# Graded covering of a supermanifold I.
The case of a Lie supergroup

Mikolaj Rotkiewicz and Elizaveta Vishnyakova

###### Abstract.

We generalize the Donagi and Witten construction of a first obstruction class
for splitting of a supermanifold via differential operators using the theory
of $n$-fold vector bundles and graded manifolds. Applying the generalized
Donagi–Witten construction we obtain a family of embeddings of the category of
supermanifolds into the category of $n$-fold vector bundles and into the
category of graded manifolds. This leads to a realization of any non-split
supermanifold in terms of a collection of vector bundles and some morphism
between them. Further we study the images of these embeddings into the
category of graded manifolds in the case of a Lie supergroup and a Lie
superalgebra. We show that these images satisfy universal property of a graded
covering or a graded semicovering.

##  1\. Introduction

This is the first part of a series of papers about $\mathbb{Z}$-graded
coverings of a supermanifold. In this paper we investigate the case of a Lie
supergroup and its Lie superalgebra. In [DW2, Section 2] Donagi and Witten
gave a description of the first obstruction class $\omega_{2}$ for a
supermanifold to be split via differential operators. More precisely, Donagi
and Witten constructed an exact sequence of locally free sheaves with Atiyah
class $\omega_{2}$. This is a very interesting and important observation that
pure even vector bundles can keep certain information about the original (non-
split) supermanifold. In the case a supermanifold has odd dimension $2$, the
exact sequence of Donagi and Witten keeps the whole information of this
supermanifold.

In this paper we generalize this idea. We show that the Donagi and Witten
exact sequence corresponds to a double vector bundle with some additional
structure, the whole information of which can be reduced to a graded manifold
of degree $2$. Further we suggest to use a dual approach which leads to
iterated differential forms, i.e. functions on the iterated antitangent bundle
(as it is explained below) instead of differential operators, which leads to a
simplification of the Donagi and Witten construction and suggests a natural
generalization. For any (non-split) supermanifold $\mathcal{M}$ of odd
dimension $\leq n$ we construct a pure even geometric object, an $n$-fold
vector bundle $\mathrm{Vb}_{n}(\mathcal{M})$ which keeps the whole information
about the original supermanifold. Moreover the obtained $n$-fold vector bundle
possesses additional symmetries. We interpret this in the following way: the
$n$-fold vector bundle arising from a supermanifold, can be reduced to a
graded manifold of degree $n$.

Summing up we construct a family $\mathrm{F}_{n}$, where $n=2,3,\ldots\infty$,
of functors from the category of supermanifolds of odd dimension $n$ to the
category of $n$-fold vector bundles and to the category of graded manifolds of
degree $n$, where the first functor $\mathrm{F}_{2}$ is a modification of the
Donagi–Witten construction. Each functor $\mathrm{F}_{n}$ is a composition of
four functors: the $(n-1)$-iteratation of the antitangent functor
$\mathbf{T}$, the functor split $\mathrm{gr}$ ($\mathrm{gr}\mathcal{M}$
denotes the retract of the supermanifold $\mathcal{M}$) , which is defined in
the category of supermanifolds, the functor parity change $\pi$, which is
defined in the category of $n$-fold vector bundles and another functor
$\iota$, which we call inverse that was discovered independently in [BGR]
(under the name the diagonalization functor) and [Vi1]. We show that the
functor $\mathrm{F}_{n}$ determines an embedding of the category of
supermanifolds of odd dimension $n$ (or smaller) into the category of graded
manifolds of degree $n$. The functor $\mathrm{F}_{\infty}$ is a limit of
functors $\mathrm{F}_{n}$. This functor defines an embedding of the category
of supermanifolds into the category of graded manifolds of any degree. This
means that the graded manifold $\mathrm{F}_{\infty}(\mathcal{M})$, the image
of a (non-split) supermanifold $\mathcal{M}$, contains the whole information
about the original supermanofold $\mathcal{M}$.

Moreover, for a Lie supergroup $\mathcal{G}$ we prove that
$\mathrm{F}_{\infty}(\mathcal{G})$ satisfies universal properties. These
properties of $\mathrm{F}_{\infty}(\mathcal{G})$ led use to introduce the
notion of a graded covering of $\mathcal{G}$. That is: the covering space is
unique up to isomorphism and every homomorphism from a graded Lie supergroup
to the Lie supergroup $\mathcal{G}$ can be lifted to
$\mathrm{F}_{\infty}(\mathcal{G})$. We also introduce a notion of a graded
semicovering of $\mathcal{G}$. This explains the meaning of other functors
$\mathrm{F}_{n}$. Our covering (semicovering) is in some sense a ”local
diffeomorphism” of Lie supergroups. Further we show that the Lie superalgebra
of $\mathrm{F}^{\prime}_{\infty}(\mathfrak{g})$ is an example of a loop
algebra construction by Allison, Berman, Faulkner, and Pianzola [ABFP], see
also [Eld]. The loop algebra was used by several authors to investigate
graded-simple (Lie) algebras. Therefore results of our paper establish a
connection between Donagi–Witten observation and pure algebraic results. Our
method leads to a notion of a graded covering and semicovering of any
supermanifold as well. However due to technical difficulties of all proofs in
this case we present the general theory of covering and semicovering spaces
for any supermanifold in the second part of this research [RV].

Our results are especially interesting in the complex-analytic (and algebraic)
category. The reason is the following. According to the Batchelor–Gawedzki
Theorem any smooth supermanifold is (non-canonically) split, that is its
structure sheaf is isomorphic to the wedge power of a certain vector bundle.
Therefore very often we can study geometry of a split supermanifold using
geometry of vector bundles. This is not the case in the complex-analytic
situation. The study of non-split supermanifolds was initiated in [Ber, Gr],
where the first non-split supermanifold was described. Significant advances in
this direction were achieved in the work of A.L. Onishchik. Interest in this
problem arose again after Donagi and Witten’s papers [DW1, DW2], where they
proved that the moduli space of super Riemann surfaces is non-split (or more
generally not projected). To each (non-split) supermanifold $\mathcal{M}$ our
functor $\mathrm{F}_{n}$ associates a graded manifold
$\mathrm{F}_{n}(\mathcal{M})$ and the image $\mathrm{F}_{n}(\mathcal{M})$
keeps the whole information about the (non-split) supermanifold $\mathcal{M}$
for sufficiently large $n$. Therefore we can study a non-split supermanifold
in the category of graded manifolds using the tools of classical complex
geometry due the fact that geometrically a graded manifold is a family of
vector bundles.

Our method studying a supermanifold is interesting in the case of a split
supermanifold as well, for instance if a split supermanifold possesses an
additional structure. Any Lie supergroup $\mathcal{G}$ is totally split, i.e.
as a supermanifold it is a cartesian product of its underlying space
$\mathcal{G}_{0}$ with a fixed odd vector space $\mathfrak{g}_{\bar{1}}$ — the
odd part of the Lie superalgebra of $\mathcal{G}$. However the supermanifold
$\mathcal{G}$ has an additional structure which can be non-trivial: the Lie
supergroup structure. We show that any graded Lie supergroup
$\mathrm{F}_{n}(\mathcal{G})$ contains the whole information about the
original supergroup $\mathcal{G}$ for any $n$, non only for sufficiently large
$n$. Moreover our result is the first step toward a graded covering of a
homogeneous supermanifold, which is very often non-split. For instance all
(except of few exceptional cases) super-grassmannians and flag supermanifolds
are non-split.

Acknowledgements: M. R. was partially supported by Capes/PrInt, UFMG. E. V.
was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - Brasil (CAPES) – Finance Code 001, (Capes-Humboldt Research
Fellowship) and by Tomsk State University, Competitiveness Improvement
Program. M. R. thanks UFMG Department of Mathematics for its kind hospitality
during the realization of this work. The authors thank Peter Littelmann for
his suggestion to investigate the case of a Lie supergroup and a Lie
superalgebra. This helped us to understand the meaning of the functor
$\mathrm{F}_{\infty}$. We also thank Alexey Sharapov and Mikhail Borovoi for
useful comments.

##  2\. Preliminaries

###  2.1. Lie superalgebras, graded Lie superalgebras and graded Lie
superalgebras of type $\Delta$

Throughout the paper we work over the field $\mathbb{K}=\mathbb{C}$ or
$\mathbb{R}$. However the algebraic part of the paper holds true over any
field of characteristic $0$. We shall work in a supergeometry context. Our
vector spaces are $\mathbb{Z}_{2}$-graded, $V=V_{\bar{0}}\oplus V_{\bar{1}}$,
elements of $V_{\bar{i}}\setminus\\{0\\}$ are called homogenous of _parity_
$\bar{i}\in\mathbb{Z}_{2}$. For $v\in V_{\bar{i}}$ we write $|v|=\bar{i}$. Our
main references on supergeomtery, in particular on Lie superalgebras, are
[Kac, BLMS, Ber, L] and [Var].

######  Definition 1.

A Lie superalgebra is a $\mathbb{Z}_{2}$-graded vector superspace
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ with a
graded bilinear operation $[\,,]$ satisfying the following conditions:

(1) $[x,y]=-(-1)^{|x||y|}[y,x],$ (2)
$[x,[y,z]]-(-1)^{|x||y|}[y,[x,z]]=[[x,y],z],$

where $x$, $y$, and $z$ are homogeneous elements.

In more general setting, if $A$ is an abelian group, an $A$-graded Lie
superalgebra is a vector space $\mathfrak{g}$ graded by an abelian group $A$,
$\mathfrak{g}=\oplus_{a\in A}\mathfrak{g}_{a}$, the bracket on $\mathfrak{g}$
is $A$-graded, i.e.
$[\mathfrak{g}_{a},\mathfrak{g}_{b}]\subset\mathfrak{g}_{a+b}$, and satisfies
a generalized Jacobi identity in which the sign $(-1)^{|x||y|}$ is replaced
with $(-1)^{\delta(a)\delta(b)}$, where $\delta:A\to\mathbb{Z}_{2}$ is a group
homomorphism.

######  Definition 2.

Let $\mathfrak{g}$ be an $A$-graded Lie superalgebra. Then

$supp(\mathfrak{g})=\\{\alpha\in
A\,\,|\,\,\mathfrak{g}_{\alpha}\neq\\{0\\}\\}.$

In this paper most of the time we will work with $\mathbb{Z}^{r}$-graded Lie
superalgebras. Let $\alpha_{1},\ldots,\alpha_{r}$ are formal variables.

######  Definition 3.

A weight system is a pair $(\Delta,\chi)$, where $\Delta$ is a subset in
$\mathbb{Z}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}\alpha_{r}$ satisfying the
following properties

  1. (1)

$0\in\Delta$ and $\alpha_{i}\in\Delta$, where $i=1,\ldots,r$;

  2. (2)

if $\delta\in\Delta$ and $\delta=\sum_{i}a_{i}\alpha_{i}$, where
$a_{i}\in\mathbb{Z}$, then $a_{i}\geqslant 0$,

and $\chi:A\to\mathbb{Z}_{2}$, $\delta\mapsto|\delta|$, is a group
homomorphism. We call $|\delta|$ the parity of $\delta$. A weight system
$(\Delta,\chi)$ is called multiplicity free, if $\delta=\sum
a_{i}\alpha_{i}\in\Delta$ implies $a_{i}\in\\{0,1\\}$.

Let $(\Delta,\chi)$ be a weight system. Then
$\Delta=\Delta_{\bar{0}}\cup\Delta_{\bar{1}}$, where $\Delta_{\bar{0}}$
contains all even weights, while $\Delta_{\bar{1}}$ contains all odd ones.
Note that $0$ is always in $\Delta_{\bar{0}}$. A Lie superalgebra
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ is called a
graded Lie superalgebra of type $(\Delta,\chi)$, if $\mathfrak{g}$ is a
$\mathbb{Z}^{r}$-graded Lie superalgebra of type $\Delta$ and

$\mathfrak{g}_{\bar{0}}=\bigoplus\limits_{\delta\in\Delta_{\bar{0}}}\mathfrak{g}_{\delta},\quad\mathfrak{g}_{\bar{1}}=\bigoplus\limits_{\delta\in\Delta_{\bar{1}}}\mathfrak{g}_{\delta}.$

Very often we will omit $\chi$ and write a Lie superalgebra of type $\Delta$
meaning a Lie superalgebra of type $(\Delta,\chi)$.

###  2.2. Supermanifolds, graded manifolds and $n$-fold vector bundles

####  2.2.1. Supermanifolds

We consider a complex-analytic supermanifold in the sense of Berezin and
Leites [Ber, L], see also [BLMS]. Thus, a supermanifold
$\mathcal{M}=(\mathcal{M}_{0},\mathcal{O}_{\mathcal{M}})$ of dimension $n|m$
is a $\mathbb{Z}_{2}$-graded ringed space that is locally isomorphic to a
superdomain in $\mathbb{C}^{n|m}$. Here the underlying space $\mathcal{M}_{0}$
is a complex-analytic manifold. The dimension $n$ of the underlying manifold
$\mathcal{M}_{0}$ is called even dimension of $\mathcal{M}$, while $m$ is
called odd dimension of $\mathcal{M}$. A morphism $\mathcal{M}\to\mathcal{N}$
between two supermanifolds is a morphism between $\mathbb{Z}_{2}$-graded
ringed spaces, this is, a pair $F=(F_{0},F^{*})$, where
$F_{0}:\mathcal{M}_{0}\to\mathcal{N}_{0}$ is a holomorphic mapping and
$F^{*}:\mathcal{O}_{\mathcal{N}}\to(F_{0})_{*}(\mathcal{O}_{\mathcal{M}})$ is
a homomorphism of sheaves of $\mathbb{Z}_{2}$-graded ringed spaces. A morphism
$F$ is called an isomorphism if $F$ is invertible. A supermanifold
$\mathcal{M}$ is called split, if its structure sheaf is isomorphic to
$\bigwedge\mathcal{E}^{*}$, where $\mathcal{E}$ is a sheaf of sections of a
certain vector bundle $\mathbb{E}[\bar{1}]$. (Here $\mathbb{E}[\bar{1}]$ means
a usual vector bundle $\mathbb{E}$ additionally assumed that its local
sections are odd.) We see that in this case the structure sheaf is
$\mathbb{Z}$-graded. A morphism of split supermanifolds is called split if it
preserves the fixed $\mathbb{Z}$-gradings. Split supermanifolds with split
morphisms form a category of split supermanifolds.

According to the Batchelor–Gawedzki Theorem any smooth supermanifold is split.
This is not true in the complex-analytic case, see [Ber, Gr]. However we can
define a functor from the category of supermanifols to the category of split
supermanifolds. Let $\mathcal{M}=(\mathcal{M}_{0},\mathcal{O})$ be a
supermanifold. (Sometimes we will omit $\mathcal{M}$ in
$\mathcal{O}_{\mathcal{M}}$ if the meaning of $\mathcal{O}$ is clear from the
context.) Then its structure sheaf possesses the following filtration

(3)
$\mathcal{O}=\mathcal{J}^{0}\supset\mathcal{J}\supset\mathcal{J}^{2}\supset\cdots\supset\mathcal{J}^{p}\supset\cdots,$

where $\mathcal{J}$ is the sheaf of ideals generated by odd elements in
$\mathcal{O}$. We define

$\mathrm{gr}(\mathcal{M}):=(\mathcal{M}_{0},\mathrm{gr}\mathcal{O}),$

where

$\mathrm{gr}\mathcal{O}=\bigoplus_{p\geq 0}\mathcal{J}^{p}/\mathcal{J}^{p+1}.$

The supermanifold $\mathrm{gr}(\mathcal{M})$ is split, that is its structure
sheaf is isomorphic to $\bigwedge\mathcal{E}^{*}$, where
$\mathcal{E}^{*}=\mathcal{J}/\mathcal{J}^{2}$ is a locally free sheaf. Since
any morphism $F$ of supermanifolds preserves the filtration (3), the morphism
$\mathrm{gr}(F)$ is defined. Summing up, the functor $\mathrm{gr}$ is a
functor from the category of supermanifolds to the category of split
supermanifolds, see for example [Vi2, Section 3.1] for details. We can apply
the functor $\mathrm{gr}$ to a Lie supergroup $\mathcal{G}$ and we will get a
split Lie supergroup $\mathrm{gr}(\mathcal{G})$. Later we also will define a
corresponding functor $\mathrm{gr}^{\prime}$ for Lie superalgebras so that
$\operatorname{Lie}\circ\mathrm{gr}(\mathcal{G})=\mathrm{gr}^{\prime}\circ\operatorname{Lie}(\mathcal{G})$
for any Lie supergroup $\mathcal{G}$.

####  2.2.2. Graded manifolds of type $\Delta$, graded manifolds of degree $n$
and $r$-fold vector bundles

Let us start with a notion of a graded manifold of type $(\Delta,\chi)$, where
$(\Delta,\chi)$ is as in Definition 3. Again very often we will omit $\chi$,
and write a graded manifold of type $\Delta$. Let

$V=\bigoplus_{\delta\in\Delta}V_{\delta},\quad
V_{\bar{0}}=\bigoplus_{\delta\in\Delta_{\bar{0}}}V_{\delta},\quad
V_{\bar{1}}=\bigoplus_{\delta\in\Delta_{\bar{1}}}V_{\delta}.$

Consider a $\mathbb{Z}^{r}$-graded ringed space
$\mathcal{U}=(\mathcal{U}_{0},\mathcal{O}_{\mathcal{U}})$, where
$\mathcal{U}_{0}\subset V^{*}_{0}$ is an open set,
$\mathcal{O}_{\mathcal{U}}:=\mathcal{F}_{\mathcal{U}_{0}}\otimes_{\mathbb{K}}S^{*}(V)$
and $\mathcal{F}_{\mathcal{U}_{0}}$ is the sheaf of smooth or holomorphic
functions on $\mathcal{U}_{0}$.

######  Definition 4.

We call the ringed space $\mathcal{U}$ a graded domain of type $(\Delta,\chi)$
and of dimension $\\{n_{\delta}\\}_{\delta\in\Delta}$, where $n_{\delta}=\dim
V_{\delta}$.

Let us choose a basis $(x_{i})$ in $V_{0}$ and a basis $(\xi_{j}^{\delta})$ in
any $V_{\delta}$. Then the set
$(x_{i},\xi_{j}^{\delta})_{\delta\in\Delta\setminus\\{0\\}}$ is a system of
local coordinates on $\mathcal{U}$. We assign the weight $0$ and the parity
$\bar{0}$ to $x_{i}$ and the weight $\delta$ and the parity $|\delta|$ to
$\xi_{j}^{\delta}$. Such coordinates $(x_{i},\xi_{j}^{\delta})$ are called
graded.

