Polarisation of Graded Bundles

Polarisation of Graded Bundles

Andrew James BRUCE †, Janusz GRABOWSKI † and Mikołaj ROTKIEWICZ ‡

A.J. Bruce, J. Grabowski and M. Rotkiewicz

† Institute of Mathematics, Polish Academy of Sciences, Poland
<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,
Poland<EMAIL_ADDRESS>
Received December 14, 2015, in final form October 25, 2016; Published online
November 02, 2016

We construct the _full linearisation functor_ which takes a graded bundle of
degree $k$ (a particular kind of graded manifold) and produces a $k$-fold
vector bundle. We fully characterise the image of the full linearisation
functor and show that we obtain a subcategory of $k$-fold vector bundles
consisting of _symmetric $k$-fold vector bundles_ equipped with a family of
morphisms indexed by the symmetric group ${\mathbb{S}}_{k}$. Interestingly,
for the degree 2 case this additional structure gives rise to the notion of a
_symplectical double vector bundle_ , which is the skew-symmetric analogue of
a  _metric double vector bundle_. We also discuss the related case of fully
linearising $N$-manifolds, and how one can use the full linearisation functor
to “superise” a graded bundle.

graded manifolds; $N$-manifolds; $k$-fold vector bundles; polarisation;
supermanifolds

55R10; 58A32; 58A50

###### Contents

  1. 1 Introduction and background
    1. 1.1 Motivation and summary of results
    2. 1.2 Graded bundles and $n$-fold graded bundles
    3. 1.3 Some constructions
    4. 1.4 The naïve approach to superisation
    5. 1.5 The linearisation of a graded bundle
  2. 2 Graded bundles vs $k$-fold vector bundles
    1. 2.1 The full linearisation functor
    2. 2.2 Characterisation of the full linearisation for the degree two case
    3. 2.3 Relation with symplectical double vector bundles
    4. 2.4 Lie algebroid structures
    5. 2.5 The equivalence between graded bundles of degree $k$ and symmetric $k$-fold vector bundles
  3. 3 $N$-manifolds and their linearisation
    1. 3.1 The linearisation of $N$-manifolds
    2. 3.2 Parity reversion functor for double vector bundles
    3. 3.3 $N$-manifolds of degree 2 and metric double vector bundles
  4. 4 Superisation of graded bundles
    1. 4.1 Using the parity shifted full linearisation
    2. 4.2 Using $\mathbb{Z}_{2}^{k}$-supermanifolds
    3. Acknowledgements

##  1 Introduction and background

###  1.1 Motivation and summary of results

The term ‘graded manifold’ appears in the literature in various meanings,
including those related just to $\mathbb{Z}_{2}$-gradation and parity, i.e.,
to _supermanifolds_ (see, e.g., [25]). Our general understanding is in the
spirit of Th. Voronov [35], who defines _graded manifolds_ as supermanifolds
equipped with a privileged class of atlases in which coordinates are assigned
weights in ${\mathbb{Z}}$, which in general is independent of the Grassmann
parity. Moreover, the coordinate changes are decreed to be polynomial in non-
zero weight coordinates and respect the weight. An additional condition is
that all the non-zero weight coordinates that are Grassmann even must be
‘cylindrical’. In precise terms, it means that the associated _weight vector
field is $h$-complete_ (cf. [16]).

Within the category of graded manifolds, the most important seem to be the
non-negatively graded manifolds. If the Grassmann parity of the coordinates
coincides with the weight (mod 2), then we have a $N$-manifold (cf. [31, 32]).
In the purely even setting, Grabowski and Rotkiewicz [18] define what they
referred to as _graded bundles_ , being a particular class of non-negatively
graded manifolds. They showed that a graded bundle, understood as a manifold
with a non-negative grading on its structure sheaf, is equivalent to a
manifold equipped with a smooth action of the multiplicative monoid
$({\mathbb{R}},\cdot)$ of reals. Such actions they call _homogeneity
structures_ (we will be more precise shortly). As this action reduced to
$({\mathbb{R}}_{>0},\cdot)$ is just generated by the weight vector field, it
means, actually by definition, that the weight vector field is _$h$
-complete_.

In this paper we examine how to canonically pass from a graded bundle of
degree $k$ (cf. [18]) to a $k$-fold vector bundle and back. We construct the
_full linearisation functor_ as the iteration of the linearisation functor
(cf. [6]), and completely characterise its image as a subcategory of the
category of $k$-fold vector bundles. A little more carefully, to every graded
bundle of degree $k$ we can associate a $k$-fold vector bundle equipped with a
system of morphisms $\sigma_{g}$, parametrized by the elements of the
symmetric group $\mathbb{S}_{k}$, satisfying an additional property; we will
call such $k$-fold vector bundles _symmetric_.

Heuristically, one should view the full linearisation as a _polarisation_ of
the admissible changes of local coordinates. That is, we adjoin new
coordinates in a natural way as to linearise the polynomial changes of fibre
coordinates. The new coordinates essentially come from the procedure of
repeated differentiation (i.e., repeated application of the tangent functor).
In this way we obtain a new enlarged manifold that has the structure of a
$k$-fold vector bundle. The original graded bundle can then be recovered from
the ‘linearised version’ as the ‘diagonal’ of _holonomic vectors_. The
constructions we present mimic the well-known relationship between higher
tangent bundles ${{T}}^{k}M$ and iterated tangent bundles. We remark that the
deceptively simple notion of polarising homogeneous polynomials has been
exploited in algebraic geometry, invariant theory, representation theory and
dynamical systems, for example. In retrospect, it is not surprising that
polarisation plays an important rôle in the general theory of graded bundles.

Motivation for considering the full linearisation functor comes from the
theory of Lie algebroids that carry an extra compatible grading and the
corresponding higher Lagrangian mechanics [5, 6], the linearisation functor is
essential in those works. In addition, we must point out the result of Jotz
Lean [22] who showed that $N$-manifolds of degree $2$ are categorically
equivalent to what she calls _metric double vector bundles_. For the specific
case of degree 2, the linearisation as presented in [6] coincides exactly with
the full linearisation presented in this paper. Moreover, for the degree 2
case the additional morphism $\sigma=\sigma_{(12)}$ leads to the construction
of a _symplectical double vector bundle_ (we will define this notion carefully
in due course). Symplectical double vector bundles are essentially the same as
metric double vector bundles, the former being defined in terms of a skew-
symmetric pairing while the latter a symmetric pairing. In light of the
constructions of Jotz Lean [22], we see that the subtle difference in symmetry
of the pairing is really due to the difference between purely even graded
bundles and $N$-manifolds. In particular, for $N$-manifolds of degree two the
weight one coordinates are anticommuting, while graded bundles live in the
strictly commutative world. Thus we have to interchange ‘symmetric’ and ‘skew-
symmetric’ in the right places when switching between graded bundles and
$N$-manifolds of degree 2.

We also draw the reader’s attention to the PhD thesis of del Carpio-Marek
[12], who (independently) established results equivalent to that that Jotz
Lean, but in terms of double vector bundles equipped with an involution.
Conceptually this approach is similar to our use of symmetric $k$-fold vector
bundles. The work of del Carpio-Marek starts from some of the results of
Bursztyn et al. [7]. According to del Carpio-Marek, they show that the
category of $N$-manifolds of degree two is equivalent to the category of
_involutive double vector bundle sequences_. The notion of a vector bundle
sequence is due to Chen et al. [8]. We further remark Jotz Lean and del
Carpio-Marek both concentrate solely on $N$-manifolds of degree 2, where our
constructions naturally cover higher degree $N$-manifolds (upon minor
modifications). The only other work we are aware of that deals with
‘linearising’ higher degree $N$-manifolds is that of Vishnyakova [34]: she
establishes a categorical equivalence of $N$-manifolds of degree $k$ and
$k$-fold vector bundles (in the category of supermanifolds) equipped with a
family of $k$ odd vector fields. Vishnyakova uses repeated application of the
antitangent functor and substructures thereof to build $k$-fold vector bundles
from $N$-manifolds. In particular, the family of odd vector fields are
essentially de Rham differentials associated with each antitangent bundle. We
must remark that we became aware of the works of del Carpio-Marek and
Vishnyakova only towards the end of completing this paper.

Philosophically, graded bundles are a natural generalisation of vector bundles
(in the category of smooth manifolds). It is thus natural to wonder if there
is some analogue of the parity reversion functor for graded bundles. The
obvious difficulty is that the admissible fibre coordinate transformations on
a graded bundle are in general non-linear (they are polynomial) and so
directly mimicking the parity reversion functor for vector bundles fails.

However, one can use the full linearisation functor, coupled with the standard
parity reversion functor, to construct a supermanifold from a graded bundle.
Alternatively, the full linearisation functor allows one to canonically
associate with an arbitrary graded bundle a $\mathbb{Z}_{2}^{k}$-supermanifold
in the sense of Covolo, Grabowski and Poncin [9, 10], and Molotkov [28]. From
many perspectives, the natural superisation of a $k$-fold vector bundle is a
$\mathbb{Z}_{2}^{k}$-supermanifold rather than a standard supermanifold (see
[9, Proposition 6.1]). Note that the $\mathbb{Z}_{2}^{k}$-superisation is
different to the $\mathbb{Z}_{2}$-superisation of $k$-fold vector bundles as
defined by Th. Voronov [37]. As the parity reversion functor for vector
bundles has turned out to be a very important notion, it is hoped that the
$\mathbb{Z}_{2}^{k}$-superisation of a graded bundle will also develop into a
useful concept.

A further remark is that the notion of the full linearisation should not be
confused with the notion of a splitting of a graded bundle into a direct sum
of graded vector bundles. Moreover, the resulting $k$-fold vector bundles or
the related $\mathbb{Z}_{2}^{k}$-superisations are not (generally) canonically
split. Of course, at each ‘stage’ we have a Batchelor–Gawȩdzki-like theorem
(cf. [1, 15]), but in general the splittings are non-canonical. The existence
of Batchelor–Gawȩdzki-like theorems in various categories has been folklore
for quite some time. According to our knowledge, the first proper proof for
$N$-manifolds can be found in [3], for graded bundles see [6] and for
$\mathbb{Z}_{2}^{k}$-supermanifolds [10].

