $L=L_{-1}\supset
L_{0}\supset\dots$ is a transitive filtered Lie superalgebra such that
$L_{0}\supset L_{\bar{0}}$ and $\dim L_{-1}/L_{0}=(0|n)$, then $L$ can be
embedded into $W(n)$.

Second, the Levi-Malcev theorem for Lie algebras claiming that any semisimple
subalgebra of the Lie algebra ${\mathfrak{g}}$ can by means of conjugations be
embedded into a fixed Levi subalgebra (i.e., the complement to the radical) of
${\mathfrak{g}}$. Hence, in the case where $\mathop{\mathrm{gr}}\nolimits
L=\widetilde{H}(n)$ or $H(n)$, the Lie superalgebra $L$ contains
${\mathfrak{o}}(n)$, whereas in the case where $\mathop{\mathrm{gr}}\nolimits
L=S(n)$, the Lie superalgebra $L$ contains ${\mathfrak{sl}}(n)$. The rest is a
routine application of the representation theory of simple Lie algebras. ∎

Acknowledgements. I am thankful to J. Bernstein, D. Fuchs, A. Lebedev, Yu.
Manin, I. Shchepochkina, and A. Tikhomirov for education, P. Grozman and A.
Krutov for help; to the grant AD 065 NYUAD for financial support.

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