######  Definition 5.

A graded manifold of type $(\Delta,\chi)$ and of dimension
$\\{n_{\delta}\\}_{\delta\in\Delta}$ is a $\mathbb{Z}^{r}$-graded ringed space
$\mathcal{N}=(\mathcal{N}_{0},\mathcal{O}_{\mathcal{N}})$, that is locally
isomorphic to a graded domain of type $(\Delta,\chi)$ and of dimension
$\\{n_{\delta}\\}_{\delta\in\Delta}.$

A morphism of graded manifolds of type $(\Delta,\chi)$ is a morphism of the
corresponding $\mathbb{Z}^{r}$-graded ringed spaces.

If $\Delta=\\{0,1,\ldots,n\\}$ and $\chi(1)=\bar{1}$, a graded manifold of
type $\Delta$ is also called a graded manifold of degree $n$. (See [R] for
more information about graded manifolds of degree $n$.) If $\Delta$ is
multiplicity free, this is if $\delta=\sum_{i}a_{i}\alpha_{i}\in\Delta$ then
$a_{i}\in\\{0,1\\}$, a graded manifold of type $\Delta$ is called an $r$-fold
vector bundle of type $\Delta$. This definition of an $r$-fold vector bundle
is equivalent to a classical one as it was shown in [GR, Theorem 4.1].

######  Remark 6.

Now let $\Delta$ be as in Definition 3 and $\Delta^{\prime}\subset\Delta$
satisfies the following property: if $\delta\in\Delta^{\prime}$ and
$\delta=\sum_{i}\delta_{i}$ for $\delta_{i}\in\Delta$, then $\delta_{i}$ are
also in $\Delta^{\prime}$. In this case to any graded manifold $\mathcal{N}$
of type $\Delta$ we can assign a graded manifold $\mathcal{N}^{\prime}$ of
type $\Delta^{\prime}$. A detailed description of this well-known construction
can be found for example in [Vi1, Section 4.1]. For instance to any graded
manifold of degree $n$ we can assign a graded manifold of degree $0\leq
n^{\prime}<n$. In this case we have a natural morphism
$\mathcal{N}\to\mathcal{N}^{\prime}$, which is called projection.

###  2.3. Lie supergroups and super Harish-Chandra pairs

####  2.3.1. Lie supergroups

A Lie supergroup is a group object in the category of supermanifolds. In other
words a Lie supergroup is a supermanifold
$\mathcal{G}=(\mathcal{G}_{0},\mathcal{O}_{\mathcal{G}})$ with three
morphisms: $\mu:\mathcal{G}\times\mathcal{G}\to\mathcal{G}$ (the
multiplication morphism), $\kappa:\mathcal{G}\to\mathcal{G}$ (the inversion
morphism), $\varepsilon:(\mathrm{pt},\mathbb{C})\to\mathcal{G}$ (the identity
morphism). Moreover, these morphisms have to satisfy the usual conditions,
modeling the group axioms. The underlying manifold $\mathcal{G}_{0}$ of
$\mathcal{G}$ is a smooth or complex-analytic Lie group. As in the theory of
Lie groups and Lie algebras, we can define the Lie superalgebra
$\mathfrak{g}=\operatorname{Lie}(\mathcal{G})$ of a Lie group $\mathcal{G}$.
By definition, $\mathfrak{g}$ is the subalgebra of the Lie superalgebra of
vector fields on $\mathcal{G}$ consisting of all right invariant vector fields
on $\mathcal{G}$. Any right invariant vector field $Y\in\mathfrak{g}$ has the
following form

(4) $Y=(Y_{e}\otimes\mathrm{id})\circ\mu^{*},$

where $Y_{e}\in(\mathfrak{m}_{e}/\mathfrak{m}^{2}_{e})^{*}$ and
$\mathfrak{m}_{e}$ is the maximal ideal of $(\mathcal{O}_{\mathcal{G}})_{e}$.
Note that as the case of classical geometry the vector superspace
$(\mathfrak{m}_{e}/\mathfrak{m}^{2}_{e})^{*}$ may be identified with the
vector superspace of all maps $D:(\mathcal{O}_{\mathcal{G}})_{e}\to\mathbb{C}$
satisfying the Leibniz rule. Further the map $Y_{e}\mapsto
Y=(Y_{e}\otimes\mathrm{id})\circ\mu^{*}$ is an isomorphism of
$(\mathfrak{m}_{e}/\mathfrak{m}^{2}_{e})^{*}$ onto $\mathfrak{g}$.

Similarly we can define a graded Lie supergroup of type $\Delta$ as a group
object in the category of graded manifolds of type $\Delta$ or a graded Lie
supergroup of degree $n$ as a group object in the category of graded manifolds
of degree $n$. Let us consider some examples of Lie supergroups with an
additional gradation in the structure sheaf.

######  Example 7 (Lie supergroup $GL(V)$).

The space of endomorphisms of a vector superspace $V=V_{\bar{0}}\oplus
V_{\bar{1}}$ has a natural gradation by integer numbers $-1,0,1$:

$\operatorname{End}(V)=\bigoplus\limits_{i=-1,0,1}\operatorname{End}^{i}(V),$

where

$\displaystyle\operatorname{End}(V)^{0}=\operatorname{End}(V_{\bar{0}})$
$\displaystyle\oplus\operatorname{End}(V_{\bar{1}}),\quad\operatorname{End}(V)^{1}=\operatorname{Hom}(V_{\bar{0}},V_{\bar{1}}),$
$\displaystyle\operatorname{End}(V)^{-1}=\operatorname{Hom}(V_{\bar{1}},V_{\bar{0}})$

giving rise to a $\mathbb{Z}$-graded Lie superalgebra $\mathfrak{gl}(V)$. It
integrates to the $\mathbb{Z}$-graded Lie supergroup $\mathrm{GL}(V)$.
Consider the case $\dim V=(1|1)$. The entries of a matrix

$A=\begin{bmatrix}a&\alpha\\\
\beta&b\end{bmatrix}\in\mathrm{GL}(\mathbb{K}^{1|1}),$

constitute a graded coordinate domain, the subdomain of $\mathbb{K}^{2|2}$,
the coordinates $a$, $b$ are of degree $0$, and $\alpha$, $\beta$ are of
degrees $-1$ and $1$, respectively. The matrix multiplication respects the
assigned grading, e.g.
$m^{*}(\alpha)=a^{\prime}\alpha^{\prime\prime}+\alpha^{\prime}b^{\prime\prime}$
is of degree $0=0+0=1+(-1)$. Note that $\mathrm{GL}(V)$ with this gradation is
not a graded manifold of degree $n$, since this supermanifold possesses graded
coordinates of negative degree.

######  Remark 8.

In this paper we denote by $\mathbf{T}$ the antitangent functor. This means
that $\mathbf{T}\mathcal{M}$ is the vector bundle obtained from the tangent
bundle $\mathrm{T}\mathcal{M}$ by means of the parity functor. If $(x^{A})$
are even/odd coordinates on $\mathcal{M}$ then the fiber coordinates on
$\mathbf{T}\mathcal{M}$ will be denoted by $(\mathrm{d}x^{A})$ and the parity
of $\mathrm{d}x^{A}$ is the parity of $x^{A}$ plus one modulo two.

A canonical example of graded Lie supergroups of degree $n$ comes from the
tangent lifts.

######  Example 9.

Higher order tangent bundles, not to be confused with the related iterated
tangent bundles, are the natural geometric home of higher derivative
Lagrangian mechanics [B]. A higher order ($k$-order) tangent bundle
$\mathrm{T}^{k}\mathcal{M}$ of a supermanifold $\mathcal{M}$ can be defined in
a natural way. By applying the functor $\mathrm{T}^{k}$ to the Lie supergroup
$\mathcal{G}$ and the structure morphisms of $\mathcal{G}$ we get a graded Lie
supergroup of type $(\Delta,\chi)$, where $\Delta=\\{0,1,\ldots,k\\}$ and
$\chi(1)=\bar{0}$.

Another interesting example of $\mathbb{Z}$-gradation on a Lie supergroup was
noticed by V. Kac:

######  Example 10.

Let $V=V_{\bar{0}}\oplus V_{\bar{1}}$ be a vector superspace equipped with a
non-degenerate even symmetric bilinear form $Q$, i.e.
$Q(x,y)=(-1)^{|x||y|}Q(y,x)$ for homogeneous $x,y\in V$ and $Q:V\times
V\to\mathbb{K}$ is an even map. It follows that $V_{\bar{0}}\perp V_{\bar{1}}$
and $Q$ restricted to $V_{\bar{0}}$ (resp. $V_{\bar{1}}$) is symmetric (resp.
skew-symmetric). Thus the dimension of $V_{\bar{1}}$ is even and one can find
lagrangian subspaces $L,L^{\prime}\subset V_{\bar{1}}$ such that
$V_{\bar{1}}=L\oplus L^{\prime}$, where $L^{\prime}$ is naturally identified
with the dual to $L$ and $Q|_{V_{\bar{1}}}$ has the following natural form
with respect to this decomposition:

$Q((l_{1},l_{1}^{\prime}),(l_{2},l_{2}^{\prime}))=l_{1}^{\prime}(l_{2})-l_{2}^{\prime}(l_{1}),$

where $l_{i}\in L$ and $l^{\prime}_{i}\in L^{\prime}$. The orthosymplectic Lie
superalgebra $\mathfrak{osp}(V,Q)$ is spanned by homogeneous endomorphism
$T:V\to V$ such that

$Q(Tx,y)+(-1)^{|T||x|}Q(x,Ty)=0$

for any homogeneous $x,y\in V$. It follows that
$\mathfrak{osp}(V,Q)_{\bar{0}}=\mathfrak{so}(V_{\bar{0}})\oplus\mathfrak{sp}(V_{\bar{1}})$
and $\mathfrak{osp}(V,Q)_{\bar{1}}=V_{\bar{0}}\otimes V_{\bar{1}}$. There are
well known isomorphisms $\mathfrak{so}(W)=W\wedge W$ and
$\mathfrak{sp}(W)=S^{2}W$. Assigning the weight $1$ to $L$ and the weight $-1$
to $L^{\prime}$ one find a canonical $\mathbb{Z}$-grading on
$\mathfrak{osp}(V)$ with non-zero parts in degrees $\pm 2$, $\pm 1$ and $0$:

$\mathfrak{osp}(V)=S^{2}(L^{\prime})\oplus(V_{\bar{0}}\otimes
L^{\prime})\oplus(L^{\prime}\otimes L\oplus\mathrm{V}_{\bar{0}}\wedge
V_{\bar{0}})\oplus(V_{\bar{0}}\otimes L)\oplus S^{2}L.$

Thus the corresponding Lie supergroup, the orthosymplectic Lie supergroup, is
also naturally $\mathbb{Z}$-graded.

####  2.3.2. Super Harish-Chandra pairs

To study a Lie supergroup one uses the theory of super Harish-Chandra pairs,
see [Ber] and also [BCC, CCF, Vi3].

######  Definition 11.

A super Harish-Chandra pair $(G,\mathfrak{g})$ consists of a Lie superalgebra
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$, a Lie group
$G$ such that $\mathfrak{g}_{\bar{0}}=\mathrm{Lie}(G)$, and an action $\alpha$
of $G$ on $\mathfrak{g}$ by authomorphisms such that

  * •

for any $g\in G$ the action $\alpha(g)$ restricted to $\mathfrak{g}_{\bar{0}}$
coincides with the adjoint action $Ad(g)$;

  * •

the differential
$(d\alpha)_{e}:\mathfrak{g}_{0}\to\mathrm{T}_{e}\operatorname{Aut}(\mathfrak{g})=\operatorname{Der}(\mathfrak{g})$
at the identity $e\in G$ coincides with the adjoint representation
$\operatorname{ad}$ of $\mathfrak{g}_{\bar{0}}$ in $\mathfrak{g}$.

Super Harish-Chandra pairs form a category. The following theorem was proved
in [Vi3] for complex-analytic Lie supergroups.

######  Theorem 12.

The category of complex-analytic Lie supergroups is equivalent to the category
of complex-analytic super Harish-Chandra pairs.

A similar result in the real case was obtained in [Kos], the algebraic case
was treated by several authors, see for example [Gav, Mas, MS] and references
therein. Theorem 12 implies that any Lie supergroup is globally split, that
is, its structure sheaf is isomorphic to a wedge product of a certain trivial
vector bundle. More explanation of this fact will be given later.

The notion of super Harish-Chandra pair has an obvious $A$-graded
generalization, where $A$ is an abelian group. Namely, we assume that
$\mathfrak{g}$ is an $A$-graded Lie superalgebra, the Lie algebra of $G$ is
$\mathfrak{g}_{0}$, where $\mathfrak{g}_{0}$ is the part of $\mathfrak{g}$ in
degree 0, $\alpha(g):\mathfrak{g}\to\mathfrak{g}$, $g\in G$, is an
automorphism of $\mathfrak{g}$, in particular it preserves the $A$-gradation
and $\mathrm{d}_{e}\alpha(X)=[X,\cdot]_{\mathfrak{g}}$ for any
$X\in\mathfrak{g}_{0}$. In [JKPS, Theorem 5.6] it was noticed that an analogue
of Theorem 12 holds true for graded Lie supergroups of degree $n$. We will
also work with super Harish-Chandra pairs of type $\Delta$. A super Harish-
Chandra pairs of type $\Delta$ is a $\mathbb{Z}^{r}$-graded super Harish-
Chandra pair $(G,\mathfrak{g})$, where $\mathfrak{g}$ is a
$\mathbb{Z}^{r}$-graded Lie superalgebra of type $\Delta$. Recall that in this
case $\mathbb{Z}^{r}$ is generated by the set $\Delta$, with additional
agreement about parities of generators of $\Delta$, see Definition 3. In this
case repeating argument [JKPS, Theorem 5.6], we get an analogue of Theorem 12
as well.

######  Example 13.

Let $V=\oplus_{k=0}^{n}V_{k}$ be a non-negatively $\mathbb{Z}$-graded vector
space with $\dim V<\infty$. Denote by $\operatorname{End}_{q}(V)$, where $q\in
Z$, the space of endomorphisms $T:V\to V$ increasing the degree by $q$, i.e.
$T:V\to V$ such that $T|_{V_{i}}:V_{i}\to V_{i+q}$. Thus
$\underline{\operatorname{End}}(V)=\oplus_{q=-n}^{n}\operatorname{End}_{q}(V)$
is a $\mathbb{Z}$-graded vector space. It will be considered as a
$\mathbb{Z}$-graded Lie superalgebra with the bracket given by the graded
commutator. The corresponding $\mathbb{Z}$-graded Lie supergroup
$\mathrm{GL}(V)$ has the body $G_{0}=\times_{i=0}^{n}GL(V_{i})$ and its sheaf
of functions is given by the super symmetric algebra over $F=\oplus_{0\leq
i\neq j\leq n}V_{i}\otimes V_{j}^{*}$, as $V_{i}\otimes
V_{j}^{*}=\operatorname{Hom}(V_{i},V_{j})^{*}$, i.e.
$\mathcal{O}(U)=\mathcal{C}^{\infty}(U)\otimes\operatorname{Sym}F$. The
gradation in $F$ is obvious, $|V_{i}\otimes V_{j}^{*}|=i-j$ and the group
multiplication preserve this gradation. The graded Harish-Chandra pair
corresponding to $GL(V)$ is $(G_{0},\underline{\operatorname{End}}(V))$
equipped with the adjoint action of $G_{0}$ on
$\underline{\operatorname{End}}(V)$. The language of Harish-Chandra pairs
allows us to study some infinite dimensional generalizations of Lie
supergroups in a convienient way. We simply drop the assumption that
$\mathfrak{g}$ has finite dimension. For example, if
$V=\oplus_{k=0}^{\infty}V_{k}$ is a $\mathbb{Z}$-graded vector space such that
$\dim V_{k}<\infty$, but the dimension of $V$ can be infinite, then $GL(V)$
can be understood as a pair consisting of the direct sum of $GL(V_{i})$,
$G_{0}=\oplus_{i=0}^{\infty}GL(V_{i})$ and the natural action of $G_{0}$ on
$\underline{\operatorname{End}}(V)=\oplus_{i,j\geq
0}\operatorname{Hom}(V_{i},V_{j})$.

####  2.3.3. Construction of the Lie supergroup corresponding to a super
Harish-Chandra pair

Let us remind a reader how to construct a Lie supergroup $\mathcal{G}$ (or a
graded Lie supergroup of type $\Delta$) using a given super (or of type
$\Delta$) Harish-Chandra pair $(G,\mathfrak{g})$. Denote by
$\mathcal{U}(\mathfrak{h})$ the universal enveloping algebra of Lie
superalgebra $\mathfrak{h}$. We need to define a structure sheaf $\mathcal{O}$
of $\mathcal{G}$. In the super and graded cases we, respectively, put

(5)
$\begin{split}\mathcal{O}=\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathfrak{g}),\mathcal{F}_{G}),\quad\mathcal{O}=\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{0})}(\mathcal{U}(\mathfrak{g}),\mathcal{F}_{G}).\end{split}$

Here $\mathcal{F}_{G}$ is the sheaf of (holomorphic) functions on $G$. Using
the Hopf algebra structure on $\mathcal{U}(\mathfrak{g})$ we can define
explicitly the multiplication morphism $\mu$, the inversion morphism $\kappa$
and the identity $\varepsilon$, see for instance [Vi3]. Indeed, assume that a
super or graded of type $\Delta$ Harish-Chandra pair $(G,\mathfrak{g})$ is
given. Let us define the supergroup structure of the corresponding Lie
supergroup or graded Lie supergroup $\mathcal{G}$. Let $X\cdot
Y\in\mathcal{U}(\mathfrak{g}\oplus\mathfrak{g})\simeq\mathcal{U}(\mathfrak{g})\otimes\mathcal{U}(\mathfrak{g})$,
where $X$ is from the first copy of $\mathcal{U}(\mathfrak{g})$ and $Y$ from
the second one, $f\in\mathcal{O}$, see (5), and $g,\,h\in G$. The following
formulas define a multiplication morphism, an inverse morphism and an identity
morphism respectively:

(6) $\begin{split}\mu^{*}(f)(X\cdot Y)(g,h)&=f(\alpha(h^{-1})(X)\cdot
Y)(gh);\\\ \kappa^{*}(f)(X)(g^{-1})&=f(\alpha(g^{-1})(S(X)))(g);\\\
\varepsilon^{*}(f)&=f(1)(e).\end{split}$

Here $S$ is the antipode map in $\mathcal{U}(\mathfrak{g})$ considered as a
Hopf algebra and $\alpha$ is as in the definition of a super Harish-Chandra
pair.

###  2.4. (Skew-symmetric) double vector bundles

A double vector bundle (DVB , in short) is a graded manifold of type
$\Delta=\\{0,\alpha,\beta,\alpha+\beta\\}$. In this subsection we assume that
the weights $\alpha$ and $\beta$ are even, however later on we shall drop this
assumption. Geometrically we can see a double vector bundle as a quadruple
$(D;A,B;M)$ with the following diagram of morphisms

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where all maps are bundle projections and there are imposed some natural
compatibility conditions, see [P, M].