It is desirable to generalise all of the considerations of this paper to more
general (purely even) $\mathbb{Z}$-graded manifolds, at least as far as
possible. However, manifolds that have both positive and negative gradings are
intrinsically harder to understand and work with: basic questions about their
topological properties and differential calculus remain. $\mathbb{Z}$-graded
manifolds in general are not fibre bundles and we lose much of our intuition
gained from the study of non-negatively graded structures. Even if we are
given an atlas with polynomial change of coordinates, the weight vector field
itself does not in general carry this information. Moreover, there is a lack
of illustrative examples to help give insight into the general theory. Graded
bundles, in contrast, have much better topological properties and there exists
many natural examples. Moreover, if one looses the non-negative grading then
there is no possibility of describing the geometry in terms of a homogeneity
structure: for example homogeneity structures play a fundamental rôle in
understanding Lie groupoids in the category of graded bundles [4]. For these
reasons, we will not touch upon the linearisation or polarisation of
$\mathbb{Z}$-graded manifolds outside the non-negatively graded case.

We summarise the main results of this paper as follows:

  * •

We present the full linearisation functor, its inverse (the diagonalisation
functor), and show the categorical equivalence between graded bundles of
degree k and symmetric $k$-fold vector bundles (cf. Theorem 2.24).

  * •

The relation between the linearisation of a graded bundle of degree 2 and
symplectical double vector bundles is carefully explained (cf. Theorem 2.13)
and canonical Lie algebroid structures on the latter are discovered.

  * •

The analogous result for $N$-manifolds of degree 2 is found in Proposition
3.3, which conceptually simplifies the result of Jotz Lean [22, Theorem 3.17].

  * •

The question of the superisation of a graded bundle is answered through the
full linearisation functor and then passing canonically to a
$\mathbb{Z}_{2}^{k}$-supermanifold (cf. Theorem 4.2).

######  Remark 1.1.

It is possible to consider non-negatively graded manifolds in terms of
consistently defined homogeneous local coordinates that do _not_ lead to
$h$-complete weight vector fields, i.e., in a broader sense than in [35]. In
our opinion, the term “graded manifolds” should be reserved for this general
concept as illustrated by the following examples. Consider
${\mathbb{R}}^{2}\supset F=(0,+\infty)\times(0,+\infty)$ and let $x,y\colon
F\rightarrow{\mathbb{R}}$ be the standard projections on $(0,+\infty)$. Say,
we want to assign weights $1$, $1$ to the coordinate functions $x$, $y$ on
$F$, so the weight vector field $\Delta\in\mathfrak{X}(F)$ is
$\Delta=x\partial_{x}+y\partial_{y}$ which is not $h$-complete ($\Delta$
integrates to $({\mathbb{R}}_{+},\cdot)$-action which can not be extended to
the action of the whole monoid $({\mathbb{R}},\cdot)$). Then
$(x,y)\mapsto(x^{\prime}=x^{2}/y,y^{\prime}=y)$ is a diffeomorphism
$F\rightarrow F$ and $x^{\prime}$, $y^{\prime}$ are homogeneous weight $1$
functions with respect to $\Delta$. Thus we are forced to accept
$(x^{\prime},y^{\prime})$ as a graded coordinate system on $F$ equally ‘good’
as $(x,y)$. However, the manifold $F$ equipped with the class of graded
coordinates represented by $(x,y)$, both of weight 1, and the associated
weight vector field $\Delta$, inherits now almost no properties of a vector
space. For example, we see that affine combinations can not be defined
intrinsically on $F$.

Another toy example of a non-negatively graded manifold modelled on the same
manifold $F$, is constructed by assigning weights $1$, $2$ to the coordinates
$x$, $y$, respectively. Then $(x^{\prime}=x^{1/2}y^{1/4},y^{\prime}=y)$ is
another coordinate system in the same class as $(x,y)$. Note that we cannot
kill intrinsically coordinates of highest weight, i.e., the coordinate $y$ in
the example, so $F$ does not give a tower of fibrations (1.1) (see below).

In particular, it will be clear that the full linearisation functor, the
principal construction of the paper, can _not_ be applied to such graded
manifolds. From our perspective, such graded manifolds exhibit pathological
behaviour and so we will not consider them in the remainder of this paper.

On the other hand, all our constructions will remain valid if we assume that
the transformation functions in homogeneous coordinates are polynomial, so for
purely even _non-negatively graded manifolds_ in the sense of [35]. The
assumption about polynomiality is automatically satisfied for graded bundles,
so for simplicity we will work only with this class of graded manifolds.

Arrangement. In the remainder of this section we recall the basic theory of
graded bundles, $n$-fold graded bundles and the linearisation functor. In
Section 2 we describe the full linearisation functor, its characterisation and
the categorical equivalences it establishes. In Section 3 the methods of the
previous sections are slighted modified to cope with $N$-manifolds. The
question of superisation of graded bundles via the full linearisation is
addressed in Section 4.

###  1.2 Graded bundles and $\boldsymbol{n}$-fold graded bundles

An important class of graded manifolds are those that carry non-negative
grading. For the moment we will consider only purely even manifolds
explicitly, although the statements in this subsection generalise to the
supercase. We will furthermore require that this grading is associated with a
smooth action $h\colon{\mathbb{R}}\times F\to F$ of the monoid
$({\mathbb{R}},\cdot)$ of multiplicative reals on a manifold $F$; a
_homogeneity structure_ in the terminology of [18]. This action reduced to
${\mathbb{R}}_{>0}$ is the one-parameter group of diffeomorphism integrating
the _weight vector field_ , thus the weight vector field is in this case _$h$
-complete_ [16] and only _non-negative integer weights_ are allowed. Thus the
algebra $\mathcal{A}(F)\subset C^{\infty}(F)$ spanned by homogeneous functions
is $\mathcal{A}(F)=\bigoplus_{i\in\mathbb{N}}\mathcal{A}^{i}(F)$, where
$\mathcal{A}^{i}(F)$ consists of homogeneous function of degree $i$.

Importantly, we have that for $t\neq 0$ the action $h_{t}$ is a diffeomorphism
of $F$ and, when $t=0$, it is a smooth surjection $\tau=h_{0}$ onto $F_{0}=M$,
with the fibres being diffeomorphic to $\mathbb{R}^{N}$ (cf. [18]). Thus, the
objects obtained are particular kinds of _polynomial bundles_ $\tau\colon F\to
M$ (e.g., [2, 36]), i.e., fibrations which locally look like
$U\times{\mathbb{R}}^{N}$ and the change of coordinates (for a certain choice
of an atlas) are polynomial in ${\mathbb{R}}^{N}$. For this reason graded
manifolds with non-negative weights _and_ $h$-complete weight vector fields
$\mathchar 28673\relax$ are also known as _graded bundles_ [18].

######  Example 1.2.

If the weight is constrained to be either zero or one, then the weight vector
field is precisely a vector bundle structure on $F$ and will be generally
referred to as an _Euler vector field_.

######  Example 1.3.

The principle canonical example of a graded bundle is the higher tangent
bundle ${{T}}^{k}M$; i.e., the $k$-th jets (at zero) of curves
$\gamma\colon\mathbb{R}\rightarrow M$. Given a smooth function $f$ on a
manifold $M$, one can construct functions $f^{(\alpha)}$ on ${{T}}^{k}M$,
where $0\leq\alpha\leq k$, the so called _$(\alpha)$ -lifts_ of $f$ (see
[29]). They are defined by

$\displaystyle
f^{(\alpha)}([\gamma]_{k}):=\left.\frac{\operatorname{d}^{\alpha}}{\operatorname{d}t^{\alpha}}\right|_{t=0}f(\gamma(t)).$

where $[\gamma]_{k}\in{{T}}^{k}M$ is the class of the curve
$\gamma\colon{\mathbb{R}}\rightarrow M$. The functions
$f^{(k)}\colon{{T}}^{k}M\to{\mathbb{R}}$ and
$f^{(1)}\colon{{T}}M\to{\mathbb{R}}$ are called the _$k$ -complete lift_ and
the _tangent lift_ of $f$, respectively. A coordinate system $(x^{a})$ on $M$
gives rise to so called _adapted coordinate systems_
$\big{(}x^{a,(\alpha)}\big{)}_{0\leq\alpha\leq k}$ on ${{T}}^{k}M$ in which
$x^{a,(\alpha)}$ is of degree $\mathchar 28939\relax$, i.e.,

$\displaystyle\mathchar 28673\relax=\sum_{a,\mathchar 28939\relax}\mathchar
28939\relax x^{a,(\alpha)}\partial_{x^{a,(\alpha)}}.$

On a general graded bundle, one can always pick an atlas of $F$ consisting of
charts for which we have homogeneous local coordinates
$\big{(}x^{A},y_{w}^{i}\big{)}$, where
$\operatorname{w}\\!\big{(}x^{A}\big{)}=0$ and $\operatorname{w}(y_{w}^{i})=w$
with $1\leq w\leq k$, for some $k\in\mathbb{N}$ known as the _degree_ of the
graded bundle. Note that, according to this definition, a graded bundle of
degree $k$ is automatically a graded bundle of degree $l$ for $l\geq k$.
However, there is always a _minimal degree_.

It will be convenient to group all the coordinates with non-zero weight
together, as these form a basis of the function algebra of the graded bundle.
The index $i$ should be considered as a “generalised index” running over all
the possible weights. The label $w$ in this respect largely redundant, but it
will come in very useful when checking the validity of various expressions.
The local changes of coordinates respect the weight and hence are polynomial
for non-zero weight coordinates. A little more explicitly, the changes of
local coordinates are of the form

$\displaystyle x^{A^{\prime}}=x^{A}(x),\qquad
y^{i^{\prime}}_{w}=\sum\frac{1}{n!}y^{j_{1}}_{w_{1}}y^{j_{2}}_{w_{2}}\cdots
y^{j_{n}}_{w_{n}}T_{j_{n}\cdots
j_{2}j_{1}}^{\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>i^{\prime}}(x),$

where $w=w_{1}+\cdots+w_{n}$ and we assume the tensors $T_{j_{n}\cdots
j_{2}j_{1}}^{\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>i^{\prime}}$ to be symmetric in
lower indices. Naturally, changes of coordinates are invertible and so,
automatically, $\big{(}T_{j}^{\>\>i^{\prime}}(x)\big{)}$ is an invertible
matrix.