In particular, $D$ has two vector bundle structures, one over the manifold $A$
and the second over $B$. Moreover, $A$, $B$ are vector bundles over $M$ called
_the side bundles_ of $D$. The compatibility condition can be easily expressed
using Euler vector fields: $[\mathbbm{E}_{A},\mathbbm{E}_{B}]=0$ [GR].
Essentially, a DVB can be consider as a manifold equipped with two Euler
vector fields, $D=(D;\mathbbm{E}_{A},\mathbbm{E}_{B})$, describing the vector
bundle structure on _the legs_ $D\to A$, $D\to B$ of $D$. This lead to many
simplifications in the theory of DVBs.

The core $C$ of DVB is defined as intersection of the kernels of two bundle
projections $D\to A$ and $D\to B$. It is a vector bundle over $M$ with the
Euler vector field defined by the restriction,
$\mathbbm{E}_{A}|_{C}=\mathbbm{E}_{B}|_{C}$.

The _flip_ of a DVB is obtained by interchanging the legs of $D$, ie. the flip
of $D=(D;\mathbbm{E}_{A},\mathbbm{E}_{B})$ is
$(D;\mathbbm{E}_{B},\mathbbm{E}_{A})$.

Symmetric and skew-symmetric DVBs are DVBs $(D;A,B;M)$ with an involution
$\sigma:D\to D$ satisfying a certain addition condition. They were introduced
in [BGR] in order to recognize the image of _the linearization functor_ which
associates with a graded manifold of type $\Delta=\\{0,\alpha,2\alpha\\}$ a
DVB. Under this association graded manifolds of type $\Delta$ with even
$\alpha$ (i.e., purely even graded bundles of degree 2) are in one-to-one
correspondence with symetric DVBs, while graded manifolds of type $\Delta$
with odd $\alpha$ (i.e., $N$-manifolds of degree 2) — with skew-symmetric
DVBs. The results are extended to any order. For the purposes of this
manuscript it is enough to recall a definition of skew-symmetric DVBs.

######  Definition 14.

A skew-symmetric DVB  is a pair $(D,\sigma)$ consisting of a DVB $D$ and an
involution $\sigma:D\to D$ (i.e. $\sigma\circ\sigma=\mathrm{id}$) such that

  1. (i)

$\sigma$ exchanges the legs of $D$, i.e. $\sigma$ is a DVB morphism from $D$
to the flip of $D$,

  2. (ii)

the restriction of $\sigma$ to the core is minus the identity.

A morphism between skew-symmetric DVB s is assumed to intertwines with their
involutions.

It follows immediately that the side bundles of a skew-symmetric DVB are
isomorphic and that any skew-symmetric DVB admits an atlas with graded
coordinates $(x^{a};y^{i}_{\alpha},Y^{j}_{\beta};z^{\mu}_{\alpha+\beta})$ such
that

  * •

the transition functions for $y^{i}$ and $Y^{i}$ are the same and

$z^{\mu}_{ij}=Q^{\mu}_{\nu}z^{\nu}+\frac{1}{2}Q^{\mu}_{ij}y^{i}Y^{j}$

with $Q^{\mu}_{ij}=-Q^{\mu}_{ji}$,

  * •

the involution $\sigma$ has a form: $\sigma^{*}(y^{i})=Y^{i}$ (hence
$\sigma^{*}(Y^{i})=y^{i}$) and
$\sigma^{*}(z^{\mu})=-z^{\mu}+\sigma^{\mu}_{ij}(x)y^{i}Y^{j}$ with
$\sigma^{\mu}_{ij}=\sigma^{\mu}_{ji}$.

To any double vector bundle $D$ (with the core $C$ and side bundles $A$, $B$)
we can assign a short exact sequence

(8) $0\to C\to\widehat{D}\to A\otimes B\to 0,$

which is obtained by dualizing the short exact sequence

$0\to\ker\mathrm{pr}_{C}\to\mathcal{A}^{\alpha+\beta}(D)\xrightarrow{\mathrm{pr}_{C}}C^{*}\to
0$

where $\mathcal{A}^{\alpha+\beta}(D)$ is the $\mathcal{F}(M)$-module of
homogeneaus functions on $D$ of weight $\alpha+\beta$ and $\mathrm{pr}_{C}$
denotes the restriction of a function $f$ to the core $C$, ie.
$\mathrm{pr}_{C}(f)=f|_{C}$. Locally, $\mathcal{A}^{\alpha+\beta}(D)$ is
generated by the functions $y_{\alpha}^{i}$, $Y^{j}_{\beta}$ and
$z^{\mu}_{\alpha+\beta}$. In other words, the dual to $\widehat{D}$ is the
vector bundle whose space of sections is $\mathcal{A}^{\alpha+\beta}(D)$.

If $(D,\sigma)$ is skew-symmetric then we can decompose

$\mathcal{A}^{\alpha+\beta}(D)=\mathcal{A}^{\alpha+\beta}_{+}(D)\oplus\mathcal{A}^{\alpha+\beta}_{-}(D),$

where $\mathcal{A}^{\alpha+\beta}_{\varepsilon}(D)$ consists of those
$f\in\mathcal{A}^{\alpha+\beta}(D)$ such that $\sigma^{*}(f)=\varepsilon f$
and $\varepsilon$ is plus or minus.

######  Theorem 15.

Let $(D,\sigma)$ be a skew-symmetric DVB  with side bundle $E\to M$. Then
$\mathcal{A}^{\alpha+\beta}_{+}(D)$ coincides with the space of sections of
$S^{2}E^{*}$ while $\mathcal{A}^{\alpha+\beta}_{-}(D)$ gives rise to a short
exact sequence

(9)
$0\to\Gamma(\bigwedge^{2}E^{*})\to\mathcal{A}^{\alpha+\beta}_{-}(D)\to\Gamma(C^{*})\to
0.$

The skew-symmetric DVB  $(D,\sigma)$ can be reconstructed completely from the
above sequence.

###### Proof.

Let $(x^{a},y^{i},Y^{j},z^{\mu})$ be adopted coordinates for a skew-symmetric
DVB $(D,\sigma)$. In these coordinates the decomposition of
$\mathcal{A}^{\alpha+\beta}(D)$ is clear: $\mathcal{A}^{\alpha+\beta}_{+}(D)$
is locally generated by functions $y^{i}Y^{j}+y^{j}Y^{i}$ and
$\mathcal{A}^{\alpha+\beta}_{-}(D)$ by $z^{\mu}$ and $y^{i}Y^{j}-y^{j}Y^{i}$.
The projection on $\Gamma(C^{*})$ is the same, it is given as the restriction
to the core bundle of $D$. ∎

###  2.5. Inverse limit of supermanifolds and Lie supergroups

We shall work with some types of infinite dimensional supermanifolds and Lie
supergroups. There is no need to present a general theory as all examples we
shall work with have a form of an inverse limit of supermanifolds (even with
the same body $\mathcal{M}_{0}$):

$\mathcal{M}_{1}\xleftarrow{\mathrm{pr}^{2}_{1}}\mathcal{M}_{2}\xleftarrow{\mathrm{pr}^{3}_{2}}\mathcal{M}_{3}\xleftarrow{}\ldots$

where $\mathcal{M}_{i}=(\mathcal{M}_{0},\mathcal{O}_{\mathcal{M}_{i}})$ and
$\mathrm{pr}^{k+1}_{k}:\mathcal{M}_{k+1}\rightarrow\mathcal{M}_{k}$ is the
projection of the graded manifold $\mathcal{M}_{k+1}$ of degree $k+1$ to the
graded manifold $\mathcal{M}_{k}$ of degree $k$, see Remark 6. We define

$\mathcal{M}_{\infty}=(\mathcal{M}_{0},\mathcal{O}_{\mathcal{M}_{\infty}}),\quad\text{
where
}\mathcal{O}_{\mathcal{M}_{\infty}}=\bigcup_{k=1}^{\infty}\mathcal{O}_{\mathcal{M}_{k}},$

i.e. $f\in\mathcal{O}_{\mathcal{M}_{\infty}}$ if and only if there exist $k$
such that $f\in\mathcal{O}_{\mathcal{M}_{k}}$. A morphism
$f:\mathcal{M}_{\infty}\to\mathcal{N}_{\infty}$ is a family of morphisms
$f=(f_{k}:\mathcal{M}_{k}\to\mathcal{N}_{a_{k}})$ where $(a_{k})$ is a non-
decreasing sequence of positive integers and the family $(f_{k})$ is
compatible with the projections:

$\textstyle{\mathcal{M}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{k+1}}$$\scriptstyle{\mathrm{pr}^{k+1}_{k}}$$\textstyle{\mathcal{N}_{a_{k+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}^{a_{k+1}}_{a_{k}}}$$\textstyle{\mathcal{M}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{k}}$$\textstyle{\mathcal{N}_{a_{k}}}$

where $\mathrm{pr}^{a_{k+1}}_{a_{k}}$ is the composition of projections
$\mathrm{pr}^{i+1}_{i}$ for $i=a_{k+1}-1$ to $a_{k}$.

The inverse limit of Lie supergroups has a Lie supergroup structure. We
additionally have to assume that the projections $\mathrm{pr}^{i+1}_{i}$ are
Lie supergroup homomorphisms. Also the inverse limit of Lie (super)algebras
has a Lie (super)algebra structure thus a Lie functor makes sense for inverse
limit of Lie supergroups. For example a $\mathbb{Z}$-graded Lie superalgebra
$\mathfrak{g}=\bigoplus_{k=0}^{\infty}\mathfrak{g}_{k}$ is an inverse limit of
Lie algebras $\mathfrak{g}/I_{k}$ where
$I_{k}=\mathfrak{g}_{k+1}\oplus\mathfrak{g}_{k+2}\oplus\ldots$. There is one-
to-one correspondence of the inverse limit of graded Lie supergroups and the
inverse limit of their graded Harish-Chandra pairs.

##  3\. The Donagi–Witten construction, supermanifolds, double vector bundles
and graded manifolds of degree $2$

In [DW2, Section 2.1] Donagi and Witten gave a description of the first
obstruction class $\omega=\omega_{2}$ via differential operators. In this
section we remind the definition of $\omega_{2}$ and the Donagi–Witten
construction. Further we give an interpretation of the Donagi–Witten
construction using the language of double vector bundles and graded manifolds
of degree $2$. At the end using differential forms instead of differential
operators we simplify this construction, which allows us to find its higher
analogue.

###  3.1. First obstruction class $\omega_{2}$

Let us describe the first obstruction class to splitting a supermanifold using
results [Ber, Gr, Oni, R]. We follow the exposition of [Oni]. First of all
consider a split supermanifold $\mathcal{M}=(\mathcal{M}_{0},\mathcal{O})$,
where $\mathcal{O}=\bigwedge\mathcal{E}^{*}$ and $\mathcal{E}$ is the sheaf of
sections of $\mathbb{E}[\bar{1}]$. Denote by $\mathcal{D}er\mathcal{O}$ the
sheaf of vector fields on $\mathcal{M}$ and by $\mathcal{D}er\mathcal{F}$ the
sheaf of vector fields on the underlying space $\mathcal{M}_{0}$. The sheaf
$\mathcal{D}er\mathcal{O}=\bigoplus_{p\geq-1}\mathcal{D}er_{p}\mathcal{O}$ is
naturally $\mathbb{Z}$-graded since $\bigwedge\mathcal{E}^{*}$ is
$\mathbb{Z}$-graded. We have the following exact sequence

(10)
$0\to\bigwedge^{3}\mathcal{E}^{*}\otimes\mathcal{E}\to\mathcal{D}er_{2}\mathcal{O}\to\bigwedge^{2}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F}\to
0,$

see [Oni, Formula (5)].

According Green [Gr] we can describe all non-split supermanifolds
$\mathcal{N}$ such that $\mathrm{gr}(\mathcal{N})\simeq\mathcal{M}$ using the
sheaf of automorphisms $\mathcal{A}ut\mathcal{O}$ of $\mathcal{O}$. More
precisely consider the following subsheaf of $\mathcal{A}ut\mathcal{O}$:

$\mathcal{A}ut_{(2)}\mathcal{O}=\\{a\in\mathcal{A}ut\mathcal{O}\,\,|\,\,a(u)-u\in\mathcal{J}^{2}\,\,\text{for
any}\,\,u\in\mathcal{O}\\},$

see also [Oni, Formula (17)]. Recall that $\mathcal{J}$ is the sheaf of ideals
generated by odd elements in $\mathcal{O}$. Denote by
$\mathrm{Aut}(\mathbb{E}^{*})$ the group of automorthisms of $\mathbb{E}^{*}$.
There is a natural action of $\mathrm{Aut}(\mathbb{E}^{*})$ on the sheaf
$\mathcal{A}ut_{(2)}\mathcal{O}$, see [Oni, Section 1.4]. This action induces
an action of $\mathrm{Aut}(\mathbb{E}^{*})$ on the set
$H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})$. By Green [Gr] points
of the set of orbits
$H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})/\mathrm{Aut}(\mathbb{E}^{*})$
are in one-to-one correspondence with isomorphism classes of supermanifolds
$\mathcal{N}$ such that $\mathrm{gr}(\mathcal{N})\simeq\mathcal{M}$. More
precisely to any supermanifold $\mathcal{N}$ such that
$\mathrm{gr}(\mathcal{N})\simeq\mathcal{M}$ we can assign a class $\gamma\in
H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})$. If $\gamma_{i}$ is the
class corresponding to a supermanifold $\mathcal{N}_{i}$ such that
$\mathrm{gr}(\mathcal{N}_{i})\simeq\mathcal{M}$, where $i=1,2$. Then
$\mathcal{N}_{1}\simeq\mathcal{N}_{2}$ if and only if $\gamma_{1}$ and
$\gamma_{2}$ are in the same orbit of $\mathrm{Aut}(\mathbb{E}^{*})$.

In [R] the following map of sheaves was defined

(11) $\mathcal{A}ut_{(2)}\mathcal{O}\to\mathcal{D}er_{2}\mathcal{O},$

see also [Oni, Formula (19)]. Combining the map (11) and the map
$\mathcal{D}er_{2}\mathcal{O}\to\bigwedge^{3}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F}$
from (10) we get the following map of cohomology sets

$H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})\to
H^{1}(\mathcal{M}_{0},\bigwedge^{2}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F})$

and the corresponding map of $\mathrm{Aut}(\mathbb{E}^{*})$-orbits

$H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})/\mathrm{Aut}(\mathbb{E}^{*})\to
H^{1}(\mathcal{M}_{0},\bigwedge^{2}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F})/\mathrm{Aut}(\mathbb{E}^{*}).$

If $\gamma\in H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})$
corresponds to a non-split supermanifold $\mathcal{N}$, the image of $\gamma$
in
$H^{1}(\mathcal{M}_{0},\bigwedge^{2}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F})$
is called the first obstruction class to splitting of $\mathcal{N}$ and
following [DW2, Section 2.1] we denote this class by $\omega_{2}$.

Consider the case when $\mathcal{M}$ has odd dimension $2$ in details. Since
$\bigwedge^{3}\mathcal{E}^{*}=\\{0\\}$, from (10) it follows that

$\mathcal{D}er_{2}\mathcal{O}\simeq\bigwedge^{2}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F}.$

In this case the map (11) is an isomorphism. Therefore we have the following
set bijection

$H^{1}(\mathcal{M}_{0},\mathcal{A}ut_{(2)}\mathcal{O})\simeq
H^{1}(\mathcal{M}_{0},\bigwedge^{2}\mathcal{E}^{*}\otimes\mathcal{D}er\mathcal{F})$

and the corresponding bijection of the sets of orbits. Now Green’s result [Gr]
implies that $\omega_{2}$ is the only obstruction for a supermanifold to be
split in this case. In other words a supermanifold $\mathcal{N}$ of odd
dimension $2$ such that $\mathrm{gr}(\mathcal{N})\simeq\mathcal{M}$ is split
if and only if $\omega_{2}=0$. Note that the notion of a split and a
projectable, see [DW1] for details, supermanifold coincide in this case.

###  3.2. The Donagi and Witten construction

In [DW2, Section 2.1] the first obstruction class $\omega_{2}$ to splitting a
supermanifold $\mathcal{M}=(\mathcal{M}_{0},\mathcal{O})$ was interpreted in
terms of a certain sheaf of differential operators on $\mathcal{M}$. Namely,
the obstruction class $\omega$ defined above is the Atiyah class of the
extension

$0\to\mathrm{T}M\to D_{\omega}\to\bigwedge^{2}E\to 0$

where sections of $D_{\omega}$ are identified with some factor of
$\mathcal{D}^{2}_{-}|_{\mathcal{M}_{0}}$ where the meaning of the latter is
explained below.

Let us remind this construction using charts and local coordinates. Consider
two charts $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ on $\mathcal{M}$ with non-
empty intersection and with local coordinates $(x_{i},\xi_{a})$ and
$(y_{j},\eta_{b})$, where $i,j=1,\ldots,n$ and $a,b=1,\ldots,m$. Let in
$\mathcal{U}_{1}\cap\mathcal{U}_{2}$ we have the following transition
functions

(12)
$\begin{split}&y_{j}=F_{j}+\frac{1}{2}G_{j}^{a_{1}a_{2}}\xi_{a_{1}}\xi_{a_{2}}+\cdots\,\,\,j=1,\ldots,n;\\\
&\eta_{b}=H_{b}^{a}\xi_{a}+\cdots,\,\,\,b=1,\ldots,m,\end{split}$

where $F_{j},G^{a_{1}a_{2}}_{j},H_{i}^{b}$ are (holomorphic) functions
depending only on even coordinates $(x_{i})$.

If $D:\mathcal{O}\to\mathcal{O}$ is a differential operator let
$\left[{D}\right]:\mathcal{O}\to\mathcal{F}$ denotes the composition of $D$
with the projection $\mathcal{O}\to\mathcal{O}/\mathcal{J}=\mathcal{F}$.

Following Donagi and Witten, see [DW2, Section 2.1], we define the sheaf
$\mathcal{D}^{2}_{-}|_{\mathcal{M}_{0}}$ on the underlying manifold
$\mathcal{M}_{0}$ as a sheaf locally generated over $\mathcal{F}$ by

$\langle 1,\left[{\frac{\partial}{\partial
x_{i}}}\right],\left[{\frac{\partial}{\partial\xi_{j}}}\right],\left[{\frac{\partial^{2}}{\partial\xi_{a}\partial\xi_{b}}}\right]\rangle.$

In [DW2, Theorem 2.5] it was shown that this definition does not depend on
local coordinates. Note that the operators $\frac{\partial}{\partial\xi_{a}}$
anticommute, ie.
$\frac{\partial^{2}}{\partial\xi_{a}\partial\xi_{b}}=-\frac{\partial^{2}}{\partial\xi_{b}\partial\xi_{a}}$.
We can get transition function for generators of
$\mathcal{D}^{2}_{-}|_{\mathcal{M}_{0}}$ writing down the transition function
for $\frac{\partial}{\partial x_{i}}$, $\frac{\partial}{\partial\xi_{j}}$,
$\frac{\partial^{2}}{\partial\xi_{a}\partial\xi_{b}}$ and then factorizing
them by $\mathcal{J}$. Indeed,

$\displaystyle\frac{\partial}{\partial x_{i}}=\frac{\partial y_{j}}{\partial
x_{i}}\frac{\partial}{\partial y_{j}}+\frac{\partial\eta_{\alpha}}{\partial
x_{i}}\frac{\partial}{\partial\eta_{\alpha}};\quad\frac{\partial}{\partial\xi_{k}}=\frac{\partial
y_{j}}{\partial\xi_{k}}\frac{\partial}{\partial
y_{j}}+\frac{\partial\eta_{\alpha}}{\partial\xi_{k}}\frac{\partial}{\partial\eta_{\alpha}}.$

Therefore modulo $\mathcal{J}$ we have:

$\displaystyle\left[{\frac{\partial^{2}}{\partial\xi_{a_{1}}\partial\xi_{a_{2}}}}\right]=\Big{(}\frac{\partial\eta_{\alpha}}{\partial\xi_{a_{1}}}\frac{\partial^{2}(y_{j})}{\partial\xi_{a_{1}}\partial\xi_{a_{2}}}\Big{)}_{\mathrm{red}}\left[{\frac{\partial}{\partial
y_{j}}}\right]-\Big{(}\frac{\partial\eta_{\alpha}}{\partial\xi_{a_{1}}}\frac{\partial\eta_{\beta}}{\partial\xi_{a_{2}}}\Big{)}_{\mathrm{red}}\left[{\frac{\partial^{2}}{\partial\eta_{\alpha}\partial\eta_{\beta}}}\right].$

Compare with [DW2, Formula 2.13]. Using (12) we can write the transition
functions for $\mathcal{D}^{2}_{-}|_{\mathcal{M}_{0}}$ explicitly.

(13) $\begin{split}&x_{i}=F^{-1}_{i}(y_{j});\\\
&\left[{\frac{\partial}{\partial x_{i}}}\right]=\frac{\partial F_{j}}{\partial
x_{i}}\left[{\frac{\partial}{\partial
y_{j}}}\right];\quad\left[{\frac{\partial}{\partial\xi_{a}}}\right]=H_{a}^{b}\left[{\frac{\partial}{\partial\eta_{b}}}\right];\\\
&\left[{\frac{\partial^{2}}{\partial\xi_{b_{1}}\partial\xi_{b_{2}}}}\right]=-G_{j}^{b_{1}b_{2}}\left[{\frac{\partial}{\partial
y_{j}}}\right]-H_{\alpha}^{b_{1}}H_{\beta}^{b_{2}}\left[{\frac{\partial^{2}}{\partial\eta_{\alpha}\partial\eta_{\beta}}}\right].\end{split}$

Note that in (13) it is more correct to use the index $red$ for even
coordinates, for example $(x_{i})_{\mathrm{red}}$. However we omit
$\mathrm{red}$ for simplicity of notations. We conclude this subsection with
the following important remark.

######  Remark 16.

Comparing Formulas (12) and (13), we see that Formulas (13) contain the whole
information about Formulas (12) modulo $\mathcal{J}^{3}$. In other words using
Formulas (13) we can reconstruct Formulas (12) modulo $\mathcal{J}^{3}$.

One of purposes of this paper is to develop this observation. This leads to an
idea to use the theory of $n$-fold vector bundles and the theory of graded
manifolds of degree $n$ to recover a supermanifolds of odd dimension greater
than $2$.

###  3.3. A geometric interpretation of the Donagi and Witten construction

In this subsection we give a description of the geometric object with
transition functions (13). We use the theory of double vector bundles and
graded manifolds of degree $2$.

######  Lemma 17.

Let $\mathbb{D}$ be a vector bundle over $M$ characterized by the space of
sections,
$\mathrm{\Gamma}(\mathbb{D})=\mathcal{D}^{2}_{-}|_{\mathcal{M}_{0}}$. Then the
dual bundle $\mathbb{D}^{*}$ has transition functions of a skew-symmetric DVB
. More precisely, there exist a skew-symmetric DVB $(D,\sigma)$ such that the
space $\mathcal{A}^{\alpha+\beta}_{-}(D)$ coincides with the space of sections
of the vector bundle $\mathbb{D}^{*}$.

###### Proof.

We simply transpose and reverse the formulas (13). ∎

The skew-symmetric DVB arising from the above lemma will be denoted by
$\mathrm{Vb}_{2}(\mathcal{M})$.

The Atiyah class of the exact sequence (8), that is the obstruction class of
splitting of this sequence, is an element in $H^{1}(M;A^{*}\otimes
B^{*}\otimes\ C)$. It will be also called the Atiyah class associated with the
DVB $D$ and denoted by $\mathrm{At}(D)$. Note that the Atiyah class associated
with any dual (vertical or horizontal) of $D$ coincides with $\mathrm{At}(D)$.
Conversely if we have the sequence (8) we can reconstruct the double vector
bundle $D$.

Now let $\mathcal{M}$ be a supermanifold with transition function (12) and let
$E=\mathrm{Vb}_{1}(\mathcal{M})$. The Atiyah sequence associated with
$D=\mathrm{Vb}_{2}(\mathcal{M})$ (which is a double vector bundle with the
same side bundles $E$ and the core $\mathrm{T}\mathcal{M}_{0}$) is

$0\to\mathrm{T}\mathcal{M}_{0}\to\widehat{D}\to E\otimes E\to 0.$

The obstruction for splitting of this exact sequence is

$\mathrm{At}(\mathrm{Vb}_{2}(\mathcal{M}))\in
H^{1}(\mathcal{M}_{0},\mathrm{T}(\mathcal{M}_{0})\otimes E^{*}\otimes E^{*}).$

Due to the decomposition $E\otimes
E=\operatorname{Sym}^{2}E\oplus\bigwedge^{2}E$, the Atiyah class
$\mathrm{At}(D)$, where $D=\mathrm{Vb}_{2}(\mathcal{M})$, decomposes to
$\mathrm{At}_{+}(D)+\mathrm{At}_{-}(D)$, where $\mathrm{At}_{+}(D)\in
H^{1}(\mathcal{M}_{0},\mathrm{T}(\mathcal{M}_{0})\otimes\operatorname{Sym}^{2}E^{*})$
and $\mathrm{At}_{-}(D)\in H^{1}(M;\mathrm{T}M\otimes\bigwedge^{2}E^{*})$. We
are only interested with $\mathrm{At}_{-}(D)$ because our DVB
$\mathrm{Vb}_{2}(\mathcal{M})$ is a skew-symmetric DVB , hence
$\mathrm{At}_{+}(D)=0$.

The class $\mathrm{At}_{-}(\mathcal{M})$ is here the same as the Atiyah class
of the sequence (9) associated with the skew-symmetric DVB $(D,\sigma)$. It
also coincides with the first obstruction class to splitting of the
supermanifold $\mathcal{M}$ as in [DW2, Section 2].

Later we will give a geometric interpretation of this fact.

###  3.4. A modification of the Donagi–Witten construction

In this subsection we suggest a different way how to obtain a double vector
bundle with obstruction the class $\omega_{2}$. First of all to write Formulas
(13) we need the transition functions (12) and their inverse. To avoid this
inconvenience we can use differential forms instead of differential operators.

Let $\mathcal{M}$ be a supermanifold as above. Consider as above two charts
$\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ with non-empty intersection on
$\mathcal{M}$ with local coordinates $(x_{i},\xi_{a})$ and $(y_{j},\eta_{b})$
and transition functions (12). Further consider the antitangent bundles
$\mathbf{T}(\mathcal{M})$ of $\mathcal{M}$ and two charts
$\mathbf{T}(\mathcal{U}_{1})$ and $\mathbf{T}(\mathcal{U}_{2})$ with standard
coordinates $(x_{i},\xi_{a},\mathrm{d}x_{i},\mathrm{d}\xi_{a})$ and
$(y_{j},\eta_{b},\mathrm{d}y_{j},\mathrm{d}\eta_{b})$, respectively. Thus
$x_{i}$, $\mathrm{d}\xi_{a}$ are even local coordinates in
$\mathbf{T}(\mathcal{U}_{1})$, while $\xi_{a}$,$\mathrm{d}x_{i})$ are odd
ones. In $\mathbf{T}(\mathcal{U}_{1})\cap\mathbf{T}(\mathcal{U}_{2})$ we get
the following transition functions.

$\displaystyle
y_{j}=F_{j}+G_{j}^{a_{1}a_{2}}\xi_{a_{1}}\xi_{a_{2}}+\cdots,\quad\eta_{a}=H_{a}^{b}\xi_{b}+\cdots;$
$\displaystyle\mathrm{d}y_{j}=\sum_{b=1}^{n}(F_{j})_{x_{b}}\mathrm{d}x_{b}+\sum_{b=1}^{n}(G_{j}^{a_{1}a_{2}})_{x_{b}}\mathrm{d}x_{b}\xi_{1}\xi_{2}+G_{j}^{a_{1}a_{2}}(\mathrm{d}(\xi_{a_{1}})\xi_{a_{2}}-\xi_{a_{1}}\mathrm{d}(\xi_{a_{2}}))+\cdots;$
$\displaystyle\mathrm{d}\eta_{a}=\sum_{b=1}^{n}(H_{a}^{i})_{x_{b}}\mathrm{d}x_{b}\xi_{j}+H_{a}^{b}\mathrm{d}\xi_{b}+\cdots.$

Here we denoted by $(F_{j})_{x_{b}}$ the derivation of $F_{j}$ by $x_{b}$. Now
we apply the functor split $\mathrm{gr}$ to $\mathbf{T}(\mathcal{M})$. In
local coordinates this means that we factorize our transition functions by all
terms that contain more than one odd variable. Then
$\mathrm{gr}\mathbf{T}(\mathcal{M})$ has the following transition functions

(14) $\begin{split}&y_{j}=F_{j}(x);\\\ &\eta_{a}=H_{a}^{b}\xi_{b};\\\
&\mathrm{d}y_{j}=\sum_{b=1}^{n}(F_{j})_{x_{b}}\mathrm{d}x_{b}+G_{j}^{a_{1}a_{2}}(\mathrm{d}(\xi_{a_{1}})\xi_{a_{2}}-\xi_{a_{1}}\mathrm{d}(\xi_{a_{2}}));\\\
&\mathrm{d}\eta_{a}=H_{a}^{b}\mathrm{d}\xi_{b}.\end{split}$

If we compare Formulas (14) with [Vo, Section 2.2, Formulas (9)-(12)], we see
that a manifold with such transition functions is a double vector bundle,
which we denote by $\mathbb{D}$. We can see
$\mathrm{gr}\mathbf{T}(\mathcal{M})$ as a graded manifold of type
$\Delta=\\{0,\alpha,\beta,\alpha+\beta\\}$, where $\alpha$ is odd (the weight
of $\xi_{a}$) and $\beta$ is even (the weight of $\mathrm{d}\xi$). Formulas
(14) shows that all weights are well-defined. The double vector bundle

(15) ${\mathbb{D}^{\prime}}$${E[\bar{1}]}$${E}$${\mathcal{M}_{0}}$

is only slightly different from the double vector bundle
$\mathrm{Vb}_{2}(\mathcal{M})$. In our case one side bundle is pure even.

######  Remark 18.

By definition the composition of functors $\mathrm{gr}\circ\mathbf{T}$ is a
functor from the category of supermanifolds to the category of (split)
supermanifolds. However above implies that we can see the image of
$\mathrm{gr}\circ\mathbf{T}$ as the category of double vector bundles (with
some additional structure). For simplicity we will use the same notation
$\mathrm{gr}\circ\mathbf{T}$ meaning a functor from the category of
supermanifolds to the category of double vector bundles. Similarly in next
sections we will consider the functor $\mathrm{gr}\circ\mathbf{T}^{(n)}$ as a
functor from the category of supermanifolds to the category of $n$-fold vector
bundles.

Now we can go further and give an interpretation of $\omega_{2}$ as the
obstruction class of splitting of a graded manifold of degree $2$. First of
all let us change that parity of the side bundle $\mathbb{E}$. (In other words
we apply the functor parity change, see [Vo].) To do this we need to rewrite
(14) in the following form

(16) $\begin{split}&y_{j}=F_{j};\\\ &\eta_{a}=H_{a}^{b}\xi_{b};\\\
&\mathrm{d}y_{j}=\sum_{b=1}^{n}(F_{j})_{x_{b}}\mathrm{d}x_{b}+G_{j}^{a_{1}a_{2}}(\xi_{a_{2}}\mathrm{d}(\xi_{a_{1}})-\xi_{a_{1}}\mathrm{d}(\xi_{a_{2}}));\\\
&\mathrm{d}\eta_{a}=H_{a}^{b}\mathrm{d}\xi_{b}.\end{split}$

and to change the parity of the weight $\beta$.

Now we use a result obtained by [JL, CM] for double vector bundles and by
[BGR, Vi1] for $n$-fold vector bundles. Later we will call this step ”to apply
the functor inverse”. In more details in [JL, CM] a functor was constructed
from the category of graded manifolds of degree $2$ to the category of double
vector bundles with some additional structures. In [BGR, Vi1] an analogue of
this result was obtained for graded manifolds of degree $n$ and for graded
manifolds of type $\Delta$. (Note that in all these papers [JL, CM, BGR, Vi1]
the categories of double vector bundles with additional structures are
different. In this paper we follow approaches of [BGR, Vi1].) In [JL, CM] it
was shown that this functor is an equivalence of the category of double vector
bundles with some additional structures and the category of graded manifolds
of degree $2$. In [BGR, Vi1] it was shown that this functor is an equivalence
of the category of $n$-fold vector bundles with some additional structures and
the category of graded manifolds of degree $n$ or of type $\Delta$. The
inversion of this functor, that is the functor from the category of $n$-fold
vector bundles with some additional structures to the category of graded
manifolds of degree $n$, we call the functor inverse. We will denote the
functor inverse by $\iota$.

In terms of local coordinates ”to apply the functor inverse” means that we
identify $\xi_{a}$ with $\mathrm{d}\xi_{a}$ in (16). We get

(17) $\begin{split}&y_{j}=F_{j};\\\ &\theta_{a}=H_{a}^{i}\zeta_{i};\\\
&t_{j}=\sum_{b=1}^{n}(F_{j})_{x_{b}}z_{b}+2G_{j}^{a_{1}a_{2}}\zeta_{a_{1}}\zeta_{a_{2}}.\end{split}$

We obtained transition functions of a graded manifold $\mathcal{N}$ of degree
$2$, which we will denote also by $\mathrm{F}_{2}(\mathcal{M})$. We assign the
following weights to our local coordinates: $x_{i}$ (weight $0$); $\zeta_{j}$
(weight $\alpha$); $z_{s}$ (weight $2\alpha$). In other words, the transition
functions (17) defines a graded manifold of type
$\Delta=\\{0,\alpha,2\alpha\\}$, where $\alpha$ is odd. Note that we can
remove the coefficient $2$ in (17). Indeed, it is enough to replace
$\zeta_{i}$ by $\frac{1}{\sqrt{2}}\zeta^{\prime}_{i}$ in any chart.

######  Remark 19.

Let us give another explanation of the procedure ”to apply the functor
inverse”. Consider two graded domain $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$
with graded coordinates $(x_{i},\zeta_{j},z_{i})$ and
$(y_{i},\theta_{j},t_{i})$, respectively, and with weights as above. Define
transition functions $\mathcal{V}_{1}\to\mathcal{V}_{2}$ by (17). (Let us omit
the coefficient $2$.) Now we apply the tangent functor $\mathbf{T}$ to
$\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ and to the morphism (17) and factorize
the result by the sheaf of ideals locally generated by $\mathrm{d}x_{i}$ (or
by $\mathrm{d}y_{i}$). We get

$\displaystyle y_{j}=F_{j};$
$\displaystyle\theta_{a}=H_{a}^{i}\zeta_{i};\quad\mathrm{d}\theta_{a}=H_{a}^{i}\mathrm{d}\zeta_{i};$
$\displaystyle\mathrm{d}t_{j}=\sum_{b=1}^{n}(F_{j})_{x_{b}}\mathrm{d}z_{b}+G_{j}^{a_{1}a_{2}}(\zeta_{a_{2}}\mathrm{d}(\zeta_{a_{1}})-\zeta_{a_{1}}\mathrm{d}(\zeta_{a_{2}})).$

We obtain Formulas (16) up to appropriate change of variables. The inversion
of this procedure is called ”to apply the functor inverse”.

The structure sheaf $\mathcal{O}_{\mathcal{N}}=\bigoplus_{q\geq
0}(\mathcal{O}_{\mathcal{N}})_{q}$ of $\mathcal{N}$ is $\mathbb{Z}$-graded.
Clearly
$(\mathcal{O}_{\mathcal{N}})_{1}\cdot(\mathcal{O}_{\mathcal{N}})_{1}\subset(\mathcal{O}_{\mathcal{N}})_{2}$.
Therefore we can assign to $\mathcal{N}$ the following exact sequence

$\displaystyle
0\to(\mathcal{O}_{\mathcal{N}})_{1}\cdot(\mathcal{O}_{\mathcal{N}})_{1}\to(\mathcal{O}_{\mathcal{N}})_{2}\to(\mathcal{O}_{\mathcal{N}})_{2}/(\mathcal{O}_{\mathcal{N}})_{1}\cdot(\mathcal{O}_{\mathcal{N}})_{1}\to
0.$

The Atiyah class of this sequence is represented by a cocycle
$(G_{J}^{a_{1}a_{2}})$ and coincides with the obstruction class of the double
vector bundle (16). It is called the obstruction class of the graded manifold
$\mathrm{F}_{2}(\mathcal{M})$ of degree $2$.

Let us summarize our results in the following theorem.

######  Theorem 20 (Main theorem about the first obstruction class).

The first obstruction class $w_{2}$ for a supermanifold $\mathcal{M}$
coincides with the obstruction class
$\mathrm{At}_{-}(\mathrm{Vb}_{2}(\mathcal{M}))$ of the skew-symetric double
vector bundle $\mathrm{Vb}_{2}(\mathcal{M})$ and with the obstruction class of
the graded manifold $\mathrm{F}_{2}(\mathcal{M})$ of degree $2$.