Importantly, a graded bundle of degree $k$ admits a sequence of surjections

$\displaystyle
F=F_{k}\stackrel{{\scriptstyle\tau^{k}_{k-1}}}{{\longrightarrow}}F_{k-1}\stackrel{{\scriptstyle\tau^{k-1}_{k-2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\tau^{3}_{2}}}{{\longrightarrow}}F_{2}\stackrel{{\scriptstyle\tau^{2}_{1}}}{{\longrightarrow}}F_{1}\stackrel{{\scriptstyle\tau^{1}}}{{\longrightarrow}}F_{0}=M,$
(1.1)

where $F_{l}$ itself is a graded bundle over $M$ of degree $l$, obtained from
the atlas of $F_{k}$ by removing all coordinates of degree greater than $l$
(see the next paragraph).

Note that $F_{1}\rightarrow M$ is a linear fibration and the other fibrations
$F_{l}\rightarrow F_{l-1}$ are affine fibrations in the sense that the changes
of local coordinates for the fibres are linear plus an additional additive
terms of appropriate weight. The model fibres here are $\mathbb{R}^{n}$.

Morphisms between graded bundles necessarily preserve the weight; in other
words, morphisms relate the respective homogeneity structures, or equivalently
morphisms relate the respective weight vector fields. Evidently, morphisms of
graded bundles can be composed as standard maps between smooth manifolds and
so we obtain the category of graded bundles. We will denote the category of
graded bundles of degree $k$ by $\textnormal{{GrB}}[k]$.

######  Remark 1.4.

We will also encounter $N$-manifolds in this paper, which are very similar to
graded bundles in many respects, and with hindsight are special examples of
graded manifolds as defined by Voronov [35]. Ševera [32] defines an
$N$-manifold as a supermanifold equipped with an action of
$(\mathbb{R},\cdot)$ such that $-1\in\mathbb{R}$ acts as the parity operator
(it flips sign of any Grassmann odd coordinate). Roytenberg [31] defines an
$N$-manifold in terms of an atlas of charts consisting of homogeneous
coordinates for which coordinates of even weight are Grassmann even and
coordinates of odd weight are Grassmann odd. It turns out that, under the
additional assumption that even coordinates of non-zero degree are
‘cylindrical’, both these notions of $N$-manifolds are equivalent, though the
first (sketch) of a proof we know of is to be found in [4] and a complete
proof can be found in [23].

The notion of a double vector bundle [30] (or an $n$-fold vector bundle [20,
21, 26, 37]) is conceptually clear in the graded language in terms of mutually
commuting weight vector fields; see [17, 18]. This leads to the higher
analogues known as _$n$ -fold graded bundles_, which are manifolds for which
the structure sheaf carries an $\mathbb{N}^{n}$-grading such that all the
weight vector fields are $h$-complete and pairwise commuting. The local
triviality of $n$-fold graded bundles means that we can always equip an
$n$-fold graded bundle with an atlas such that the charts consist of
coordinates that are simultaneously homogeneous with respect to the weights
associated with each weight vector field (cf. [18]). If all the weight vector
fields are in fact Euler vector fields, then we have an _$n$ -fold vector
bundle_. The changes of local coordinates on an $n$-fold graded bundle must
respect the weights. Similarly, morphisms between $n$-fold graded bundles
respect the weights of the local coordinates.

We will denote a $k$-fold vector bundle as
${{D}}=(D;\Delta^{1},\ldots,\Delta^{k})$, where $\Delta^{i}$’s are Euler
vector fields on an underlying manifold $D$. We will often use the short hand
‘DVB’ for double vector bundle. We shall use a similar notation for a $k$-fold
graded bundle, but shall use the letter
${{F}}=(F;\Delta^{1},\ldots,\Delta^{k})$ to indicate that weights greater than
one are allowed. We remark that the ordering of the weight vector fields is a
part of the definition of a $k$-fold graded bundle and that morphisms of
$k$-fold vector bundles respect this ordering. Thus, any permutation of the
ordering of the weight vector fields gives a different $k$-fold vector bundle
structure on the same underlying manifold. The $k$-fold graded bundle ${{F}}$
gives rise to a number of graded bundles $\tau_{i}\colon F\rightarrow F_{i}$
defined by pairs $(F;\Delta^{i})$ called _legs_ of ${{F}}$, and a number of
$(k-1)$-fold graded bundles
${{F}}_{i}=(F_{i};\Delta^{1},\ldots,\Delta^{i-1},\Delta^{i+1},\ldots,\Delta^{k})$
which we call _feet_ of ${{F}}$.

Important examples are: the _iterated tangent bundle_
${{T}}^{(k)}M:={{T}}\cdots{{T}}M$ (a $k$-fold vector bundle) and
${{T}}^{k}{{T}}^{l}M$ (a double graded bundle).

By iterating the construction of $(\mathchar 28939\relax)$-lifts, we obtain
functions $f^{(\beta,\alpha)}:=(f^{(\beta)})^{(\alpha)}$ on
${{T}}^{k}{{T}}^{l}M$ for $0\leq\alpha\leq k$, $0\leq\beta\leq l$, and,
generally, functions $f^{(\varepsilon_{r},\ldots,\varepsilon_{1})}$ on
${{T}}^{n_{1}}\cdots{{T}}^{n_{r}}M$ for $0\leq\varepsilon_{j}\leq n_{j}$,
$1\leq j\leq r$. A coordinate system $(x^{a})$ on $M$ gives rise to so called
_adapted coordinate systems_ $(x^{a,(\varepsilon)})_{\varepsilon}$ on
${{T}}^{(k)}M$, where the multi-index
$\varepsilon=(\varepsilon_{r},\ldots,\varepsilon_{1})$ is from $\\{0,1\\}^{k}$
and $x^{a,(\alpha)}$, $x^{a,(\varepsilon)}$ are obtained from $x^{a}$ by the
above lifting procedure.

######  Definition 1.5.

Consider a double vector bundle, which following classical notation we denote
as ${{D}}=(D;A,B;M)$, where $A$ and $B$ are the side bundles. If we are given
a vector bundle isomorphism $A\simeq B$ then ${{D}}$ is called _balanced_. We
may simply write ${{D}}=(D;A,A;M)$ but usually will avoid this. In more
generality, a $k$-fold vector bundle ${{D}}$ with legs $D\rightarrow D_{i}$,
$i=1,\ldots,k$, is said to be a _balanced $k$-fold vector bundle_ if all feet
$D_{i}$ are pairwise-isomorphic as $(k-1)$-fold vector bundles in some
canonical way.

###  1.3 Some constructions

By saying that a vector field $\Delta$ on a manifold $F$ _is a weight vector
field_ we mean that $F$ can be given a graded bundle structure which defines
$\Delta$ as its weight vector field. In particular, $\Delta$ has to be
complete and the linear part of $\Delta$ in any of singular point of $\Delta$
has to have integer, non-negative eigenvalues. A important remark [18, Remark
5.2] is that the sum (unlike the difference) of two commuting weight vector
fields is again a weight vector field, hence a linear combination of weight
vector fields with non-negative integer coefficients $a_{s}$,
$\Delta=\sum_{s}a_{s}\Delta^{s}$ is a weight vector field. We shall use this
fact to set some further useful notation.

######  Definition 1.6 (removal coordinates of weight $>l$).

Let ${{F}}=(F;\Delta)$ be a graded bundle. Denote by $F[\Delta\leq l]$ the
base manifold of the locally trivial fibration defined by taking the weight
$>l$ coordinates with respect to this complementary weight to be the fibre
coordinates. We have a natural projection that we will denote as

$\displaystyle\textnormal{p}^{F}_{[\Delta\leq l]}\colon\ F\rightarrow
F[\Delta\leq l].$

Note that, as the transformation rules of coordinates of weights $\leq l$
involve only coordinates of weights $\leq l$, the above definition is correct.
Since compositions of commuting homogeneity structures are homogeneity
structures, we immediately get the following.

######  Proposition 1.7.

Linear combinations of commuting weight vector fields with non-negative
integer coefficients are weight vector fields. If at start
${{F}}=(F;\Delta^{1},\ldots,\Delta^{k})$ is a $k$-fold graded bundle and
$\Delta=\sum a_{s}\Delta^{s}$, where $a_{i}\in\textnormal{N}$, then
$F[\Delta\leq l]$ has the induced $k$-fold graded bundle structure such that
$\textnormal{p}^{F}_{[\Delta\leq l]}$ is a $k$-fold graded bundle morphism.

The next construction is in a sense dual and depends on putting to zero
coordinates of negative weights with respect to the vector field $X=\sum
a_{s}\Delta^{s}$. Here, we assume that $a_{s}\in{\mathbb{Z}}$ can be negative
(see Remark 1.9).

######  Definition 1.8 (putting to zero coordinates of $X$-negative weights).

Consider a $k$-fold graded bundle $(F;\Delta^{1},\ldots,\Delta^{k})$ and let
$X=\sum_{s}a_{s}\Delta^{s}$ be a linear combination of weight vector fields of
$F$ with integer coefficients. Define locally a submanifold $F[X\geq 0]\subset
F$ as

$\displaystyle F[X\geq 0]:=\bigg{\\{}\big{(}y^{i}_{w}\big{)}\colon
y^{i}_{w}=0\ \text{for all}\ w=(w_{1},\ldots,w_{k})\ \text{such that}\
\sum_{s}a_{s}w_{s}<0\bigg{\\}}.$

Note first that $F[X\geq 0]$ is a well-defined submanifold of $F$. Indeed, set
$X$-weight $\operatorname{w}_{X}(f)$ for the weight with respect to the vector
field $X$ of a homogeneous function $f$, thus
$\operatorname{w}_{X}(y^{i}_{w})=\sum_{s}a_{s}w_{s}$ if
$w=(w_{1},\ldots,w_{s})$, and consider another coordinate system
$(y^{i^{\prime}})$. Since coordinate changes preserve $\operatorname{w}_{X}$
and $y^{i^{\prime}}$ is a linear combination of monomials of type
$y^{i_{1}}\cdots y^{i_{j}}$, we see that if
$\operatorname{w}_{X}(y^{i^{\prime}})$ is negative then at least one of the
weight $\operatorname{w}_{X}(y^{i_{1}})$,
$\operatorname{w}_{X}(y^{i_{2}}),\ldots,\operatorname{w}_{X}(y^{i_{j}})$ has
to be negative, and thus $y^{i^{\prime}}$ vanishes on $F[X\geq 0]$, hence the
latter is invariantly defined. As the weight vector fields
$\Delta^{1},\ldots,\Delta^{k}$ preserve the $X$-weight, they are tangent to
the submanifold $F[X\geq 0]$ which therefore has the induced $k$-fold graded
bundle structure.