Note that the map $\mathcal{M}\mapsto\mathrm{F}_{2}(\mathcal{M})$ is a functor
from the category of supermanifolds of odd dimension $2$ to the category of
graded manifolds of degree $2$. Indeed, $\mathrm{F}_{2}$ is a composition of
four functors: $\mathbf{T}$, $\mathrm{gr}$, $\pi$ and $\iota$. This functor
defines an equivalence of the category of supermanifolds of odd dimension $2$
and the category of graded manifolds of degree $2$ with the following
additional condition. If
$\mathcal{N}=(\mathcal{N}_{0},\mathcal{O}_{\mathcal{N}})$ is a graded manifold
of degree $2$, we additionally assume that the locally free sheaf
$\mathcal{E}:=(\mathcal{O}_{\mathcal{N}})_{2}/(\mathcal{O}_{\mathcal{N}})_{1}\cdot(\mathcal{O}_{\mathcal{N}})_{1}$
is isomorphic to the sheaf of sections of
$\mathrm{T}[\bar{1}](\mathcal{N}_{0})$. In this case we can choose local
coordinates in the form (17). Using (17) we can write transition function for
a supermanifold of odd dimension $2$

$\displaystyle y_{j}=F_{j}+G_{j}^{a_{1}a_{2}}\zeta_{a_{1}}\zeta_{a_{2}};$
$\displaystyle\theta_{a}=H_{a}^{i}\zeta_{i}.$

These transition functions satisfy the cocycle condition. Indeed, consider
three charts $\mathcal{U}_{1},\mathcal{U}_{2}$ and $\mathcal{U}_{3}$ with
$\mathcal{U}_{1}\cap\mathcal{U}_{2}\cap\mathcal{U}_{3}\neq\emptyset$. Denote
by $T_{ij}$ the transition function
$T_{ij}:\mathcal{U}_{i}\to\mathcal{U}_{j}$. Consider the following composition
of maps

$T_{31}\circ T_{23}\circ T_{12}=:R.$

Since $\mathrm{F}_{2}$ is a functor, we get

$\mathrm{F}_{2}(T_{31})\circ\mathrm{F}_{2}(T_{23})\circ\mathrm{F}_{2}(T_{12})=\mathrm{F}_{2}(R).$

The composition
$\mathrm{F}_{2}(T_{31})\circ\mathrm{F}_{2}(T_{23})\circ\mathrm{F}_{2}(T_{12})$
is equal to $id$, since the cocycle condition for the graded manifold
$\mathcal{N}$ holds true. Therefore $\mathrm{F}_{2}(R)=id$. The morphism
$R:\mathcal{U}_{1}\to\mathcal{U}_{1}$ is completely defined by its image
$\mathrm{F}_{2}(R)=id$. Therefore, $R=id$.

Denote the category of supermanifold of odd dimension $2$ by
$\mathrm{Smf_{2}}$ and the category of graded manifolds of degree $2$ and of
odd dimension $2$ with the additional condition for
$(\mathcal{O}_{\mathcal{N}})_{2}/(\mathcal{O}_{\mathcal{N}})_{1}\cdot(\mathcal{O}_{\mathcal{N}})_{1}$
as above, by $\mathrm{GrMan_{2}T}$. Now we can summarize our results in the
following theorem.

######  Theorem 21.

The categories $\mathrm{Smf_{2}}$ and $\mathrm{GrMan_{2}T}$ are equivalent.

In [RV] we generalize this theorem to the category of supermanifolds of any
odd dimension.

######  Remark 22.

If our supermanifold $\mathcal{M}$ has the odd dimension $m>2$, we still can
repeat the procedure above. Therefore the functor $\mathrm{F}_{2}$ is a
functor from the category of supermanifolds to the category of graded manifold
of degree $2$. However for any dimension the functor $\mathrm{F}_{2}$ is not
an embedding. In this case from a graded manifold of degree $2$ we can recover
a supermanifold module $\mathcal{J}^{3}$.

Summing up, in this section we showed that the first obstruction class to
splitting of the supermanifold in the sense of [DW2, Section 2] coincides with
the obstruction class of the splitting of the double vector bundle
$\mathrm{gr}\mathbf{T}(\mathcal{M})$ and with the obstruction class of the
splitting of the graded manifold $\mathrm{F}_{2}(\mathcal{M})$. A general
splitting theory for supermanifolds of odd dimension $m$ based on splitting of
the corresponding $m$-fold vector bundles and graded manifolds of degree $m$
will be developed in our oncoming paper.

##  4\. A generalization of the Donagi–Witten construction

In this section we give a construction of functors $\mathrm{F}_{n}$, where
$n=2,3,\ldots,\infty$, from the category of supermanifolds to the category of
graded manifolds of degree $n$. Due to the size of this paper and technical
difficulty of the proof of the main result, the existence of the functor
inverse $\iota$ for any supermanifold of the form
$\mathrm{gr}\circ\mathbf{T}(\mathcal{M})$, where $\mathcal{M}$ is a
supermanifold, we leave details of this proof to Part $II$ of this paper. Here
we give only the idea of the proof.

The functor $\mathrm{F}_{n}$ is again a composition of four functors: the
$(n-1)$-times (or infinity many times for $n=\infty$) iterated antitangent
functor $\mathbf{T}^{(n-1)}:=\mathbf{T}\circ\cdots\circ\mathbf{T}$, the
functor split $\mathrm{gr}$, the functor parity change $\pi$ and the functor
inverse $\iota$. If $\mathcal{M}$ is a supermanifold, then
$\mathbf{T}^{(n-1)}(\mathcal{M})$,
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$ are already defined. Denote
by $\Delta$ the maximal multiplicity free weight system generated by an odd
weight $\alpha$ and by even weights $\beta_{1},\ldots,\beta_{n-1}$, see
Definition 3. We need the following propositions.

######  Proposition 23.

The supermanifold $\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$ is an
$n$-fold vector bundle of type $\Delta$, where $\Delta$ is the maximal
multiplicity free system generated by an odd weight $\alpha$ and by even
weights $\beta_{1},\ldots,\beta_{n-1}$.

###### Proof.

Let $\mathrm{d}_{i}$, where $i=1,\cdots,n-1$, be the iterated de Rham
differentials. They are vector fields in the structure sheaf of
$\mathbf{T}^{(n-1)}(\mathcal{M})$. Denote by $\mathrm{d}_{I}$, where
$I=(i_{1},\ldots,i_{k})$, the composition
$\mathrm{d}_{i_{1}}\circ\ldots\circ\mathrm{d}_{i_{k}}$. Let $\mathcal{I}$ be
the ideal generated by odd elements in
$\mathcal{O}_{\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})}$ and let us
choose a chart $\mathcal{U}$ on $\mathcal{M}$ with local coordinates
$(x_{a},\xi_{b})$. Then the corresponding standard local coordinates in the
chart $\mathbf{T}^{(n-1)}(\mathcal{U})$ are
$(\mathrm{d}_{I}(x_{a}),\mathrm{d}_{J}(\xi_{b}))_{C(I),C(J)\leq n-1}$, where
$C(I)$ is the cardinality of $I$. Hence the corresponding local coordinates in
the chart $\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{U})$ are
$(\mathrm{d}_{I}(x_{a})+\mathcal{I}^{2},\mathrm{d}_{J}(\xi_{b})+\mathcal{I}^{2})_{C(I),C(J)\leq
n-1}$. We assign the weight $\alpha+\beta_{i_{i}}+\cdots+\beta_{i_{k}}$ to
$\mathrm{d}_{I}(Z)+\mathcal{I}^{2}$, where $Z\in\\{x_{a},\xi_{b}\\}$, if
$\mathrm{d}_{I}(Z)$ is odd and the weight $\beta_{i_{i}}+\cdots+\beta_{i_{k}}$
to $\mathrm{d}_{I}(Z)+\mathcal{I}^{2}$ if $\mathrm{d}_{I}(Z)$ is even.

It remains to prove that transition functions in
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$ preserve our weights. Let
$\mathcal{U}^{\prime}$ be another chart on $\mathcal{M}$ with coordinates
$(x^{\prime}_{a},\xi^{\prime}_{b})$ such that
$\mathcal{U}\cap\mathcal{U}^{\prime}\neq\emptyset$. Let

$x_{a}^{\prime}=\sum_{C(K)=2k}F_{K}(x)\xi_{K}$

be the expression of $x_{a}^{\prime}$ in coordinates of $\mathcal{U}$. Then
$\mathrm{d}_{I}(x^{\prime}_{a})+\mathcal{I}^{2}$ in coordinates of
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{U})$ is a sum of monomials in the
following form.

(18)
$H_{I,K}(x)\mathrm{d}_{I_{1}}(x_{a_{1}})\cdots\mathrm{d}_{I_{p}}(x_{a_{p}})\mathrm{d}_{I_{p+1}}(\xi_{b_{1}})\cdots\mathrm{d}_{I_{q}}(\xi_{b_{q}})+\mathcal{I}^{2},$

where $H_{I,K}(x)$ is a certain derivative of $F_{K}(x)$, which has weight
$0$, and $(I_{1},\ldots,I_{q})$ is a decomposition of the sequence $I$ into
$q$ parts. We see that the weights of
$\mathrm{d}_{I}(x^{\prime}_{a})+\mathcal{I}^{2}$ and of (18) coincide.
Similarly for $\mathrm{d}_{I}(\xi^{\prime}_{b})+\mathcal{I}^{2}$. This
completes the proof. ∎

######  Proposition 24.

The functor $\mathrm{gr}\circ\mathbf{T}^{(n-1)}$ is an embedding of the
category of supermanifolds of odd dimension $n$ into the category of $n$-fold
vector bundles of type $\Delta$.

###### Proof.

Let $\mathcal{N}$ be an $n$-fold vector bundle. Let us show that if a preimage
$\mathcal{M}$ of $\mathcal{N}$ exists, it is unique. The idea of the proof of
similar to the idea of the proof of Theorem 21. Assume that there exists
another supermanifold $\mathcal{M}^{\prime}$ with
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})=\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M}^{\prime})$.
By construction we have
$\mathcal{M}_{0}=\mathcal{M}^{\prime}_{0}=\mathcal{N}_{0}$. Let us choose an
atlas $\\{\mathcal{U}_{i}\\}$ on $\mathcal{M}$ and an atlas
$\\{\mathcal{U}^{\prime}_{i}\\}$ on $\mathcal{M}^{\prime}$ such that
$(\mathcal{U}_{i})_{0}=(\mathcal{U}^{\prime}_{i})_{0}$. Clearly such atlases
there exist, since it is sufficient to choose any atlas on $\mathcal{M}_{0}$
of Stein domains. Now consider two charts $\mathcal{U}^{\prime}_{1}$ and
$\mathcal{U}^{\prime}_{2}$ with
$\mathcal{U}^{\prime}_{1}\cap\mathcal{U}^{\prime}_{2}\neq\emptyset$ on
$\mathcal{M}^{\prime}$. Denote by $T^{\prime}$ the transition function
$T^{\prime}:\mathcal{U}^{\prime}_{1}\to\mathcal{U}^{\prime}_{2}$ and by $T$
the transition function $T:\mathcal{U}_{1}\to\mathcal{U}_{2}$. By our
assumption
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(T)=\mathrm{gr}\circ\mathbf{T}^{(n-1)}(T^{\prime})$.
But this implies that $T=T^{\prime}$. The proof is complete. ∎

######  Remark 25.

The question when the functor $\mathrm{gr}\circ\mathbf{T}^{(n-1)}$ possesses a
preimage is treated in [RV].

Now we can use results of [BGR] and [Vi1] to define the functor inverse. Note
that these two results lead to two different approaches to the problem. Let us
start with a description of results [BGR] or [Vi1]. In [BGR] a functor
$\mathrm{G}_{n}$ was constructed from the category of graded manifolds of
degree $n$ to the category of $n$-fold symmetric vector bundles. In [Vi1] a
functor $\mathrm{V}_{n}$ was constructed from the category of graded manifolds
of degree $n$ to the category of $n$-fold vector bundles with $n-1$ odd
commutative homological vector fields. In both cases it was proven that these
functors are equivalence of categories. In Section 3 we saw that due to the
fact that the first obstruction class $\omega_{2}$ is an element in

$H^{1}(\mathcal{M}_{0},\mathrm{T}[\bar{1}](\mathcal{M}_{0})\otimes\bigwedge^{2}\mathbb{E}^{*})$

we can construct a graded manifold of degree $2$ with transition functions
(17) from a double vector bundle given by (16). In terms of [BGR] this means
that the double vector bundle with the obstruction class $\omega_{2}$ is
symmetric and $\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{M})$ is in the
image of the functor $\mathrm{G}_{2}$ constructed in [BGR]. Our functor
inverse $\iota$ is the inversion of the equivalence $\mathrm{G}_{2}$. More
generally we have

######  Theorem 26.

For a supermanifold $\mathcal{M}$ of odd dimension $n$ the image
$\pi\circ\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$ is a symmetric
$n$-fold vector bundle. Therefore there exists a graded manifold $\mathcal{N}$
of degree $n$ such that
$\mathrm{G}_{n}(\mathcal{N})\simeq\pi\circ\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$.
In other words the functor inverse $\iota=\mathrm{G}_{n}^{-1}$ is defined on
the image of the functor $\pi\circ\mathrm{gr}\circ\mathbf{T}^{(n-1)}$.

Let us describe another approach based on results of [Vi1]. The graded
supermanifold $\mathrm{gr}\circ\mathbf{T}(\mathcal{M})$ possesses a natural
odd homological vector field. Indeed, let $\mathrm{d}_{R}$ be the exterior
derivative (de Rham defferential) of the supermanifold $\mathcal{M}$. Clearly
$\mathrm{d}_{R}$ is an odd homological vector field in the structure sheaf
$\mathcal{O}_{\mathbf{T}(\mathcal{M})}$ of $\mathbf{T}(\mathcal{M})$. Denote
by $\mathcal{I}$ the sheaf of ideals locally generated by all odd variables of
$\mathcal{O}_{\mathbf{T}(\mathcal{M})}$. Then we have

$\mathrm{d}_{R}(\mathcal{I})\subset\mathcal{O}_{\mathbf{T}(\mathcal{M})}\quad\text{and}\quad\mathrm{d}_{R}(\mathcal{I}^{2})\subset\mathcal{I}.$

Therefore we have an induced operator in the structure sheaf of
$\mathrm{gr}\circ\mathbf{T}(\mathcal{M})$

$\mathcal{O}_{\mathrm{gr}\circ\mathbf{T}(\mathcal{M})}=\mathrm{gr}\mathcal{O}_{\mathbf{T}(\mathcal{M})}=\bigoplus_{p\geq
0}\mathcal{I}^{p}/\mathcal{I}^{p+1}.$

Clearly this induced operator is odd and homological. Hence the double vector
bundle $\mathrm{gr}\circ\mathbf{T}(\mathcal{M})$ is a double vector bundle
with a homological vector field. Therefore by [Vi1] there exists a graded
manifold $\mathrm{V}_{2}^{-1}(\mathrm{gr}\circ\mathbf{T}(\mathcal{M}))$ of
degree $2$. This procedute can be generalized. Now we can formulate a general
result.

######  Theorem 27.

For a supermanifold $\mathcal{M}$ of odd dimension $n$ the image
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$ is an $n$-fold vector bundle
with $n-1$ commuting odd homological vector fields. Therefore there exists a
graded manifold $\mathcal{N}$ of degree $n$ such that
$\mathrm{V}_{n}(\mathcal{N})\simeq\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$.
In other words the functor $\iota^{\prime}=\mathrm{V}_{n}^{-1}$ is defined on
the image of $\mathrm{gr}\circ\mathbf{T}^{(n-1)}$.

The idea of the proof is simple: the $(n-1)$-times iterated tangent functor
leads to $n-1$ commuting de Rham differentials, which induce $(n-1)$ commuting
odd homological vector fields on
$\mathrm{gr}\circ\mathbf{T}^{(n-1)}(\mathcal{M})$. We leave details of the
proof to Part II of this paper. Note that in this case the functor parity
change $\pi$ is included in the functor $\mathrm{V}_{n}$, therefore we do not
need to apply it explicitly. We conclude this section with the following
proposition

######  Proposition 28.

The functors $\iota\circ\pi\circ\mathrm{gr}\circ\mathbf{T}^{(n-1)}$ and
$\iota^{\prime}\circ\mathrm{gr}\circ\mathbf{T}^{(n-1)}$ are embeddings of the
category of supermanifolds of odd dimension $n$ to the category of graded
manifolds of degree $n$.

###### Proof.

The proposition follows from Proposition 24, from the fact that $\pi$ is an
equivalence of categories of $n$-fold vector bundles and from Theorems 26 and
27. ∎

The inverse limit of functors $\mathrm{F}_{n}$ is denoted by
$\mathrm{F}_{\infty}$. That is if $\mathcal{M}$ is a supermanifold, we put
$\mathrm{F}_{\infty}(\mathcal{M}):=\varprojlim\mathrm{F}_{n}(\mathcal{M})$ and
the same for morphisms.

##  5\. Donagi–Witten functor for Lie superalgebras and
Lie supergroups

In this section, we adapt the results of Section 3 to the case of Lie
superagebras and Lie supergroups. We show that there is an analogue of the
functor $\mathrm{F}_{2}$ in the category of Lie superalgebras. We denote by
$\mathrm{sLieAlg}$ the category of Lie superalgebras, by
$\mathrm{grLieAlg}_{n}$ the category of $\mathbb{Z}$-graded Lie superalgebras
of type $\\{0,1,\ldots,n\\}$ with $|1|=\bar{1}$ and by
$\mathrm{grLieAlg}_{\infty}$ the category of non-negatively
$\mathbb{Z}$-graded Lie superalgebras.

###  5.1. Donagi and Witten construction for Lie superalgebras

In this section we will construct a functor
$\mathrm{F}^{\prime}_{2}:\mathrm{sLieAlg}\to\mathrm{grLieAlg}_{2}$. In next
sections we will use this construction to obtain a functor from the category
of Lie superalgebras $\mathrm{sLieAlg}$ to the category
$\mathrm{grLieAlg}_{\infty}$. As in 3 the functor $\mathrm{F}^{\prime}_{2}$ is
a composition of four functors, which we denote by $\mathbf{T}^{\prime}$
(tangent), $\mathrm{gr}^{\prime}$ (split), $\pi^{\prime}$ (parity change) and
$\iota^{\prime}$ (inverse). (To distinguish the category of Lie superalgebras
in the case we will use superscript. That is $\mathbf{T}^{\prime}$ instead of
$\mathbf{T}$ and so on.) Let
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a Lie
superalgebra. Recall that to define a Lie superalgebra we need to define a Lie
algebra $\mathfrak{g}_{\bar{0}}$, a $\mathfrak{g}_{\bar{0}}$-module
$\mathfrak{g}_{\bar{1}}$ and a symmetric $\mathfrak{g}_{\bar{0}}$-module map
$[\cdot,\cdot]:\mathfrak{g}_{\bar{1}}\otimes\mathfrak{g}_{\bar{1}}\to\mathfrak{g}_{\bar{0}}$
such that $[[Y,Y],Y]=0$, for any $Y\in\mathfrak{g}_{\bar{1}}$.