Considering $-X$ instead of $X$, we obtain graded subbundles $F[-X\geq 0]$,
thus we can put to zero invariantly all coordinates of positive $X$-weight,
and, eventually, graded subbundle

$\displaystyle F[X=0]=F[X\geq 0]\cap F[-X\geq 0].$

Is is clear that $F[X=0]$ consist of singular points of the vector field
$X=\sum a_{s}\Delta^{s}$.

######  Remark 1.9.

Formally we could admit even irrational coefficient in $X$ to define $F[X\geq
0]$ or $F[X=0]$ but this does not enlarge possible constructions, e.g., if
$\Delta^{1}=x\partial_{x}$, and $\Delta^{2}=y\partial_{y}$ are weight vector
fields of a double graded bundle $F$ then
$F[x\partial_{x}+\sqrt{2}y\partial_{y}\geq 0]$ is the same as
$F[x\partial_{x}+qy\partial_{y}\geq 0]$ for some rational number $q$ close to
$\sqrt{2}$ since there is only a finite number of weights assigned to
coordinates; moreover, $F[X\geq 0]=F[kX\geq 0]$, $k\neq 0$, so we can forget
about non-integer coefficients in $X$.

######  Example 1.10.

If $(F,\Delta)$ is a graded bundle, then $F[\Delta=0]$ is the base of $F$.

######  Example 1.11.

If ${{D}}=(D;\Delta^{1},\Delta^{2})=(D;A,B;M)$ is a double vector bundle, then
$C:={{D}}[\Delta^{1}-\Delta^{2}=0]$ is the core bundle of ${{D}}$. Besides,
${{D}}[\Delta^{1}-\Delta^{2}\geq 0]$ is the kernel of the vector bundle
morphism $D\rightarrow B$, so the intersection
${{D}}[\Delta^{1}-\Delta^{2}\geq 0]\cap{{D}}[\Delta^{2}-\Delta^{1}\geq 0]$
coincides with the core bundle.

######  Remark 1.12.

The projection $\textnormal{p}^{F}_{[\Delta\leq l]}$ from Definition 1.6 and
the inclusions $F[X=0]\rightarrow F$ and $F[X\geq 0]\rightarrow F$ are
functorial. We shall use this fact later on.

###  1.4 The naïve approach to superisation

Let us briefly discuss the problem of the ‘superisation’ of a graded bundle
via the explicit example of the degree 2 case: this will be sufficient to
outline the problem. Consider $F_{2}$ equipped with local coordinates
$\big{(}x^{A},~{}y^{a},~{}z^{i}\big{)}$ of weight $0$, $1$ and $2$,
respectively. The admissible changes of local coordinates are of the form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad
y^{a^{\prime}}=y^{b}T_{b}^{\>\>\>a^{\prime}}(x),\qquad
z^{i^{\prime}}=z^{j}T_{j}^{\>\>\>i^{\prime}}(x)+\frac{1}{2}y^{a}y^{b}T_{ba}^{\>\>\>\>i^{\prime}}(x),$

where we take $T_{ba}^{\>\>\>\>a^{\prime}}$ to be symmetric in the lower
indices.

Any meaningful notion of ‘superisation’ should be in terms of an invertible
functor and thus establish a categorical equivalence between the category of
graded bundles and some subcategory of the category of supermanifolds. Let us
by brute force ‘superise’ $F_{2}$ by declaring the Grassmann parity of the
coordinates to be equal to the weight. However, notice that this procedure
does not work properly as the changes of coordinates on “$\Pi F_{2}$” would
not contain the tensor $T_{ba}^{\>\>\>\>i^{\prime}}$ due to symmetry of the
tensor in the lower indices. Therefore, some information about $F_{2}$ is lost
and so we are unable to recover $F_{2}$ from such a ‘superisation’. This
example illustrates a ‘no-go theorem’.

_There is no canonical nontrivial direct procedure for constructing a one-to-
one correspondence between graded bundles and $N$-manifolds._

That is, there is no direct analogue of the parity reversion function $\Pi$
for graded bundles. One cannot in general simply declare (maybe up to signs)
some coordinates to be Grassmann odd. However, that is not to say that one can
never superise graded bundles. An obvious exception here are $k$-fold graded
bundles for which one or more of the graded structures is linear. In such
cases one can apply the standard parity reversion functor; a great example
here are $k$-fold vector bundles or graded-linear bundles.

###  1.5 The linearisation of a graded bundle

In this subsection we shall recall the construction of the _linearisation
functor_ given in [6]. In order to avoid some notational clashes, we have
changed some of the notation as compared with previous works.

The tangent bundle ${{T}}F_{k}$ of a graded bundle naturally has the structure
of a double graded bundle. The first weight vector field is simply the
_tangent lift_ [19, 38] $\Delta^{1}:=\textnormal{d}_{{{T}}}\Delta_{F_{k}}$ of
the weight vector field $\Delta_{F_{k}}$ on $F_{k}$, and the second being the
natural Euler vector field on the tangent bundle
$\Delta^{2}:=\Delta_{{{T}}F_{k}}$. The tangent bundle of a graded bundle is a
canonical example of a graded-linear bundle, i.e., a graded bundle equipped
with a compatible linear structure on the total space.

######  Definition 1.13 ([6]).

A double graded bundle ${{F}}=(F;\Delta^{1},\Delta^{2})$ such that $\mathchar
28673\relax^{1}$ is of degree $k-1$ and $\mathchar 28673\relax^{2}$ is of
degree 1, i.e., $\Delta^{2}$ is an Euler vector field, will be referred to as
a _graded-linear bundle_ , or for short a GrL-bundle.

It follows that ${{F}}$ is of total weight $\leq k$ with respect to
$\Delta:=\Delta^{1}+\Delta^{2}$ and a vector bundle structure

$\displaystyle\textnormal{p}^{F}_{[\Delta^{2}\leq 0]}\colon\ F\rightarrow
F\big{[}\Delta^{2}\leq 0\big{]}.$

with respect to the projection onto the submanifold $F[\Delta^{2}\leq 0]$
which inherits a graded bundle structure of degree $k-1$.

######  Example 1.14.

Consider the vertical bundle with respect to the projection $\tau\colon
F_{k}\rightarrow M$. The weight vector fields on the vertical bundle
${{V}}F_{k}\subset{{T}}F_{k}$ are simply the appropriate restrictions of those
on the tangent bundle. _Via_ passing to the total weight, we can view
${{V}}F_{k}$ as a graded bundle of degree $k+1$. However, it will be useful to
shift the first component of bi-weight to allow us to consider the vertical
bundle as a graded bundle of degree $k$. We will always consider this shifted
weight when encountering a vertical bundle of a graded bundle. In other words,
if we take $\Delta^{1}\in\mathfrak{X}({{T}}F_{k})$ the tangent lift of the
weight vector field and $\Delta^{2}\in\mathfrak{X}({{T}}F_{k})$ the Euler
vector field associated with the tangent bundle structure on $F_{k}$ we get

$\displaystyle{{V}}F_{k}={{T}}F_{k}\big{[}\Delta^{1}-\Delta^{2}\geq 0\big{]},$

equipped with the weight vector field
$\Delta^{1}_{{{V}}F_{k}}:=(\Delta^{1}-\Delta^{2})|_{{{V}}F_{k}}$ of degree $k$
and the Euler vector field $\Delta^{2}|_{{{V}}F_{k}}$. It is crucial here
that, although it is not a combination with non-negative coefficients,
$\Delta^{1}-\Delta^{2}$ is still a weight vector field on ${{V}}F_{k}$.

######  Example 1.15.

Consider a degree two graded bundle $F_{2}$ which we equip with homogeneous
local coordinates $(x,y,z)$ of weight $0$, $1$ and $2$, respectively. Taking
the tangent functor combined with the necessary shift in the weight, produces
homogeneous local coordinates

$\displaystyle\Big{(}\underbrace{x}_{(0,0)},\hskip
5.0pt\underbrace{y}_{(1,0)},\hskip 5.0pt\underbrace{z}_{(2,0)};\hskip
5.0pt\underbrace{\dot{x}}_{(-1,1)},\hskip
5.0pt\underbrace{\dot{y}}_{(0,1)},\hskip
5.0pt\underbrace{\dot{z}}_{(1,1)}\Big{)}.$

The removal of the coordinate $\dot{x}$ is well defined (see Definition 1.8).
Doing so produces a double graded bundle which we recognize as the standard
vertical bundle of $F_{2}$.

The _linearisation functor_ takes a graded bundle and produces a GrL-bundle.
The basic idea is to mimic the canonical embedding

$\displaystyle{{T}}^{k}M\hookrightarrow{{T}}\big{(}{{T}}^{k-1}M\big{)},$

which sends a point in ${{T}}^{k}M$ represented by a curve $t\mapsto\gamma(t)$
to a vector tangent to ${{T}}^{k-1}M$ represented by the curve
$s\mapsto[t\mapsto\gamma(s+t)]_{k-1}\in{{T}}^{k-1}M$. The linearisation of a
graded bundle $F_{k}$ is then viewed as a reduction of the tangent bundle
${{T}}F_{k}$.

######  Definition 1.16 ([6]).

The _linearisation of a graded bundle_ $F_{k}$ is the GrL-bundle defined as

$\displaystyle{\mathbf{l}}(F_{k}):={{V}}F_{k}\big{[}\Delta_{{{V}}F_{k}}^{1}\leq
k-1\big{]},$

so that we have the natural projection
$\textnormal{p}^{{{V}}F_{k}}_{{\mathbf{l}}(F_{k})}\colon{{V}}F_{k}\rightarrow{\mathbf{l}}(F_{k})$.
With a small abuse of notation, we would write
${\mathbf{l}}(F_{k})={{T}}F_{k}[0\leq\Delta^{1}-\Delta^{2}\leq k-1]$, i.e., we
get rid of coordinates on ${{T}}F_{k}$ of negative weights and of weight $k$
with respect to the vector field $\Delta^{1}-\Delta^{2}$.

######  Theorem 1.17 ([6]).

Linearisation is a functor from the category of graded bundles to the category
of double graded bundles.

A full proof of the above theorem can be found in [6]. Essentially, the
functorial properties of the linearisation follow from the fact that it is
constructed from the tangent functor.

Let us briefly describe the local structure of the linearisation. Consider
$F_{k}$ equipped with local coordinates
$\big{(}x^{A},y_{w}^{a},z^{i}_{k}\big{)}$, where the weights are assigned as
$\operatorname{w}\big{(}x^{A}\big{)}=0$, $\operatorname{w}(y^{a}_{w})=w$
$(1\leq w<k)$ and $\operatorname{w}(z^{i}_{k})=k$. It will be convenient to
single out the highest weight coordinates, as well as the zero weight
coordinates, and so we change our notation slightly. In any natural
homogeneous system of coordinates on the vertical bundle ${{V}}F_{k}$, one
projects out the highest weight coordinates on $F_{k}$ to obtain
${\mathbf{l}}(F_{k})$. Thus, on ${\mathbf{l}}(F_{k})$ we have local
homogeneous coordinates

$\displaystyle\Big{(}\underbrace{x^{A}}_{(0,0)},\hskip
5.0pt\underbrace{y^{a}_{w}}_{(w,0)};\hskip
5.0pt\underbrace{\dot{y}^{b}_{w}}_{(w-1,1)},\hskip
5.0pt\underbrace{\dot{z}_{k}^{i}}_{(k-1,1)}\Big{)}.$

Note that with this assignment of the weights, the linearisation of a graded
bundle of degree $k$ is itself a graded bundle of degree $k$ when passing from
the bi-weight to the total weight. It is important to note that the
linearisation of a graded bundle has the structure of a vector bundle
${\mathbf{l}}(F_{k})\rightarrow F_{k-1}$, hence the nomenclature
“linearisation”. The vector bundle structure is clear from the construction by
examining the bi-weight. The second leg of GrL-bundle ${\mathbf{l}}(F_{k})$ is
a degree $k-1$ graded bundle with base $F_{1}$:

$\textstyle{{\mathbf{l}}(F_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M.}$

The basic properties of the linearisation functor are summarized in the
following.

######  Proposition 1.18.

  1. $(a)$

There exist a canonical vector bundle morphism
$p=\textnormal{p}^{{{V}}F_{k}}_{{\mathbf{l}}(F_{k})}\colon{{V}}F_{k}\rightarrow{\mathbf{l}}(F_{k})$
covering the projection $\tau^{k}_{k-1}\colon F_{k}\rightarrow F_{k-1}$ such
that ${{V}}F_{k}$ is the pullback of the vector bundle
${\mathbf{l}}(F_{k})\rightarrow F_{k-1}$ by $p$:

$\textstyle{p^{*}{\mathbf{l}}(F_{k})\simeq{{V}}F_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{\mathbf{l}}(F_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{k}_{k-1}}$$\textstyle{F_{k-1}.}$

  2. $(b)$

The linearisation ${\mathbf{l}}({{T}}^{k}M)$ of the $k$-th order tangent
bundle of $M$ is isomorphic with ${{T}}{{T}}^{k-1}M$ as GrL-bundle. In
particular, there is a canonical epimorphism
$p\colon{{V}}{{T}}^{k}M=\ker({{T}}\tau^{k}_{M}\colon{{T}}{{T}}^{k}M$
$\rightarrow{{T}}M)\twoheadrightarrow{{T}}{{T}}^{k-1}M$. In coordinates,

$\displaystyle p^{*}\big{(}x^{a,(\alpha)}\big{)}=x^{a,(\alpha)},\qquad
p^{*}\big{(}dx^{a,(\beta-1)}\big{)}=dx^{a,(\beta)},$

where $0\leq\alpha\leq k-1$, $1\leq\beta\leq k$.

  3. $(c)$

There is a canonical $($total$)$ weight preserving embedding

$\displaystyle\iota\colon\ F_{k}\hookrightarrow{\mathbf{l}}(F_{k})$

given by the composition of the weight vector field
$\Delta_{F_{k}}\in\mathfrak{X}(F_{k})$ considered as a section of ${{V}}F_{k}$
with the natural projection $p\colon{{V}}F_{k}\rightarrow{\mathbf{l}}(F_{k})$.
In natural local coordinates, the nontrivial part of the embedding is given by

$\displaystyle\iota^{*}(\dot{y}^{a}_{w})=wy^{a}_{w},\qquad\iota^{*}(\dot{z}_{k}^{i})=kz_{k}^{i}.$

  4. $(d)$

The induced morphism
${\mathbf{l}}(\tau^{k}_{k-r})\colon{\mathbf{l}}(F_{k})\rightarrow{\mathbf{l}}(F_{k-r})$
coincides with
${\mathbf{l}}(F_{k})\rightarrow{\mathbf{l}}(F_{k})[\Delta^{1}_{{\mathbf{l}}(F_{k})}\leq
k-r-1]$, where $\tau^{k}_{k-r}=p^{F_{k}}_{[\Delta_{F_{k}}\leq k-r]}\colon
F_{k}\to F_{k-r}$ is the canonical projection on a lower graded bundle.

  5. $(e)$

Given an injective graded bundle morphism $\phi\colon F_{k}\rightarrow
F_{k}^{\prime}$, the associated morphism ${\mathbf{l}}(\phi)$ is injective, as
well.

###### Proof.

We shall only give a geometrical explanation of (b). Another proof can be
found in the arXiv preprint version of [5]. The statement (a) follows directly
from the definition of the linearisation functor as both vector bundles in
interest are of the same rank. The statements (d) and (e) are very easy to
proof directly, while (c) was carefully proved in [6].

Recall [11, 14], a $k$-th tangent bundle ${{T}}^{k}M$ is equipped with an
endomorphism $J$ of its tangent bundle, called the canonical _almost tangent
structure of order $k$_, which has the form

$\displaystyle
J\partial_{x^{A,(\alpha)}}=\partial_{x^{A,(\alpha+1)}}\quad(\text{for}\
0\leq\alpha\leq k-1),\qquad J\partial_{x^{A,(k)}}=0,$

so the image of $J^{i}=\underbrace{J\circ\cdots\circ J}_{i\text{-times}}$
coincides with the kernel of $J^{k+1-i}$ for $i=1,\ldots,k$. Thus we have an
exact sequence

$\displaystyle
0\rightarrow\operatorname{im}J^{k}\rightarrow{{T}}{{T}}^{k}M\xrightarrow{J}{{V}}{{T}}^{k}M.$

The image of $J^{k}$ is given in ${{T}}{{T}}^{k}M$ by vanishing all
differentials $dx^{A,(\alpha)}$ with $0\leq\alpha<k$, hence
${{T}}\tau^{k}_{k-1}\colon{{T}}{{T}}^{k}M\twoheadrightarrow{{T}}{{T}}^{k-1}M$
vanishes on the kernel of $J$, and so it factors to a vector bundle morphism
${{V}}{{T}}^{k}M\twoheadrightarrow{{T}}{{T}}^{k-1}M$. It is immediate to check
that the obtained projection coincides with
$p^{{{V}}F_{k}}_{{\mathbf{l}}F_{k}}\colon{{V}}F_{k}\rightarrow{\mathbf{l}}(F_{k})$,
where $F_{k}={{T}}^{k}M$. ∎

The linearisation functor associates in a canonical functorial way a GrL-
bundle with any graded bundle. It is clear that one can now iterate the
application of the linearisation functor and from a graded bundle of degree
$k$ produce a $k$-fold vector bundle. We describe this construction carefully
in the next section.

##  2 Graded bundles vs $\boldsymbol{k}$-fold vector bundles

###  2.1 The full linearisation functor

Recall, that the linearisation functor ${\mathbf{l}}$ takes a graded bundle
$F_{k}$ of degree $k$ to a double graded bundle of bi-degree $(k-1,1)$. We may
apply the functor ${\mathbf{l}}$ again to the graded bundle structure of
degree $k-1$ on GrL-bundle ${\mathbf{l}}(F_{k})$ to get a triple graded bundle
of multi-degree $(k-2,1,1)$. After $k-1$ iterations we arrive at a $k$-fold
vector bundle denoted by ${\mathbf{l}}^{(k-1)}(F_{k})$ which we take as a
definition of the _full linearisation functor_.

######  Definition 2.1.

The full linearisation functor

$\displaystyle{\mathbf{L}}\colon\
\textnormal{{GrB}}[k]\rightarrow\textnormal{{VB}}[k]$

is defined as the composition of the linearisation functor $(k-1)$ times,
i.e., ${\mathbf{L}}(F_{k})={\mathbf{l}}^{(k-1)}(F_{k})$.

######  Example 2.2.

${\mathbf{L}}({{T}}^{k}M)={{T}}^{(k)}M$ by Proposition 1.18.

Similarly, we can consider iterations of the vertical functor. Recall, the
vertical functor ${{V}}$ takes a graded bundle of degree $k$ to a double
graded bundle of degree $(k,1)$. Applying again the functor ${{V}}$ to
$\big{(}{{V}}(F_{k}),\Delta^{1}_{{{V}}F_{k}}\big{)}$ we obtain
${{V}}{{V}}F_{k}$ which is a $3$-fold graded bundle of degree $(k,1,1)$.
Continuing this way we find that for any $r\geq 1$,
${{V}}^{(r)}F_{k}=\underbrace{{{V}}{{V}}\cdots{{V}}}_{r\text{-times}}F_{k}$ is
a $(r+1)$-fold graded bundle of degree $(k,1,\ldots,1)$.

######  Proposition 2.3.

There is a canonical epimorphism $(0\leq r\leq k-1)$

$\displaystyle p^{{{V}}^{(r)}F_{k}}_{{\mathbf{l}}^{(r)}({F}_{k})}\colon\
{{V}}^{(r)}F_{k}\rightarrow{\mathbf{l}}^{(r)}(F_{k})={{V}}^{(r)}F_{k}\big{[}\Delta^{1}_{{{V}}^{(r)}F_{k}}\leq
k-r\big{]}$ (2.1)

and an embedding

$\displaystyle{\mathbf{l}}^{(r)}(F_{k})\hookrightarrow{{T}}^{(r)}{{T}}^{k-r}F_{k}.$
(2.2)

Both mappings respect $(r+1)$-fold graded bundle structures.