Tangent functor $\mathbf{T}^{\prime}$. Since a Lie superalgebra
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ is a vector
superspace, its antitangent bundle is the following linear superspace

$\mathbf{T}^{\prime}(\mathfrak{g})=\mathfrak{g}\oplus\mathrm{d}(\mathfrak{g}).$

Here $\mathrm{d}\mathfrak{g}$ denote a copy of $\mathfrak{g}$ with reversed
parity. In more details, $V=\mathfrak{g}\oplus\mathrm{d}(\mathfrak{g})$ is a
linear superspace with the underlying space
$V_{\bar{0}}=\mathfrak{g}_{\bar{0}}\oplus\mathrm{d}\mathfrak{g}_{\bar{1}}$ and
with the structure sheaf $\mathcal{F}_{V_{\bar{0}}}\otimes
S^{*}(\mathfrak{g}^{*}_{\bar{1}}\oplus\mathrm{d}\mathfrak{g}^{*}_{\bar{0}})$.
If $X\in\mathfrak{g}$, sometimes we will denote by $\mathrm{d}(X)$ the
corresponding element in $\mathrm{d}(\mathfrak{g})$. Further the Lie
superalgebra structure on $\mathbf{T}^{\prime}(\mathfrak{g})$ is defined as
follows. The Lie bracket on the vector subspece $\mathfrak{g}\oplus\\{0\\}$
coincides with the Lie bracket of $\mathfrak{g}$, further
$\mathrm{d}(\mathfrak{g})$ is a $\mathfrak{g}$-module, where the action of
$\mathfrak{g}$ coincide with the adjoint action up to sign

$[X,\mathrm{d}Y]:=(-1)^{|X|}\mathrm{d}([X,Y]).$

and the product $[\mathrm{d}(\mathfrak{g}),\mathrm{d}(\mathfrak{g})]=0$ is
trivial.

######  Remark 29.

The Lie superalgebra $\mathbf{T}^{\prime}(\mathfrak{g})$ sometimes is called
in the literature a Takiff superalgebra, named after the author of [T]. This
Lie superalgebra is a Lie superalgebra in the form
$\mathfrak{g}\otimes_{\mathbb{K}}\mathbb{K}[\tau]$, where $\tau$ is an odd
element and $\mathbb{K}[\tau]$ are all polynomials in $\tau$. Summing up, in
our paper $\mathbf{T}^{\prime}$ is derived from both “Takiff” and
“antitangent”.

Functor split $\mathrm{gr}^{\prime}$. The functor $\mathrm{gr}^{\prime}$ is
defined as follows. For a Lie superalgebra $\mathfrak{g}$ we put
$\mathrm{gr}^{\prime}(\mathfrak{g}):=\mathfrak{g}^{\prime}$, where
$\mathfrak{g}^{\prime}$ is obtained from the Lie superalgebra $\mathfrak{g}$
putting $[\mathfrak{g}^{\prime}_{\bar{1}},\mathfrak{g}^{\prime}_{\bar{1}}]=0.$
In [Vi2, Theorem 3], see also Therem 36 below, it was shown that
$\operatorname{Lie}(\mathrm{gr}\mathcal{G})=\mathrm{gr}^{\prime}(\operatorname{Lie}\mathcal{G})$
for any Lie supergroup $\mathcal{G}$. In other words $\mathfrak{g}^{\prime}$
is the Lie superalgebra of $\mathrm{gr}(\mathcal{G})$, where $\mathcal{G}$ is
a Lie supergroup with the Lie superalgebra $\mathfrak{g}$. Clearly
$\mathrm{gr}^{\prime}$ is defined on morphisms as well. Let us compute the
composition of the functors $\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}$.

######  Lemma 30.

Let $\mathfrak{g}$ be a Lie superalgebra. Then the Lie superalgebra
$\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g}))$ is equal to
$\mathbf{T}^{\prime}(\mathfrak{g})$ as $\mathfrak{g}_{\bar{0}}$-modules. And
the following holds true for the Lie superalgebra multiplication

$\displaystyle[\mathfrak{g}_{\bar{1}},\mathfrak{g}_{\bar{1}}]=0,\quad[\mathfrak{g}_{\bar{1}},\mathrm{d}(\mathfrak{g}_{\bar{0}})]=0,\quad[\mathrm{d}(\mathfrak{g}),\mathrm{d}(\mathfrak{g})]=0.$

and $[Y_{1},\mathrm{d}(Y_{2})]=-\mathrm{d}([Y_{1},Y_{2}])$ for
$Y_{1},Y_{2}\in\mathfrak{g}_{\bar{1}}$.

######  Remark 31.

Note that the Lie algebra $\mathrm{gr}\mathbf{T}(\mathfrak{g})$ keeps all the
information on the bracket on $\mathfrak{g}$. We can see the Lie superalgebra
$\mathfrak{a}=\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g}))$ as a
$\mathbb{Z}\times\mathbb{Z}$-graded Lie superalgebra of type
$\\{0,\alpha,\beta,\alpha+\beta\\}$, where $\alpha$ is odd and $\beta$ is
even. Indeed, we put

$\mathfrak{a}_{0}:=\mathfrak{g}_{\bar{0}},\quad\mathfrak{a}_{\alpha}:=\mathfrak{g}_{\bar{1}},\quad\mathfrak{a}_{\beta}:=\mathrm{d}\mathfrak{g}_{\bar{1}},\quad\mathfrak{a}_{\alpha+\beta}:=\mathrm{d}\mathfrak{g}_{\bar{0}}.$

Lemma 30 implies that the multiplication in $\mathfrak{a}$ is
$\mathbb{Z}\times\mathbb{Z}$-graded.

Functor parity change $\pi^{\prime}$. In previous sections we reminded how to
define the functor $\pi$ for double vector bundles. In this section we show
that on the parity reversed double vector bundle
$\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$, where $\mathcal{G}$ is a
Lie supergroup, a Lie supergroup structure can be defined. We start with the
case of Lie superalgebras.

We define the functor $\pi^{\prime}$ on the image of the composition of
functors $\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}$ as follows. The Lie
superalgebra $\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g}))$
possesses a parity reversion of the weight $\beta$. Indeed, by definition the
Lie superalgebra
$\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g})))$ is
equal to $\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g}))$ as
$\mathfrak{g}_{\bar{0}}$-modules, but now we assume that elements $X$ and
$\mathrm{d}(X)$ have the same parities. In other words we assume that
$\mathrm{d}$ is even. More precisely, let
$\mathfrak{a}=\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g}))$. Denote
by $\mathfrak{h}$ the vector superspace
$\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g})))$. We
have

$\mathfrak{h}=\bigoplus_{\delta\in\Delta}\mathfrak{h}_{\delta},$

where $\Delta=\\{0,\alpha,\beta,\alpha+\beta\\}$ with
$|\alpha|=|\beta|=\bar{1}$. Let us define a Lie superalgebra structure on
$\mathfrak{h}$ of type $\Delta$. We denote by $[\,,]_{\mathfrak{a}}$ the Lie
bracket on $\mathfrak{a}$ and by $[\,,]_{\mathfrak{h}}$ the Lie bracket on
$\mathfrak{h}$.

######  Proposition 32.

The Lie superalgebra structure on $\mathfrak{h}$ is defines by the following
data.

  * •

We set $[X,Y]_{\mathfrak{h}}=[X,Y]_{\mathfrak{a}}$ for all homogeneous $X,Y$,
except for the case $X\in\mathfrak{h}_{\alpha}$ and
$Y\in\mathfrak{h}_{\beta}$.

  * •

For $X\in\mathfrak{h}_{\alpha}$ and $Y\in\mathfrak{h}_{\beta}$ we set
$[X,Y]_{\mathfrak{h}}:=[X,Y]_{\mathfrak{a}}=-[Y,X]_{\mathfrak{a}}=:[Y,X]_{\mathfrak{h}}$.

###### Proof.

Step 1. Clearly
$\mathfrak{h}_{\bar{0}}=\mathfrak{h}_{0}\oplus\mathfrak{h}_{\alpha+\beta}$ is
a Lie algebra, namely it is a semidirect product of the Lie algebra
$\mathfrak{a}_{0}$ and $\mathfrak{a}_{0}$-module
$\mathfrak{a}_{\alpha+\beta}$.

Step 2. $\mathfrak{h}_{\bar{1}}$ is a $\mathfrak{h}_{\bar{0}}$-module.
Clearly, $\mathfrak{h}_{\bar{1}}$ is a $\mathfrak{h}_{0}$-module and
$\mathfrak{h}_{\alpha+\beta}$ acts trivially on $\mathfrak{h}_{\bar{1}}$.

Step 3. Note that the map
$\mathfrak{h}_{\bar{1}}\otimes\mathfrak{h}_{\bar{1}}\to\mathfrak{h}_{\bar{0}}$
induced by $[\,,]_{\mathfrak{h}}$ is an $\mathfrak{h}_{\bar{0}}$-module map.
It remains only to check Jacobi identity for homogeneous
$X,Y,Z\in\mathfrak{h}_{\bar{1}}=\mathfrak{h}_{\alpha}\oplus\mathfrak{h}_{\beta}$.
However the product of any three elements of this form is $0$. ∎

Later we will give another proof of this result. It is easy to check that
$\pi^{\prime}$ is defined on the morphisms of the form
$\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\phi)$, where
$\phi:\mathfrak{g}\to\mathfrak{g}_{1}$ is a morphism of Lie superalgebras.

Functor inverse $\iota^{\prime}$. Above we explained the meaning of the
functor inverse $\iota$ for supermanifolds. Now we show that an analogue of
this functor can be defined in the category of Lie superalgebras. Let
$\mathfrak{g}$ be a Lie superalgebra. Denote by
$\mathfrak{p}:=\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g})$
a $\mathbb{Z}$-graded subsuperspace of
$\mathfrak{h}:=\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g})))$
with support $\\{0,1,2\\}$, where $|1|=\bar{1}$, given by

$\displaystyle\mathfrak{p}_{0}:=\mathfrak{h}_{\bar{0}},\quad\mathfrak{p}_{1}=\\{Y+\mathrm{d}Y\,\,|\,\,Y\in\mathfrak{g}_{\bar{1}}\\}\subset\mathfrak{h}_{\alpha}\oplus\mathfrak{h}_{\beta},\quad\mathfrak{p}_{2}:=\mathfrak{h}_{\alpha+\beta}.$

######  Proposition 33.

The superspace $\mathfrak{p}$ is a $\mathbb{Z}$-graded Lie superalgebra with
support $\\{0,1,2\\}$ and with $|1|=\bar{1}$.

###### Proof.

The proposition follows from a direct calculation. For example let us show
that $[\mathfrak{p}_{1},\mathfrak{p}_{1}]\subset\mathfrak{p}_{2}$. For
$Y_{1},Y_{2}\in\mathfrak{g}_{1}$ we have

$\displaystyle[Y_{1}+\mathrm{d}Y_{1},Y_{2}+\mathrm{d}Y_{2}]=$
$\displaystyle[Y_{1},Y_{2}]+[\mathrm{d}Y_{1},Y_{2}]+[Y_{1},\mathrm{d}Y_{2}]+[\mathrm{d}Y_{1},\mathrm{d}Y_{2}]=$
$\displaystyle
d[Y_{1},Y_{2}]+d[Y_{1},Y_{2}]=2d[Y_{1},Y_{2}]\in\mathfrak{p}_{2}.$

∎

Note that the functor $\iota^{\prime}$ is defined only for Lie superalgebras
of the form
$\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g})$.
Further, if $f:\mathfrak{g}\to\mathfrak{g}_{1}$ is a morphism of Lie
superalgebras, then the morphism
$\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(f)$ can be
restricted to the subalgebras
$\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g})\to\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g}_{1})$.
Summing up, we constructed the following functor

$\mathrm{F}^{\prime}_{2}:=\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}:\mathrm{sLieAlg}\to\mathrm{grLieAlg}_{2}.$

This functor is an embedding of category $\mathrm{sLieAlg}$ into
$\mathrm{grLieAlg}_{2}$.

######  Remark 34.

(1) It is necessary to apply the functor $\pi^{\prime}$ before the functor
inverse. Indeed, if $Y$ and $\mathrm{d}Y$, where $Y\in\mathfrak{p}_{1}$, have
different parities, we will get

(19)
$\begin{split}[Y_{1}+\mathrm{d}Y_{1},Y_{2}+\mathrm{d}Y_{2}]=&[Y_{1},Y_{2}]+[\mathrm{d}Y_{1},Y_{2}]+[Y_{1},\mathrm{d}Y_{2}]+[\mathrm{d}Y_{1},\mathrm{d}Y_{2}]=\\\
&\mathrm{d}[Y_{1},Y_{2}]-\mathrm{d}[Y_{1},Y_{2}]=0.\end{split}$

(2) Another observation is the following. Lie superalgebras of the form
$\mathrm{gr}^{\prime}(\mathfrak{k})$, where $\mathfrak{k}$ is a Lie
superalgebra possesses another parity reversion

$\mathfrak{k}_{\bar{0}}\oplus\mathfrak{k}_{\bar{1}}\longmapsto\mathfrak{k}_{\bar{0}}\oplus\mathfrak{k}_{\bar{1}}[\bar{1}],$

where $[\bar{1}]$ is the shift of parity be $\bar{1}$. In other words we
assume that all odd elements are even. Clearly
$\mathfrak{k}_{\bar{0}}\oplus\mathfrak{k}_{\bar{1}}[\bar{1}]$ is a Lie algebra
(not superalgebra). In this case the same argument, see (19), shows that the
functor inverse loses information about original Lie superalgebra.

###  5.2. Donagi and Witten construction for Lie supergroups

In this section we develop the construction of Section 3 for Lie supergroups.
In details, we apply the functor $\mathrm{F}_{2}$ to a Lie supergroup
$\mathcal{G}$. Again $\mathrm{F}_{2}$ is a composition of four functors:
$\mathbf{T}$ (tangent), $\mathrm{gr}$ (split), $\pi$ (parity change) and
$\iota$ (inverse).

Tangent functor $\mathbf{T}$. Let $\mathcal{G}$ be a Lie supergroup with
multiplication $\mu$, inversion $\kappa$ and identity $e$. Clearly,
$\mathbf{T}(\mathcal{G})$ is a Lie supergroup as the functor $\mathbf{T}$
preserves products. Indeed, since $\mathbf{T}$ is a functor the morthisms
$\mathbf{T}(\mu)$, $\mathbf{T}(\kappa)$ and $\mathbf{T}(e)$ satisfy the Lie
supergroup axioms.

######  Proposition 35.

The Lie superalgebra of the Lie supergroup $\mathbf{T}(\mathcal{G})$ is equal
to $\mathbf{T}^{\prime}(\mathfrak{g})$, where
$\mathfrak{g}=\mathrm{Lie}(\mathcal{G})$.

###### Proof.

We shall follow the definition of the Lie functor of a Lie supergroup given in
[BLMS]. Let $X_{1},X_{2}\in\mathrm{T}_{e}\mathcal{G}$ be homogeneous. Any
homogeneous vector $X\in\mathrm{T}_{e}\mathcal{G}$ can be represented by a
curve $g:\mathcal{R}\to\mathcal{G}$ where $\mathcal{R}$ is $\mathbb{R}^{1|0}$
or $\mathbb{R}^{0|1}$ depending on the parity of $X$, so that

$g^{\ast}(f)=f(e)+tX(f)\bmod I_{t}^{2},$

for any $f\in\mathcal{O}_{\mathcal{G}}(U)$ defined in a neighbourhood
$U\subset\mathcal{G}_{0}$ of $e$, where $t$ is the distinguished coordinate on
$\mathcal{R}$ of parity $\bar{0}$ or $\bar{1}$, and $I_{t}=\langle t\rangle$
is the ideal generated by $t$. Say $X_{1},X_{2}\in\mathrm{T}_{e}\mathcal{G}$
are represented by curves $g_{i}:\mathcal{R}_{i}\to\mathcal{G}$, where $i=1,2$
and $s,t$ be the distinguished coordinates on $\mathcal{\mathbb{R}}_{1}$,
$\mathcal{\mathbb{R}}_{2}$, respectively. Then there exist
$Z\in\mathrm{T}_{e}\mathcal{G}$ such that for
$\Phi:=g_{1}g_{2}(g_{1})^{-1}(g_{2})^{-1}:\mathcal{R}_{1}\times\mathcal{R}_{2}\to\mathcal{G}$
we have

$\Phi^{\ast}(f)=f(e)+(-1)^{|X||Y|}tsZ(f)\,\bmod{I_{t}^{2}+I_{s}^{2}}.$

The bracket $[X_{1},X_{2}]$ is defined as $Z$.

We are ready to describe the Lie superalgebra
$\operatorname{Lie}(\mathbf{T}G)$. Let $(U,x^{A})$ be a chart around $e\in
U\subset\mathcal{G}_{0}$ with local coordinates $(x^{A})$. Without loss of
generality we may assume that $x^{A}(e)=0$. The group law assures that the
Taylor expansion of the multiplication map
$m:\mathcal{G}\times\mathcal{G}\to\mathcal{G}$ has the form

$m((x^{A}),(\underline{x}^{A}))=(x^{A}+\underline{x}^{A}+\frac{1}{2}Q^{A}_{BC}x^{C}\underline{x}^{B}\bmod
I),$

where $I=\langle x^{2},\underline{x}^{2}\rangle$. Moreover,
$Q^{A}_{BC}=-(-1)^{\tilde{B}\tilde{C}}Q^{A}_{BC}$. The inverse $(x^{A})^{-1}$
is given by $(-x^{A})(\bmod I)$. The multiplication on
$\mathbf{T}(\mathcal{G})$ is given by $\mathbf{T}(m)$, ie.

$(x^{A},\dot{x}^{A})\cdot(\underline{x}^{A},\dot{\underline{x}}^{A})=\left(x^{A}+\underline{x}^{A}+\frac{1}{2}Q^{A}_{BC}x^{C}\underline{x}^{B},\dot{x}^{A}+\dot{\underline{x}}^{A}+\frac{1}{2}Q^{A}_{BC}(\dot{x}^{C}\underline{x}^{B}+x^{C}\dot{\underline{x}}^{B})\right).$

The inverse of $(x^{A},\dot{x}^{A})$ is $(-x^{A},-\dot{x}^{A})$ modulo $I$,
hence

$\displaystyle(x^{A},\dot{x}^{A})\cdot(\underline{x}^{A},\underline{\dot{x}}^{A})\cdot(-x^{A},-\dot{x}^{A})\cdot(-\underline{x}^{A},-\underline{\dot{x}}^{A})\equiv$
$\displaystyle\left(x^{A}+\underline{x}^{A}+\frac{1}{2}Q^{A}_{BC}x^{C}\underline{x}^{B},\dot{x}^{A}+\dot{\underline{x}}^{A}+\frac{1}{2}Q^{A}_{BC}(\dot{x}^{C}\underline{x}^{B}+x^{C}\dot{\underline{x}}^{B}\right)\cdot$
$\displaystyle\cdot\left(-x^{A}-\underline{x}^{A}+\frac{1}{2}Q^{A}_{BC}x^{C}\underline{x}^{B},-\dot{x}^{A}-\dot{\underline{x}}^{A}+\frac{1}{2}Q^{A}_{BC}(\dot{x}^{C}\underline{x}^{B}+x^{C}\dot{\underline{x}}^{B})\right)$
$\displaystyle\equiv(Q^{A}_{BC}x^{C}\underline{x}^{B},Q^{A}_{BC}(\dot{x}^{C}\underline{x}^{B}+x^{C}\underline{\dot{x}}^{B}))\,(\bmod\,I).$

We read off from the last formula that
$[\partial_{x^{C}},\partial_{x^{B}}]=Q^{A}_{BC}\partial_{x^{A}}$,
$[\partial_{\dot{x}^{C}},\partial_{x^{B}}]=Q^{A}_{BC}\partial_{\dot{x}^{A}}$,
and $[\partial_{\dot{x}^{A}},\partial_{\dot{x}^{B}}]=0$. This completes the
proof. ∎

The following theorem was proved [Vi2].

######  Theorem 36.

[Vi2, Theorem 3] Let $\mathcal{G}$ be a Lie supergroup corresponding to the
super Harish-Chandra pair $(\mathcal{G}_{0},\mathfrak{g})$, where
$\mathcal{G}_{0}$ is the underlying space of $\mathcal{G}$ and
$\mathfrak{g}=\mathrm{Lie}(\mathcal{G})$. Then $\mathrm{gr}(\mathcal{G})$ is a
Lie supergroup corresponding to the following super Harish-Chandra pair
$(\mathcal{G}_{0},\mathfrak{g}^{\prime})$, where
$\mathfrak{g}^{\prime}=\mathrm{gr}^{\prime}(\mathfrak{g})$.

Proposition 35 and Theorem 36 imply the following corollary.

######  Corollary 37.

Let $\mathcal{G}$ be a Lie supergroup, then

$\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\operatorname{Lie}\mathcal{G})=\mathrm{Lie}(\mathrm{gr}\circ\mathbf{T}(\mathcal{G})).$

As a consequence we get that the Lie supergroup
$\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ is a graded Lie supergroup of type
$\\{0,\alpha,\beta,\alpha+\beta\\}$, where $|\alpha|=\bar{1}$ and
$|\beta|=\bar{0}$, with the graded Harish-Chandra pair
$(\mathcal{G}_{0},\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g}))$
of the same type $\\{0,\alpha,\beta,\alpha+\beta\\}$. (Compare also with
Proposition 23.)

Functor parity change $\pi$. Let us describe the functor $\pi$ using graded
Harish-Chandra pairs. Let
$(\mathcal{G}_{0},\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g}))$
be the graded Harish-Chandra pair of type $\\{0,\alpha,\beta,\alpha+\beta\\}$
of the Lie supergroup $\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ as above. In
this case $\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ is a group object in the
category of double vector bundles. As we have seen in Section 3, the double
vector bundle $\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ possesses the
following parity reversion: we change the parity of the weight $\beta$ from
even to odd. We denote this new vector bundle by
$\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$. Further we can define a Lie
supergroup structure on $\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$
using the graded Harish-Chandra pair
$(\mathcal{G}_{0},\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}(\mathfrak{g}))$
of type $\Delta=\\{0,\alpha,\beta,\alpha+\beta\\}$, where
$|\alpha|=|\beta|=\bar{1}$. Now the Lie supergroup
$\pi\circ\mathrm{gr}\circ\mathbf{T}^{\prime}(\mathcal{G})$ is defined.

Let us describe the Lie supergroup morphisms of
$\pi\circ\mathrm{gr}\circ\mathbf{T}^{\prime}(\mathcal{G})$ using the language
of double vector bundles. If $\mu$, $\kappa$ and $e$ are group morphisms of
the Lie supergroup $\mathcal{G}$, then
$\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ is a group object in the category of
double vector bundles with the structure morphisms
$\mathrm{gr}\circ\mathbf{T}(\mu)$, $\mathrm{gr}\circ\mathbf{T}(\kappa)$ and
$\mathrm{gr}\circ\mathbf{T}(e)$. Since the category of double vector bundles
possesses the parity change: $\beta$ even to $\beta$ odd, we denote by
$\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ with
$\pi\circ\mathrm{gr}\circ\mathbf{T}(\mu)$,
$\pi\circ\mathrm{gr}\circ\mathbf{T}(\kappa)$ and
$\pi\circ\mathrm{gr}\circ\mathbf{T}(e)$ the result of this parity change.
Formulas (6) tells us that this definition coincides with the definition in
terms of graded Harish-Chandra pairs.

Functor inverse $\iota$. In Proposition 33 we saw that the Lie superalgebra
$\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g})$
possesses a $\mathbb{Z}$-graded Lie subsuperalgebra
$\mathfrak{p}=\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g})$.
We define by $\iota\circ\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ the
corresponding Lie subsupergroup. More precisely, we define
$\iota\circ\pi\circ\mathrm{gr}\circ\mathbf{T}(\mathcal{G})$ using the graded
Harish-Chandra pairs
$(\mathcal{G}_{0},\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathbf{T}^{\prime}(\mathfrak{g})).$

##  6\. Generalized Donagi–Witten construction for
Lie superalgebras

In Section 4 we constructed an injective functor
$\mathrm{F}:=\mathrm{F}_{\infty}$ from the category of supermanifolds to the
category of graded manifolds. In this section we show that this functor can be
defined in the category of Lie supergroups. We start with Lie superalgebras.
In more details we will construct a functor $\mathrm{F}^{\prime}$ from the
category of Lie superalgebras $\mathrm{sLieAlg}$ to the category of non-
negatively $\mathbb{Z}$-graded Lie algebras $\mathrm{grLieAlg}_{\infty}$.
Further we will use these results for the category of Lie supergroups. Again
the functor $\mathrm{F}^{\prime}$ is a composition of four functors: the
iterated antitangent functor $\mathbf{T}^{\prime\infty}$, the functor split
$\mathrm{gr}^{\prime}$, the functor parity change $\pi^{\prime}$ and the
functor inverse $\iota^{\prime}$.

###  6.1. Iterated tangent functor $\mathbf{T}^{\prime\infty}$

Let $\mathfrak{g}$ be a Lie superalgebra. Let us describe the superalgebra
$\mathbf{T}^{\prime\infty}(\mathfrak{g})$, where
$\mathbf{T}^{\prime\infty}:=\mathbf{T}^{\prime}\circ\mathbf{T}^{\prime}\circ\cdots$
is the infinitely many times iterated antitangent functor. Let us consider
first twice iterated tangent functor $\mathrm{T^{\prime}}^{2}(\mathfrak{g})$.
By definition

$\mathrm{T^{\prime}}^{2}(\mathfrak{g})=\mathbf{T}^{\prime}(\mathbf{T}^{\prime}(\mathfrak{g}))=\mathbf{T}^{\prime}(\mathfrak{g}\oplus\mathrm{d}_{1}(\mathfrak{g}))=\mathfrak{g}\oplus\mathrm{d}_{1}(\mathfrak{g})\oplus\mathrm{d}_{2}(\mathfrak{g}\oplus\mathrm{d}_{1}(\mathfrak{g})).$

We replaced $\mathrm{d}$ from Section 5.1 by $\mathrm{d}_{1}$ and
$\mathrm{d}_{2}$ is the second de Rham differential. The multiplication in
$\mathrm{T^{\prime}\circ T^{\prime}}(\mathfrak{g})$ is defined in a natural
way. Moreover we can easily verify the following lemma.

######  Lemma 38.

To obtain the multiplication in $\mathrm{T^{\prime}}^{2}(\mathfrak{g})$ we can
use the following rule:
$[\mathrm{d}_{i}X,\mathrm{d}_{j}Y]=(-1)^{|X|}\mathrm{d}_{i}\mathrm{d}_{j}([X,Y])$
for any $X,Y\in\mathrm{T^{\prime}}^{2}(\mathfrak{g})$.

######  Corollary 39.

Since the operators $\mathrm{d}_{i}$ assumed to be odd, we have

$\mathrm{d}_{i}\mathrm{d}_{i}([X,Y])=0$

for any $X,Y\in\mathrm{T^{\prime}}^{2}(\mathfrak{g})$.

To define the functor $\mathrm{T^{\prime}}^{\infty}$ we use de Rham
differentials $\mathrm{d}_{1},\mathrm{d}_{2},\ldots,\mathrm{d}_{n},\ldots.$
Lemma 38 and Corrolary 39 imply the following lemma.

######  Lemma 40.

The Lie superalgebra $\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})$ is an
infinite dimensional Lie superalgebra with the underlying vector space

$\bigoplus_{p\geq
0}\bigoplus_{i_{1}<\cdots<i_{p}}\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(\mathfrak{g})$

and multiplication defined by the following formula

$[\mathrm{d}_{i}X,\mathrm{d}_{j}Y]=(-1)^{|X|}\mathrm{d}_{i}\mathrm{d}_{j}([X,Y])$

for any $X,Y\in\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})$.

###  6.2. A connection with the functor of points for Lie superalgebras

For details about the functor of points for Lie superalgebras we refer for
example to [Gav, Section 2.2.4]. Let us recall this construction. To any Lie
superalgebra $\mathfrak{g}$ we can associate a functor $L_{\mathfrak{g}}$ from
the category of supercommutative algebras to the category of Lie algebras. It
is defined as follows

$L_{\mathfrak{g}}(A)=(A\otimes\mathfrak{g})_{\bar{0}},$

where $A$ is a super-commutative algebra. The product $A\otimes\mathfrak{g}$
is a Lie superalgebra with the following multiplication

(20) $[a\otimes X,a^{\prime}\otimes
X^{\prime}]:=(-1)^{|X||a^{\prime}|}aa^{\prime}\otimes[X,X^{\prime}].$

Comparing with Lemma 40 we see that the Lie superalgebra
$\mathrm{T^{\prime}}^{k}(\mathfrak{g})$ is isomorphic to the Lie superalgebra
$\bigwedge(\xi_{1},\ldots,\xi_{k})\otimes\mathfrak{g}$. Clearly,
$\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})$ is isomorphic to
$\bigwedge(\xi_{1},\xi_{2},\ldots)\otimes\mathfrak{g}$, where
$\bigwedge(\xi_{1},\xi_{2},\ldots)$ is the Grassmann algebra with infinitely
many variables $\xi_{1},\xi_{2},\ldots$. Now we see that

$L_{\mathfrak{g}}(\bigwedge(\xi_{1},\xi_{2},\ldots))=\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})_{\bar{0}}.$

###  6.3. Functors split $\mathrm{gr}^{\prime}$ and parity change
$\pi^{\prime}$

The functor split is defined as above.

######  Remark 41.

The Lie superalgebra
$\mathrm{gr}^{\prime}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})$ is a
graded Lie superalgebra with support $\Delta$, where $\Delta$ is the maximal
multiplicity free weight system generated by
$\alpha,\beta_{1},\ldots,\beta_{k},\ldots$. Here $\alpha$ is odd, while
$\beta_{1},\ldots,\beta_{k},\ldots$ are even. The grading is defined as
follows. We assign the weight $\alpha+\beta_{i_{i}}+\cdots+\beta_{i_{p}}$ to
$\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(Z)$, if
$\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(Z)$ is odd, and we assign the
weight $\beta_{i_{i}}+\cdots+\beta_{i_{p}}$ to
$\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(Z)$ if
$\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(Z)$ is even.

For example elements of the subspace $\mathfrak{g}_{\bar{0}}$ have weight $0$
and elements of $\mathfrak{g}_{\bar{1}}$ have weight $\alpha$.

Let us define the parity change functor $\pi^{\prime}$. Let us take the Lie
superalgebra
$\mathrm{gr}^{\prime}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))$, where
$\mathfrak{g}$ is a Lie superalgebra. The functor parity change $\pi^{\prime}$
is defined as follows

$\mathfrak{h}:=\pi^{\prime}(\mathrm{gr^{\prime}}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})))=\mathrm{gr^{\prime}}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))$

as $\mathfrak{g}_{\bar{0}}$-modules. Further we assume that all operators
$\mathrm{d}_{i}$ are even and again $\mathrm{d}_{i}\circ\mathrm{d}_{i}=0$. In
other words, this means that

$\displaystyle\mathfrak{h}_{\bar{0}}:=\bigoplus_{p\geq
0}\bigoplus_{i_{1}<\cdots<i_{p}}\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(\mathfrak{g}_{\bar{0}}),\quad\mathfrak{h}_{\bar{1}}:=\bigoplus_{p\geq
0}\bigoplus_{i_{1}<\cdots<i_{p}}\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(\mathfrak{g}_{\bar{1}}).$

To simplify our presentation we will use the following notations

$\displaystyle I=\\{i_{1},\ldots,i_{p}\\},\quad
J=\\{j_{1},\ldots,j_{q}\\},\quad K=\\{k_{1},\ldots,k_{r}\\},$

and $\mathrm{d}_{I}$ for $\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}$. We
denote by $\sharp I:=C(I)\,\,mod\,\,2=\bar{p}$ the cardinality of $I$ modulo
$2$. The multiplication in
$\mathfrak{h}=\pi^{\prime}(\mathrm{gr}^{\prime}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})))$
is defined by the following rules

  1. (Rule 1)

If $I\cap J\neq\emptyset$, we have
$[\mathrm{d}_{I}(\mathfrak{g}),\mathrm{d}_{J}(\mathfrak{g})]=\\{0\\}$.

  2. (Rule 2)

If $\sharp I+\bar{i}$ and $\sharp J+\bar{j}$ are odd, we have
$[\mathrm{d}_{I}(\mathfrak{g}_{\bar{i}}),\mathrm{d}_{I}(\mathfrak{g}_{\bar{j}})]=\\{0\\}$.

  3. (Rule 3)

In other cases we have
$[\mathrm{d}_{I}(\mathfrak{g}_{\bar{i}}),\mathrm{d}_{I}(\mathfrak{g}_{\bar{j}})]=\mathrm{d}_{I\cup
J}([\mathfrak{g}_{\bar{i}},\mathfrak{g}_{\bar{j}}])$.

######  Theorem 42.

The superspace
$\mathfrak{h}=\pi^{\prime}(\mathrm{gr}^{\prime}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})))$
is a Lie superalgebra.

###### Proof.

Step 1. Let us prove first that $\mathfrak{h}_{\bar{0}}$ is a Lie algebra.
Consider the following subspace

$\mathfrak{r}:=\bigoplus_{p\geq
0}\bigoplus_{i_{1}<\cdots<i_{p}}\mathrm{d}_{i_{1}}\cdots\mathrm{d}_{i_{p}}(\mathfrak{g}_{\bar{0}})\subset\mathrm{gr}^{\prime}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})).$

Clearly, $\mathfrak{r}$ is a Lie subsuperalgebra in
$\mathrm{gr^{\prime}}(\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))$ and this
Lie superalgebra is split. This is
$\mathfrak{r}=\mathfrak{r}_{\bar{0}}\oplus\mathfrak{r}_{\bar{1}}$ with
$[\mathfrak{r}_{\bar{1}},\mathfrak{r}_{\bar{1}}]=\\{0\\}$. Such Lie
superalgebras possesses a parity reversion, see Remark 34 part (2). We can see
that in notations of Remark 34 we have
$\mathfrak{r}_{\bar{0}}\oplus\mathfrak{r}_{\bar{1}}[\bar{1}]=\mathfrak{h}_{\bar{0}}$.

Step 2. Let us prove that $\mathfrak{h}_{\bar{1}}$ is an
$\mathfrak{h}_{\bar{0}}$-module. Let us take
$\mathrm{d}_{I}(X)\in\mathrm{d}_{I}(\mathfrak{g}_{\bar{0}})$,
$\mathrm{d}_{J}(Y)\in\mathrm{d}_{J}(\mathfrak{g}_{\bar{0}})$ and
$\mathrm{d}_{K}(Z)\in\mathrm{d}_{K}(\mathfrak{g}_{\bar{1}})$ with $I\cap
J=\emptyset$, $I\cap K=\emptyset$ and $J\cap K=\emptyset$. We consider the
following cases

  1. (1)

Let $\sharp I=\sharp J=\bar{0}$, any $K$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]+[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]+[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=$
$\displaystyle\mathrm{d}_{I}\circ\mathrm{d}_{J}\circ\mathrm{d}_{K}([X,[Y,Z]])+\mathrm{d}_{J}\circ\mathrm{d}_{K}\circ\mathrm{d}_{I}([Y,[Z,X]])+\mathrm{d}_{K}\circ\mathrm{d}_{I}\circ\mathrm{d}_{J}([Z,[X,Y]])=$
$\displaystyle\mathrm{d}_{I}\circ\mathrm{d}_{J}\circ\mathrm{d}_{K}([X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]])=0.$

  2. (2)

Let $\sharp I=\bar{1}$, $\sharp J=\bar{0}$ and $\sharp K=\bar{0}$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]=[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]=[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=0.$

  3. (3)

Let $\sharp I=\bar{1}$, $\sharp J=\bar{0}$ and $\sharp K=\bar{1}$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]+[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]+[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=$
$\displaystyle\mathrm{d}_{I}\circ\mathrm{d}_{J}\circ\mathrm{d}_{K}([X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]])=0.$

  4. (4)

Let $\sharp I=\sharp J=\bar{1}$ and $\sharp K=\bar{0}$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]=[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]=[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=0.$

  5. (5)

Let $\sharp I=\sharp J=\sharp K=\bar{1}$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]=[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]=[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=0.$

Step 3. Let us check Jacobi identity for elements from
$\mathfrak{h}_{\bar{1}}$. Let us take
$\mathrm{d}_{I}(X)\in\mathrm{d}_{I}(\mathfrak{g}_{\bar{1}})$,
$\mathrm{d}_{J}(Y)\in\mathrm{d}_{J}(\mathfrak{g}_{\bar{1}})$ and
$\mathrm{d}_{K}(Z)\in\mathrm{d}_{K}(\mathfrak{g}_{\bar{1}})$ with $I\cap
J=\emptyset$, $I\cap K=\emptyset$ and $J\cap K=\emptyset$ and consider the
following cases

  1. (1)

Let $\sharp I=\sharp J=\bar{0}$, any $K$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]=[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]=[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=0.$

  2. (2)

Let $\sharp I=\bar{0}$, $\sharp J=\bar{1}$ and $\sharp K=\bar{1}$. Then

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]+[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]+[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=$
$\displaystyle\mathrm{d}_{I}\circ\mathrm{d}_{J}\circ\mathrm{d}_{K}([X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]])=0.$

  3. (3)

Let $\sharp I=\sharp J=\sharp K=\bar{1}$. Then again

$\displaystyle[\mathrm{d}_{I}(X),[\mathrm{d}_{J}(Y),\mathrm{d}_{K}(Z)]]+[\mathrm{d}_{J}(Y),[\mathrm{d}_{K}(Z),\mathrm{d}_{I}(X)]]+[\mathrm{d}_{K}(Z),[\mathrm{d}_{I}(X),\mathrm{d}_{J}(Y)]]=$
$\displaystyle\mathrm{d}_{I}\circ\mathrm{d}_{J}\circ\mathrm{d}_{K}([X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]])=0.$

The proof is complete. ∎

######  Remark 43.

It is unexpected that such a parity change can give a well-defined Lie
superalgebra.