###### Proof.

We shall prove (2.1) by induction. The case $r=0$ is trivial and the case
$r=1$ follows directly from the definition of the functor ${\mathbf{l}}$.
Denote $\tilde{F}_{k}:={{V}}^{(r)}F_{k}$ which we consider now as a graded
bundle of degree $k$. By the inductive hypothesis, the base of the canonical
projection
$\tilde{\tau}^{k}_{k-r}\colon\tilde{F}_{k}\rightarrow\tilde{F}_{k}\big{[}\Delta^{1}_{\tilde{F}_{k}\leq
k-r}\big{]}$, which is defined in (1.1), coincides with
${\mathbf{l}}^{(r)}(F_{k})$. To get ${\mathbf{l}}^{(r+1)}F_{k}$, apply the
functor ${\mathbf{l}}$ to the projection $\tilde{\tau}^{k}_{k-r}$, then
compose it with the canonical projection
$p^{{{V}}\tilde{F}_{k}}_{{\mathbf{l}}(\tilde{F}_{k})}$ on the linearisation of
$\tilde{F}_{k}$. Using Proposition 1.18(d) we arrive at the sequence

$\displaystyle{{V}}^{(r+1)}F_{k}={{V}}\tilde{F}_{k}\xrightarrow{p^{{{V}}\tilde{F}_{k}}_{{\mathbf{l}}(\tilde{F}_{k})}}({{V}}\tilde{F}_{k})\big{[}\Delta^{1}_{{{V}}^{(r+1)}F_{k}}\leq
k-1\big{]}={\mathbf{l}}(\tilde{F}_{k})\xrightarrow{{\mathbf{l}}(\tilde{\tau}^{k}_{k-r})}{\mathbf{l}}^{(r+1)}(F_{k})$
$\displaystyle\hphantom{{{V}}^{(r+1)}F_{k}={{V}}\tilde{F}_{k}\xrightarrow{p^{{{V}}\tilde{F}_{k}}_{{\mathbf{l}}(\tilde{F}_{k})}}}{}\overset{(*)}{=}{\mathbf{l}}(\tilde{F}_{k})\big{[}\Delta^{1}_{{\mathbf{l}}(\tilde{F}_{k})}\leq
k-r-1\big{]}={{V}}^{(r+1)}F_{k}\big{[}\Delta^{1}_{{{V}}^{(r+1)}F_{k}}\leq
k-r-1\big{]}$

that completes our inductive reasoning.

To prove (2.2) consider the graded bundle $\tau\colon F_{k}\rightarrow M$ as a
canonical substructure of $\tau^{k}_{N}\colon{{T}}^{k}N\rightarrow N$ with
$N=F_{k}$ (for this fundamental observation see [18]) and apply the functor
${\mathbf{l}}$ to both structures. By Proposition 1.18(b) and (e) we get a
GrL-bundle embedding
${\mathbf{l}}(\iota)\colon{\mathbf{l}}(F_{k})\rightarrow{\mathbf{l}}({{T}}^{k}N)={{T}}{{T}}^{k-1}N$,
where $\iota\colon F_{k}\rightarrow{{T}}^{k}N$ denotes the canonical
embedding. We can iterate the application of the linearisation functor and
finally get (2.2). ∎

######  Example 2.4.

Consider a graded bundle $F_{2}$ equipped with homogeneous local coordinates
$(x^{A},~{}y^{a},~{}z^{i})$ of weight $0,1$ and $2$, respectively. Then,
${{V}}F_{2}\subset{{T}}F_{2}$, ${{V}}^{(2)}F_{2}\subset{{T}}^{(2)}F_{2}$, and
${{V}}^{(3)}F_{2}\subset{{T}}^{(3)}F_{2}$ come equipped with homogeneous
coordinates

$\displaystyle\big{(}x^{A,(0)}_{0},y^{a,(0))}_{1},z^{i,(0)}_{2};y^{a,(1)}_{0},z^{i,(1)}_{1}\big{)},$
$\displaystyle\big{(}x^{A,(0,0)}_{0},y^{a,(0,0)}_{1},z^{i,(0,0)}_{2},y^{a,(0,1)}_{0},z^{i,(0,1)}_{1};y^{a,(1,0)}_{0},z^{i,(1,0)}_{1},z^{i,(1,1)}_{0}\big{)},$
$\displaystyle\big{(}x^{A,(000)}_{0},y^{a,(000)}_{1},z^{i,(000)}_{2},y^{a,(001)}_{0},z^{i,(001)}_{1},y^{a,(010)}_{0},z^{i,(010)}_{1},z^{i,(011)}_{0};$
$\displaystyle\qquad{}y^{a,(100)}_{0},z^{i,(100)}_{1},z^{i,(101)}_{0},z^{i,(110)}_{0}\big{)},$

respectively. The first component written in the bottom of the multi-weight is
that naturally inherited from $F_{2}$, while the other components come from
the vector bundle structure of the tangent bundles. In full generality,
consider a graded bundle $F_{k}$ equipped with homogeneous local coordinates
$\big{(}x^{A},y^{i}_{w}\big{)}$, then ${{V}}^{(r)}F_{k}$ can be equipped with
inherited from ${{T}}^{(r)}F_{k}$ homogeneous local coordinates

$\displaystyle\big{(}x^{A},y^{i,(\varepsilon)}_{w}\big{)},$

where $\varepsilon\in\\{0,1\\}^{r}$ and $|\varepsilon|\leq w$. The weight of
$y^{i,(\varepsilon)}_{w}$ with respect to $\Delta^{1}_{{{V}}^{(r)}}F_{k}$ is
$w-|\varepsilon|$ (not explicitly written now). Consider
$({{T}}^{(r)}F_{k};\Delta^{0},\Delta^{1},\ldots,\Delta^{r})$ as a $(r+1)$-fold
graded bundle with Euler vector fields $\Delta^{1},\ldots,\Delta^{r}$ of
$r$-th iterated tangent bundle of $F_{k}$ (namely, $\Delta^{i}$ is the Euler
vector field of the vector bundle projection
${{T}}^{(i-1)}\tau_{T^{(r-i)}F_{k}}\colon{{T}}^{(r)}F_{k}\rightarrow{{T}}^{(r-1)}F_{k}$,
$1\leq i\leq r$) and the weight vector field
$\Delta^{0}:=\operatorname{d}_{{{T}}}^{(r)}\Delta_{F_{k}}$ being the
(iterated) tangent lift of the weight vector field $\Delta_{F_{k}}$ of
$F_{k}$. Define $X_{r}\in\mathfrak{X}({{T}}^{(r)}F_{k})$ as
$X_{r}=\Delta^{0}-\sum_{1}^{k}\Delta^{k}$, so the weight of
$y^{a,(\varepsilon_{1},\ldots,\varepsilon_{r})}_{w}$ with respect to $X_{r}$
is $w-(\varepsilon_{1}+\cdots+\varepsilon_{r})$, hence

$\displaystyle{{V}}^{(r)}F_{k}={{T}}^{(r)}F_{k}[X_{r}\geq 0].$ (2.3)

Note that $\Delta^{1}_{{{V}}^{(r)}F_{k}}=X_{r}|_{{{V}}^{(r)}F_{k}}$.

######  Example 2.5.

Let us consider in some detail the generic case when we are dealing with a
graded bundle of order three. This example should be enough to highlight the
main constructions explicitly without too much notational clutter. Let us
employ homogeneous local coordinates
$\big{(}x^{A},y^{a},z^{i},w^{\kappa}\big{)}$ on $F_{3}$ where the weight is
assigned as $0$, $1$, $2$ and $3$, respectively. The admissible changes of
local coordinates are of the form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad
y^{a^{\prime}}=y^{b}T_{b}^{\>\>\>a^{\prime}}(x),\qquad
z^{i^{\prime}}=z^{j}T_{j}^{\>\>\>i^{\prime}}(x)+\frac{1}{2}y^{a}y^{b}T_{ba}^{\>\>\>\>i^{\prime}}(x),$
$\displaystyle
w^{\kappa^{\prime}}=w^{\mu}T_{\mu}^{\>\>\kappa^{\prime}}(x)+z^{i}y^{a}T_{ai}^{\>\>\>\kappa^{\prime}}(x)+\frac{1}{3!}y^{a}y^{b}y^{c}T_{cba}^{\>\>\>\>\>\>\kappa^{\prime}}(x).$

Then on ${\mathbf{l}}(F^{3})$ we can employ coordinates
$(x^{A},\dot{y}^{a},y^{a},\dot{z}^{i},z^{i},\dot{w}^{\kappa})$ of weights
$(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$, $(2,0)$ and $(2,1)$, respectively. The
additional transformation rules are

$\displaystyle\dot{y}^{a^{\prime}}=\dot{y}^{b}T^{a^{\prime}}_{b},\qquad\dot{z}^{i^{\prime}}=\dot{z}^{j}T^{i^{\prime}}_{j}+\dot{y}^{a}y^{b}T^{i^{\prime}}_{ba},$
$\displaystyle\dot{w}^{\kappa^{\prime}}=\dot{w}^{\mu}T_{\mu}^{\>\>\kappa^{\prime}}+(\dot{z}^{i}y^{a}+z^{i}\dot{y}^{a})T_{ai}^{\>\>\>\kappa^{\prime}}+\frac{1}{2!}\dot{y}^{a}y^{b}y^{c}T_{cba}^{\>\>\>\>\>\>\kappa^{\prime}}.$

By iterating this construction and differentiation above formulas, we get
${\mathbf{l}}^{(2)}F_{3}$ with coordinates $(x^{A}$, $\dot{y}^{a}=y^{a,(10)}$,
$y^{a}=y^{a,(00)}$, $\dot{z}^{i}=z^{i,(10)}$, $dy^{a}=y^{a,(01)}$,
$d\dot{z}^{i}=z^{i,(11)}$, $dz^{i}=z^{i,(01)}$,
$d\dot{w}^{\kappa}=w^{\kappa,(11)})$ of degrees $(000)$, $(010)$, $(100)$,
$(110)$, $(001)$, $(011)$, $(101)$, $(111)$, respectively, and transformation
laws for extra coordinates

$\displaystyle\allowdisplaybreaks
dy^{a^{\prime}}=dy^{b}T^{a^{\prime}}_{b}(x),$ $\displaystyle
d\dot{z}^{i^{\prime}}=d\dot{z}^{j}T^{i^{\prime}}_{j}(x)+\dot{y}^{a}dy^{b}T^{i^{\prime}}_{ba}(x),$
$\displaystyle
dz^{i^{\prime}}=dz^{j}T^{i^{\prime}}_{j}(x)+dy^{a}y^{b}T^{i^{\prime}}_{ba}(x),$
$\displaystyle
d\dot{w}^{\kappa^{\prime}}=d\dot{w}^{\mu}\,T^{\kappa^{\prime}}_{\mu}(x)+(\dot{z}^{i}dy^{a}+d\dot{z}^{i}y^{a}+dz^{i}\dot{y}^{a})T^{\kappa^{\prime}}_{ai}(x)+\dot{y}^{a}\,dy^{b}y^{c}T^{\kappa^{\prime}}_{cba}(x).$

We see that transformation laws for coordinates of the same total degree are
essentially the same. We conclude that ${\mathbf{l}}^{(2)}(F_{3})$ is a triple
vector bundle with isomorphic feet ${\mathbf{l}}(F_{2})$

$\textstyle{{\mathbf{l}}F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{L}}F_{3}={\mathbf{l}}^{(2)}F_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{l}}F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$$\textstyle{{\mathbf{l}}F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

In full generality, it is natural to use the coordinate system
$\big{(}x^{A},y^{a,(\varepsilon)}_{w}\big{)}$, $\varepsilon\in\\{0,1\\}^{r}$,
for ${\mathbf{l}}^{(r)}F_{k}$ as well (now we take only such
$y^{a,(\varepsilon)}_{w}$ that $0\leq w-|\varepsilon|\leq k-r$), as these
coordinates are constant on fibers of the fibration
$p^{{{V}}^{(r)}(F_{k})}_{{\mathbf{l}}^{(r)}(F_{k})}$.

On the other hand, consider $F_{3}$ as a substructure of ${{T}}^{3}F_{3}$
given by equations $x^{A,(\beta)}=0$, $y^{a,(\alpha)}_{w}=0$, for
$1\leq\alpha,\beta\leq 3$ such that $\alpha\neq w$. Then we recognise
${\mathbf{l}}(F_{3})$ as a subset of ${{T}}{{T}}^{2}F_{3}$ and
${\mathbf{L}}(F_{3})={\mathbf{l}}^{(2)}(F_{3})$ as a subset of
${{T}}^{(3)}F_{3}$, the latter is defined by equations
$x^{A,(\varepsilon)}=0$, for $\varepsilon\neq(0,0,0)$ and
$y_{w}^{a,(\varepsilon)}=0$ if
$\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\neq w$ in local coordinate
system $\big{(}x^{A},y_{w}^{a,(\varepsilon)}\big{)}$,
$\varepsilon=(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})\in\\{0,1\\}^{3}$,
on ${{T}}^{(3)}F_{3}$. Transformations for the remaining coordinates are the
same as given above for ${\mathbf{l}}^{(2)}F_{3}$ with the identification
$\big{(}x^{A},y_{|\varepsilon|}^{(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})}\big{)}\mapsto\big{(}x^{A},y_{|\varepsilon|}^{(\varepsilon_{2},\varepsilon_{3})}\big{)}$,
$|\varepsilon|=\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}$. The weight of
$y_{w}^{a,(\varepsilon)}$ with respect to the vector field
$X_{3}\in\mathfrak{X}(F_{3})$, defined as the difference between
$\operatorname{d}_{{T}}^{(3)}\Delta_{F_{3}}$ and the sum of three Euler vector
fields defining triple vector bundle structure on ${{T}}^{(3)}F_{3}$, is equal
$w-(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})$. Thus
${\mathbf{L}}(F_{3})$ as a subset of ${{T}}^{(3)}F_{3}$ is the set of singular
points of the vector field $X_{3}$.

In general, the embedding ${\mathbf{L}}(F_{k})\subset{{T}}^{(k)}F_{k}$ defined
in (2.2) can be given an explicit form, namely

$\displaystyle{\mathbf{L}}(F_{k})\simeq{{T}}^{(k)}F_{k}[X_{k}=0],$ (2.4)

where $X_{k}=\Delta^{0}-\sum_{1}^{k}\Delta^{k}$ is as above. (It is not worth
to consider ${\mathbf{L}}(F_{k})$ as a $(k+1)$-fold graded bundle since
$\Delta^{0}$ restricted to ${\mathbf{L}}(F_{k})$ coincides with the sum of
Euler vector fields $\Delta^{i}$, so $({{T}}^{(k)}F_{k}[X=0];\Delta^{0})$ is
just a graded bundle of degree $k$ defined by taking the total degree in
$k$-fold vector bundle ${\mathbf{L}}(F_{k})$.)

Similarly, the epimorphism ${{V}}^{(k-1)}F_{k}\rightarrow{\mathbf{L}}(F_{k})$
has the explicit form

$\displaystyle{{V}}^{(k-1)}F_{k}\rightarrow\big{(}{{V}}^{(k-1)}F_{k}\big{)}\big{[}\Delta^{1}_{{{V}}^{(k-1)}F_{k}}\leq
1\big{]}$ (2.5)

and $\Delta^{1}_{{{V}}^{(k-1)}F_{k}}=X_{k-1}|_{{{V}}^{(k-1)}F_{k}}$. The
isomorphism $\big{(}V^{(k-1)}F_{k}\big{)}[\Delta^{1}_{V^{(k-1)}F_{k}}\leq
1]\simeq{{T}}^{(k)}F_{k}[X=0]$ has the form

$\displaystyle\big{(}x^{A},y^{i,(\varepsilon)}_{w}\big{)}\mapsto\big{(}x^{A},y^{i,(w-|\varepsilon|,\varepsilon_{2},\ldots,\varepsilon_{k})}_{w}\big{)},$
(2.6)

where
$\varepsilon=(\varepsilon_{2},\ldots,\varepsilon_{k})\in\\{0,1\\}^{k-1}$, so
$|\varepsilon|=\varepsilon_{2}+\cdots+\varepsilon_{k}$, and $0\leq
w-|\varepsilon|\leq 1$.

######  Corollary 2.6.

The full linearisation of a graded bundle $F_{k}$ can be seen as a quotient
structure of the iterated vertical bundle ${{V}}^{(k-1)}F_{k}$ and as a
substructure of the iterated tangent bundle ${{T}}^{(k)}F_{k}$, as well. The
local coordinate formulas are obtained from (2.3), (2.4), (2.5) and (2.6).

In view of Corollary 2.6, it does not fundamentally matter if we chose to
interpret the induced homogeneous coordinates on ${\mathbf{L}}(F_{k})$ as
coming from ${{V}}^{(k-1)}F_{k}$ or ${{T}}^{(k)}F_{k}$, though the latter
maybe easier to comprehend. However, when examining the structure of the full
linearisation, understanding ${\mathbf{L}}(F_{k})$ as a substructure of
${{T}}^{(k)}F_{k}$ will prove useful.

###  2.2 Characterisation of the full linearisation for the degree two case

The full linearisation is a functor from the category of graded bundles of
degree $k$ to the category of $k$-fold vector bundles, however it is clear
that not all $k$-fold vector bundles arise as the full linearisation of a
graded bundle. All $k$-fold vector bundles are isomorphic, but usually non-
canonically, to a _decomposed_ or _split_ $k$-fold vector bundle which is
build out of $2^{k}-1$ vector bundles $E_{\varepsilon}\rightarrow M$, where
the weight $\varepsilon\in\\{0,1\\}^{k}$ is a nonzero sequence, known as
_building $($vector$)$ bundles._ Essentially, every $k$-fold vector bundle is
built from its building bundles, just not in a canonical fashion. For example,
for any DVB, say ${{D}}=(D;A,B;M)$ with the core bundle $C\rightarrow M$,
there exist a double vector bundle isomorphism
${{D}}\rightarrow(A\times_{M}B\times_{M}C;A,B;M)$ inducing the identity on all
the building vector bundles $A={{D}}_{(0,1)}$, $B={{D}}_{(1,0)}$ and
$C={{D}}_{(1,1)}$.

If the $k$-fold vector bundle in question is balanced, then every vector
bundle $E_{\varepsilon}$ of a given total weight are identified and thus _any_
balanced $k$-fold vector bundle is the full linearisation of some graded
bundle of degree $k$. However, this is not sufficient to completely
characterise the full linearisation. There are morphisms between the full
linearisations that are permissible and not permissible; the latter means that
they cannot be traced back to morphisms between the original graded bundles.
For concreteness let us first consider the degree two case carefully.

The full linearisation ${{D}}:={\mathbf{L}}(F_{2})$ of a graded bundle $F_{2}$
comes with an additional canonical structure: a double vector bundle morphism
$\sigma\colon{{D}}\rightarrow{{D}}^{\textnormal{f}}$, where
${{D}}^{\textnormal{f}}=(D,\mathchar 28673\relax^{2},\mathchar
28673\relax^{1})$ is the flip of ${{D}}=(D,\mathchar 28673\relax^{1},\mathchar
28673\relax^{2})$. The morphism $\sigma$ simply exchanges the legs (so the
Euler vector fields), and can be defined as the restriction of the canonical
involution $\kappa\colon{{T}}{{T}}F_{2}\rightarrow{{T}}{{T}}F_{2}$ to the
double vector subbundle $D\subset{{T}}{{T}}F_{2}={{T}}^{(2)}F_{2}$. In local
coordinates as introduced earlier, the canonical DVB morphism is given by

$\displaystyle\sigma^{*}y^{a,(10)}=y^{a,(01)},\qquad\sigma^{*}y^{b,(01)}=y^{b,(10)},\qquad\sigma^{*}z^{i,(11)}=z^{i,(11)}.$
(2.7)

######  Remark 2.7.

One must note that a general balanced DVB does _not_ admit a canonical
$\sigma\colon$ ${{D}}\rightarrow{{D}}^{\textnormal{f}}$, the local description
above is given in specific (adapted) coordinates on ${{T}}^{(2)}F_{2}$.

Now we can consider a category whose objects are double vector bundles
${{D}}=(D;A,B;M)$ equipped with a DVB morphism
$\sigma\colon{{D}}\rightarrow{{D}}^{\textnormal{f}}$, such that

$\displaystyle\sigma^{\textnormal{f}}\circ\sigma=\operatorname{id}_{D}$ (2.8)

and

$\displaystyle\sigma|_{C}=\operatorname{id}_{C},$ (2.9)

where $C\rightarrow M$ denotes the core bundle of ${{D}}$. It follows from
(2.8) that ${{D}}$ is balanced, as $\sigma|_{A}\colon A\rightarrow B$ and
$\sigma|_{B}\colon B\rightarrow A$ are vector bundle morphisms satisfying
$\sigma|_{B}\circ\sigma|_{A}=\operatorname{id}_{A}$ and
$\sigma|_{A}\circ\sigma|_{B}=\operatorname{id}_{B}$. Under the identification
$A=B$ given by $\sigma|_{A}$, it follows that the following sets coincide

$\displaystyle\\{q\in D\colon\sigma(q)=q\\}=\big{\\{}q\in
D\colon\pi^{D}_{A}(q)=\pi^{D}_{B}(q)\big{\\}}.$

######  Definition 2.8.

Such pairs $({{D}},\sigma)$ we will refer to as _symmetric double vector
bundles_ and the above set as the _diagonal_ or the set of _holonomic vectors_
of $({{D}},\mathchar 28955\relax)$. A morphism of symmetric double vector
bundles $({{D}},\sigma)\rightarrow({{D}}^{\prime},\sigma^{\prime})$ is a DVB
morphism $\phi\colon{{D}}\rightarrow{{D}}^{\prime}$ commuting with the
additional structure:

$\displaystyle\sigma^{\prime}\circ\phi=\phi^{\textnormal{f}}\circ\sigma.$
(2.10)

As the flip operation is a functor,
$\phi^{\textnormal{f}}\colon{{D}}^{\textnormal{f}}\rightarrow{{D}}^{\prime\textnormal{f}}$
coincides with $\phi$ as a map between sets; a DVB and its flip have the same
underlying manifold structure. Composition of morphisms in this category is
the standard composition of smooth maps between manifolds. We will denote this
category as $\textnormal{{SymmVB}}[2]$.