###  6.4. Functor inverse $\iota^{\prime}$

Our goal now is to define a $\mathbb{Z}$-graded subsuperalgebra
$\mathfrak{p}:=\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})$
in
$\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})$.
We put

$\displaystyle\mathfrak{p}_{0}:=\mathfrak{g}_{\bar{0}};$
$\displaystyle\mathfrak{p}_{1}:=\mathrm{diag}(\mathfrak{g}_{\bar{1}}\oplus\bigoplus_{i}\mathrm{d}_{i}(\mathfrak{g}_{\bar{1}}));$
$\displaystyle\mathfrak{p}_{2}:=\mathrm{diag}(\bigoplus_{i}\mathrm{d}_{i}(\mathfrak{g}_{\bar{0}})\oplus\bigoplus_{i<j}\mathrm{d}_{i}\mathrm{d}_{j}(\mathfrak{g}_{\bar{0}}));$
$\displaystyle\mathfrak{p}_{3}:=\mathrm{diag}(\bigoplus_{i<j}\mathrm{d}_{i}\mathrm{d}_{j}(\mathfrak{g}_{\bar{1}})\oplus\bigoplus_{i<j<k}\mathrm{d}_{i}\mathrm{d}_{j}\mathrm{d}_{k}(\mathfrak{g}_{\bar{1}}));$
$\displaystyle\cdots$
$\displaystyle\mathfrak{p}_{n}:=\mathrm{diag}(\bigoplus_{C(I)=n-1}\mathrm{d}_{I}(\mathfrak{g}_{\bar{n}})\oplus\bigoplus_{C(J)=n}\mathrm{d}_{J}(\mathfrak{g}_{\bar{n}}));$
$\displaystyle\cdots$

Here $C(I)$ is the cardinality of $I$.

######  Proposition 44.

The subsuperspace $\mathfrak{p}$ is a $\mathbb{Z}$-graded Lie subsuperalgebra.

###### Proof.

Let us prove that
$[\mathfrak{p}_{i},\mathfrak{p}_{j}]\subset\mathfrak{p}_{i+j}$. Let us take
$X\in\mathfrak{g}_{\bar{i}}$ and $Y\in\mathfrak{g}_{\bar{j}}$. Consider

$\displaystyle\left[\sum_{C(I)=i-1}\mathrm{d}_{I}(X)+\sum_{C(J)=i}\mathrm{d}_{J}(X),\sum_{C(I)^{\prime}=j-1}\mathrm{d}_{I^{\prime}}(Y)+\sum_{C(J^{\prime})=j}\mathrm{d}_{J^{\prime}}(Y)\right].$

Using Rules (2) we get

$\displaystyle\left[\sum_{C(I)=i-1}\mathrm{d}_{I}(X),\sum_{C(I)^{\prime}=j-1}\mathrm{d}_{I^{\prime}}(Y)\right]=0.$

Further,

$\displaystyle\left[\sum_{C(I)=i-1}\mathrm{d}_{I}(X),\sum_{C(J^{\prime})=j}\mathrm{d}_{J^{\prime}}(Y)\right]+\left[\sum_{C(J)=i}\mathrm{d}_{J}(X),\sum_{C(I^{\prime})=j-1}\mathrm{d}_{I^{\prime}}(Y)\right]+$
$\displaystyle\left[\sum_{C(J)=i}\mathrm{d}_{J}(X),\sum_{C(J^{\prime})=j}\mathrm{d}_{J^{\prime}}(Y)\right]=$
$\displaystyle\underbrace{\left(\binom{i+j-1}{j}+\binom{i+j-1}{j-1}\right)}_{\binom{i+j}{i}}\sum_{C(K)=i+j-1}\mathrm{d}_{K}([X,Y])+$
$\displaystyle\binom{i+j}{i}\sum_{C(K)=i+j}\mathrm{d}_{K}([X,Y]).$

The proof is complete. ∎

If we have a morphism $\psi$ of Lie superalgebras, then
$\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathrm{T^{\prime}}^{\infty}(\psi)$
preserves subalgebras $\mathfrak{p}$’s. Therefore we can define
$\iota^{\prime}$ on morphisms. Therefore the functor $\iota^{\prime}$ is
defined on the image of the functor
$\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathrm{T^{\prime}}^{\infty}$.
Summing up, we constructed the following functor

$\mathrm{F}^{\prime}:=\iota^{\prime}\circ\pi^{\prime}\circ\mathrm{gr}^{\prime}\circ\mathrm{T^{\prime}}^{\infty}:\mathrm{sLieAlg}\to\mathrm{grLieAlg}.$

######  Remark 45.

The Lie superalgebra $\mathfrak{p}$ is “locally isomorphic” to $\mathfrak{g}$
in the following sense. We have $\mathfrak{p}_{0}\simeq\mathfrak{g}_{\bar{0}}$
as Lie superalgebras and $\mathfrak{p}_{n}\simeq\mathfrak{g}_{\bar{n}}$ as
$\mathfrak{g}_{\bar{0}}$-modules for any $n$.

###  6.5. The functor $\mathrm{F}_{\infty}$ as a inverse limit

We can define the Lie superalgebras $\mathbf{T}^{\prime\infty}(\mathfrak{g})$,
$\mathrm{gr}^{\prime}(\mathbf{T}^{\prime\infty}(\mathfrak{g}))$,
$\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime\infty}(\mathfrak{g})))$
and
$\iota^{\prime}(\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime\infty}(\mathfrak{g}))))$
using the inverse limit. Indeed,

(21) $\displaystyle\mathbf{T}^{\prime\infty}(\mathfrak{g})$
$\displaystyle=\varprojlim\mathbf{T}^{\prime n}(\mathfrak{g});$ (22)
$\displaystyle\mathrm{gr}^{\prime}(\mathbf{T}^{\prime\infty}(\mathfrak{g}))$
$\displaystyle=\varprojlim\mathrm{gr}^{\prime}(\mathbf{T}^{\prime
n}(\mathfrak{g}));$ (23)
$\displaystyle\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime\infty}(\mathfrak{g})))$
$\displaystyle=\varprojlim\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime
n}(\mathfrak{g})));$ (24)
$\displaystyle\iota^{\prime}(\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime\infty}(\mathfrak{g}))))$
$\displaystyle=\varprojlim\iota^{\prime}(\pi^{\prime}(\mathrm{gr}^{\prime}(\mathbf{T}^{\prime
n}(\mathfrak{g})))).$

Lemma 40 implies (21). Further, Equality (22) follows from definition of the
functor $\mathrm{gr}^{\prime}$. To obtain (23) we can repeat Theorem 42
assuming that we have only $n$ de Rham differentials $\mathrm{d}_{i}$. Finally
(24) follows from Proposition 44 again assuming that we have only $n$ de Rham
differentials $\mathrm{d}_{i}$. Summing up,

$\mathrm{F}^{\prime}(\mathfrak{g})=\mathrm{F}^{\prime}_{\infty}(\mathfrak{g})=\varprojlim\mathrm{F}^{\prime}_{n}(\mathfrak{g}).$

##  7\. Generalized Donagi–Witten construction for
Lie supergroups

Again the functor $\mathrm{F}$ is a composition of four functors: the iterated
tangent functor $\mathbf{T}^{\infty}$, the functor split $\mathrm{gr}$, the
functor parity change $\pi$ and the functor inverse $\iota$. Let $\mathcal{G}$
be a Lie supergroup with the supergroup morphisms $\mu$, $\kappa$ and $e$.

###  7.1. Iterated antitangent functor $\mathbf{T}^{\infty}$

Above we considered the tangent functor $\mathbf{T}$ applying to a Lie
supergroup $\mathcal{G}$. We saw that $\mathbf{T}(\mathcal{G})$ is a Lie
supergroup again. Therefore we can iterate this procedure and get the Lie
supergroup

$\mathbf{T}^{2}(\mathcal{G}):=\mathbf{T}(\mathbf{T}(\mathcal{G})).$

By definition we put

$\mathbf{T}^{n}(\mathcal{G}):=\mathbf{T}\circ\cdots\circ\mathbf{T}(\mathcal{G}),$

where on the left hand side the tangent functor $\mathbf{T}$ is iterated $n$
times. Now we define

$\displaystyle\mathbf{T}^{\infty}(\mathcal{G})=\varprojlim\mathbf{T}^{n}(\mathcal{G}),$
$\displaystyle\mathbf{T}^{\infty}(\mu)=\varprojlim\mathbf{T}^{n}(\mu),\quad$
$\displaystyle\mathbf{T}^{\infty}(\kappa)=\varprojlim\mathbf{T}^{n}(\kappa),\quad\mathbf{T}^{\infty}(e)=\varprojlim\mathbf{T}^{n}(e),$

see Section 2.5. Clearly the infinite dimensional supermanifold
$\mathbf{T}^{\infty}(\mathcal{G})$ is a Lie supergroup, since the morphisms
$\mathbf{T}^{\infty}(\mu)$, $\mathbf{T}^{\infty}(\kappa)$ and
$\mathbf{T}^{\infty}(e)$ satisfy the group axioms.

Let us describe $\mathbf{T}^{\infty}(\mathcal{G})$ in terms of graded Harish-
Chandra pairs. Above we saw that

$\mathrm{T^{\prime}}^{n}(\mathfrak{g})\simeq\bigwedge(\xi_{1},\xi_{2},\ldots\xi_{n})\otimes\mathfrak{g}\quad\text{and}\quad\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})\simeq\bigwedge(\xi_{1},\xi_{2},\ldots)\otimes\mathfrak{g},$

where $\bigwedge(\xi_{1},\xi_{2},\ldots)$ is the Grassmann algebra with
infinitely many variables $\xi_{1},\xi_{2},\ldots$ labeled by natural numbers.
We can identify the structure sheaf
$\mathcal{O}_{\mathbf{T}^{\infty}(\mathcal{G})}$ of
$\mathbf{T}^{\infty}(\mathcal{G})$ with

$\mathrm{Hom}^{\prime}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathbf{T}^{\infty}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}})\subset\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathbf{T}^{\infty}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}}),$

where $\mathrm{Hom}^{\prime}$ are all
$\mathcal{U}(\mathfrak{g}_{\bar{0}})$-homomorphisms that are zero on an ideal
generated by

$\bigoplus_{q>0}\bigwedge^{q}(\xi_{n},\xi_{n+1},\ldots)\otimes\mathfrak{g}\subset\mathcal{U}(\mathbf{T}^{\infty}(\mathfrak{g}))$

for some $n\geq 1$.

###  7.2. Functor split $\mathrm{gr}$, functor parity change $\pi$ and functor
inverse $\iota$

The functor $\mathrm{gr}$ for $\mathbf{T}^{\infty}(\mathcal{G})$ is defined as
above. More precisely, let $\mathcal{J}$ be the sheaf of ideals in
$\mathcal{O}_{\mathbf{T}^{\infty}(\mathcal{G})}$ generated by odd elements. We
get a filtration in $\mathcal{O}_{\mathbf{T}^{\infty}(\mathcal{G})}$ by the
subsheaves $\mathcal{J}^{p}$, where $p\geq 0$. The corresponded graded sheaf
we denote by $\mathrm{gr}\mathcal{O}_{\mathbf{T}^{\infty}(\mathcal{G})}$. By
definition $\mathrm{gr}\mathcal{O}_{\mathbf{T}^{\infty}(\mathcal{G})}$ is the
structure sheaf of $\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))$. In terms
of graded Harish-Chandra pairs we have

$\mathcal{O}_{\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))}=\mathrm{Hom}^{\prime}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}}),$

where $\mathrm{Hom}^{\prime}$ are all
$\mathcal{U}(\mathfrak{g}_{\bar{0}})$-homomorphisms that are zero on some
ideal generated by

$\mathrm{gr^{\prime}}\Big{(}\bigoplus_{q>0}\bigwedge^{q}(\xi_{n},\xi_{n+1},\ldots)\otimes\mathfrak{g}\Big{)}\subset\mathcal{U}(\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})),\,\,n\geq
1.$

The supermanifold $\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))$ is an
$\infty$-fold vector bundle of type $\Delta$, where $\Delta$ is generated by
an odd weight $\alpha$ and even weights $\beta_{1},\beta_{2},\ldots$, see
Proposition 23. Since
$\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))$ is a
graded Lie superalgebra of type $\Delta$, the universal enveloping algebra

$\mathcal{U}(\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))=\bigoplus_{\delta\in\mathbb{Z}\alpha\times\mathbb{Z}\beta_{1}\times\cdots}\mathcal{U}(\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))_{\delta}$

is $\mathbb{Z}\times\mathbb{Z}\times\cdots$-graded. We have

(25)
$\mathcal{O}_{\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))}=\bigoplus_{\delta\in\mathbb{Z}\alpha\times\mathbb{Z}\beta_{1}\times\cdots}(\mathcal{O}_{\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))})_{\delta},$

where

$(\mathcal{O}_{\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))})_{\delta}=\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g}))_{\delta},\mathcal{F}_{\mathcal{G}_{0}}).$

Note that in this formula we can omit ′ and write simply $\mathrm{Hom}$. We
can use the equality (25) as a definition of the structure sheaf
$\mathcal{O}_{\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))}$. Since
$\mathrm{gr}$ is a functor, the structure morphisms in
$\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G}))$ are graded as well. In terms
of inverse limit we have
$\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G})=\varprojlim\mathrm{gr}(\mathbf{T}^{n}(\mathcal{G}))$.

To define the functor $\pi$ we change parities in any
$\mathrm{gr}\circ\mathbf{T}^{n}(\mathcal{G})$. In terms of graded Harish-
Chandra pairs we get

$\mathcal{O}_{\pi\circ\mathrm{gr}\circ\mathbf{T}^{\infty}(\mathcal{G})}=\mathrm{Hom}^{\prime}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\pi^{\prime}\circ\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}}),$

where $\mathrm{Hom}^{\prime}$ are all
$\mathcal{U}(\mathfrak{g}_{\bar{0}})$-homomorphisms that are zero on some
ideal generated by

$\pi^{\prime}\circ\mathrm{gr^{\prime}}\Big{(}\bigoplus_{q>0}\bigwedge^{q}(\xi_{n},\xi_{n+1},\ldots)\otimes\mathfrak{g}\Big{)}\subset\mathcal{U}(\pi^{\prime}\circ\mathrm{gr^{\prime}}\circ\mathrm{T^{\prime}}^{\infty}(\mathfrak{g})),\,\,n\geq
1.$

In terms of inverse limit again we have
$\pi\circ\mathrm{gr}(\mathbf{T}^{\infty}(\mathcal{G})=\varprojlim\pi\circ\mathrm{gr}(\mathbf{T}^{n}(\mathcal{G}))$.
The Lie supergroup $\pi\circ\mathrm{gr}\circ\mathbf{T}^{\infty}(\mathcal{G})$
is a graded Lie supergroup of type $\pi(\Delta)$, where $\pi(\Delta)$ is the
maximal multiplicity free system generated by odd weights
$\alpha,\beta_{1},\beta_{2},\ldots$.

A similar idea we use for functor $\iota$. Recall that
$\mathrm{F}:=\iota\circ\pi\circ\mathrm{gr}\circ\mathbf{T}^{\infty}$ and we
denote $\mathrm{F}_{n}:=\iota\circ\pi\circ\mathrm{gr}\circ\mathbf{T}^{n-1}$.
First of all we define the graded Lie supergroup $\mathrm{F}_{n}(\mathcal{G})$
of degree $n$ (that is of type $\\{0,1,\ldots,n\\}$ with $|1|=\bar{1}$) as the
$\mathbb{Z}$-graded Lie supergroup corresponding to the $\mathbb{Z}$-graded
Harish-Chandra pair $(\mathcal{G}_{0},\mathrm{F}^{\prime}_{n}(\mathfrak{g})).$
For any $n$ we have a natural homomorphism
$\mathrm{F}^{\prime}_{n+1}(\mathfrak{g})\to\mathrm{F}^{\prime}_{n}(\mathfrak{g}).$
This induces a homomorphism of enveloping algebras

$\Phi_{n}:\mathcal{U}(\mathrm{F}^{\prime}_{n+1}(\mathfrak{g}))\to\mathcal{U}(\mathrm{F}^{\prime}_{n}(\mathfrak{g})).$

Therefore we have a natural map of sheaves

$\mathcal{O}_{\mathrm{F}_{n}(\mathcal{G})}=\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathrm{F}^{\prime}_{n}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}})\hookrightarrow\mathcal{O}_{\mathrm{F}_{n_{+}1}(\mathcal{G})}=\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathrm{F}^{\prime}_{n+1}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}})$

Now we can identify the structure sheaf of $\mathrm{F}(\mathcal{G})$ with

$\mathcal{O}_{\mathrm{F}(\mathcal{G})}=\mathrm{Hom}^{\prime}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathrm{F}^{\prime}(\mathfrak{g})),\mathcal{F}_{\mathcal{G}_{0}}),$

where $\mathrm{Hom}^{\prime}$ are all
$\mathcal{U}(\mathfrak{g}_{\bar{0}})$-homomorphisms that are zero on some
$\mathrm{Ker}(\Phi_{n})$. Further we see that
$\mathcal{O}_{\mathrm{F}(\mathcal{G})}$ is $\mathbb{Z}$-graded. Indeed,

$\mathcal{O}_{\mathrm{F}(\mathcal{G})}=\bigoplus_{n\in\mathbb{Z}}(\mathcal{O}_{\mathrm{F}(\mathcal{G})})_{n}=\bigoplus_{n\in\mathbb{Z}}\mathrm{Hom}_{\mathcal{U}(\mathfrak{g}_{\bar{0}})}(\mathcal{U}(\mathrm{F}^{\prime}(\mathfrak{g}))_{n},\mathcal{F}_{\mathcal{G}_{0}}).$