Let us pick local coordinates
$\big{(}x^{A},y_{(1,0)}^{a},y_{(0,1)}^{b},z_{(1,1)}^{i}\big{)}$ on an
arbitrary symmetric double vector bundle $({{D}},\sigma)$ in such a way that
$\sigma^{*}y^{a}_{(0,1)}=y^{a}_{(1,0)}$, so
$\sigma^{*}y^{a}_{(1,0)}=y^{a}_{(0,1)}$ by (2.8). Then, by (2.9), the
isomorphism $\sigma\colon{{D}}\rightarrow{{D}}^{\textnormal{f}}$ is of the
form

$\displaystyle\sigma^{*}z^{i}_{(1,1)}=z^{i}_{(1,1)}+y^{a}_{(1,0)}y^{b}_{(0,1)}\sigma_{ba}^{i}(x).$

From (2.8), ‘applying sigma twice’ to the $z$ coordinate results in

$\displaystyle(\operatorname{id}_{D})^{*}z^{i}_{(1,1)}=z^{i}_{(1,1)}+y^{a}_{(1,0)}y^{b}_{(0,1)}\sigma_{ba}^{i}(x)+y^{a}_{(0,1)}y^{b}_{(1,0)}\sigma_{ba}^{i}(x),$

and thus $\sigma_{ba}^{i}={-}\sigma_{ab}^{i}$.

A morphism of double vector bundles $\phi\colon{{D}}\rightarrow{{D}}^{\prime}$
is locally of the form (we do not explicitly write the morphism $M\rightarrow
M^{\prime}$ as its exact form is not important for our discussion)

$\displaystyle\phi^{*}y^{a^{\prime}}_{(1,0)}=y^{a}_{(1,0)}Q_{a}^{\>\>\>a^{\prime}}(x),\qquad\phi^{*}y^{b^{\prime}}_{(0,1)}=y^{b}_{(0,1)}R_{b}^{\>\>\>b^{\prime}}(x),$
$\displaystyle\phi^{*}z^{i^{\prime}}_{(1,1)}=z^{i}_{(1,1)}Q_{i}^{\>\>\>i^{\prime}}(x)+y^{a}_{(1,0)}y^{b}_{(0,1)}Q_{ba}^{\>\>\>\>i^{\prime}}(x).$

The reader can quickly convince themselves that if we further insist on the
condition (2.10), then $Q_{a}^{\>\>\>a^{\prime}}=R_{a}^{\>\>\>a^{\prime}}$ and
$Q_{ba}^{\>\>\>\>i^{\prime}}=Q_{ab}^{\>\>\>\>i^{\prime}}$. We now explicitly
see the role of the morphism $\sigma$ in ensuring that morphism are
‘symmetric’. In particular, any morphism satisfying (2.10) comes from a
morphism of graded bundles $F_{2}\rightarrow F_{2}^{\prime}$, where
$F_{2}:=\\{q\in D~{}|~{}\pi^{D}_{A}(q)=\pi^{D}_{B}(q)\\}$ and similar for
$F_{2}^{\prime}$.

As a matter of formality, let us denote the full linearisation

$\displaystyle\Phi\colon\
\textnormal{{GrB}}[2]\rightarrow\textnormal{{SymmVB}}[2],$

which assigns $({\mathbf{L}}(F_{2}),\kappa|_{{\mathbf{L}}(F_{2})})$ to
$F_{2}$, and the _diagonalisation_ functor as

$\displaystyle\Psi\colon\
\textnormal{{SymmVB}}[2]\rightarrow\textnormal{{GrB}}[2],$

which assigns $F_{2}:=\\{q\in D\,|\,\pi^{D}_{A}(q)=\pi^{D}_{B}(q)\\}$ to $D$.
It is clear that $F_{2}\simeq\Psi(\Phi(F_{2}))$ canonically.

The other way round, we claim that
$\Phi\circ\Psi\simeq\textnormal{Id}_{\textnormal{{SymmVB}}[2]}$. This claim
boils down to finding a canonical isomorphism

$\displaystyle I\colon\
({{D}},\sigma)\rightarrow({\mathbf{L}}(F_{2}),\kappa|_{{\mathbf{L}}(F_{2})}),$

where $F_{2}=\Psi({{D}},\sigma)$, such that the following diagram is
commutative:

$\textstyle{{{D}}}$$\textstyle{{{D}}^{\textnormal{f}}}$$\textstyle{{\mathbf{L}}(F_{2})}$$\textstyle{({\mathbf{L}}(F_{2}))^{\textnormal{f}}.}$$\textstyle{\sigma}$$\textstyle{I}$$\textstyle{I^{\textnormal{f}}}$$\textstyle{\kappa|_{{\mathbf{L}}(F_{2})}}$

Let us denote the naturally inherited coordinates on
$F_{2}=\Psi({{D}},\sigma)$ as

$\displaystyle\big{(}x^{A},\,y^{a}=y^{a}_{(1,0)}|_{F_{2}}=y^{a}_{(0,1)}|_{F_{2}},\,z^{i}=z^{i}_{(1,1)}|_{F_{2}}\big{)}$

and on the full linearisation ${\mathbf{L}}(F_{2})\subset{{T}}^{(2)}F_{2}$ we
have the induced coordinates

$\displaystyle\big{(}x^{A},\,y^{a,(1,0)},\,y^{b,(0,1)},\,z^{i,(1,1)}\big{)}.$

It is then a matter of calculation (we omit details) to show that

$\displaystyle I^{*}y^{a,(1,0)}=y^{a}_{(1,0)},\qquad
I^{*}y^{b,(0,1)}=y^{b}_{(0,1)},\qquad
I^{*}z^{i,(1,1)}=z^{i}_{(1,1)}+\frac{1}{2}y^{a}_{(1,0)}y^{b}_{(0,1)}\sigma_{ba}^{i}(x)$
(2.11)

is the natural isomorphism we are looking for. The reader should note that $I$
does not actually depend on the coordinates employed on $D$. To sum up, we
have thus established the following:

######  Proposition 2.9.

There is an equivalence of categories between $\textnormal{{GrB}}[2]$ and the
category $\textnormal{{SymmVB}}[2]$.

######  Example 2.10.

Consider $D={{T}}^{(2)}M$ and
$\sigma=\kappa_{M}\colon{{T}}^{(2)}M\rightarrow{{T}}^{(2)}M$ which is not an
automorphism but an isomorphism from ${{T}}^{(2)}M$ to the flip
$({{T}}^{(2)}M)^{\textnormal{f}}$. Thus $({{T}}^{(2)}M,\kappa_{M})$ is a
symmetric DVB and the diagonalisation is clearly ${{T}}^{2}M$.

###  2.3 Relation with symplectical double vector bundles

Consider a symmetric double vector bundle $({{D}},\sigma)$, where
${{D}}=(D;A,B;M)$, so $\sigma\colon(D;A,B;M)$ $\rightarrow(D;B,A;M)$ is a DVB
morphism satisfying (2.8) and (2.9). Taking the dual of $\sigma$ with respect
to vector bundle structures defined by vertical arrows produces a  _map_
$D_{B}^{*}\rightarrow D^{*}_{A}$, and not just a relation, as $\sigma$ is an
isomorphism. We shall see soon that it is a DVB morphism depicted in the
diagram (2.13) which we shall denote by
$\sigma^{{v}}\colon{{D}}^{\textnormal{f}{{v}}}\rightarrow{{D}}^{{v}}$.
Moreover, $\sigma^{{{v}}}|_{C^{*}}=\operatorname{id}_{C^{*}}$ because of
(2.9),

$\textstyle{(D^{*}_{B})^{*}_{C^{*}}}$$\textstyle{C^{*}}$$\textstyle{{{E}}^{{h}}}$$\textstyle{A}$$\textstyle{M}$ |  $\stackrel{{\scriptstyle I}}{{\xleftarrow{\hskip 28.45274pt}}}$ |  $\textstyle{D^{*}_{A}}$$\textstyle{C^{*}}$$\textstyle{{{D}}^{{v}}}$$\textstyle{A}$$\textstyle{M}$ |  $\stackrel{{\scriptstyle\sigma^{{{v}}}}}{{\xleftarrow{\hskip 28.45274pt}}}$ |  $\textstyle{D^{*}_{B}}$$\textstyle{C^{*}}$$\textstyle{{{E}}={{D}}^{\textnormal{f}{{v}}}}$$\textstyle{B}$$\textstyle{M.}$
---|---|---|---|---
(2.13)

The vector bundles $D^{*}_{A}$ and $D^{*}_{B}$ are in duality [24] (over base
$C^{*}$, where $C$ is the core of ${{D}}$), which results in a DVB isomorphism
${{D}}^{{v}}=(D^{*}_{A};A,C^{*};M)\rightarrow{{D}}^{\textnormal{f}{{v}}{{h}}}=((D^{*}_{B})^{*}_{C^{*}};A,C^{*};M)$.
It is worth to remember that with the choice of the pairing

$\displaystyle\langle\langle\cdot,\cdot\rangle\rangle\colon\
D^{*}_{A}\times_{C^{*}}D^{*}_{B}\rightarrow{\mathbb{R}},\qquad\langle\langle\Phi,\Psi\rangle\rangle=\Phi(d)-\Psi(d),\qquad
d\in D,$ (2.14)

as given in [24], where $\Phi\in(D_{a})^{*}$, $\Psi\in(D_{b})^{*}$ and $d$
maps to $(a,b)$ under the canonical projection $D\rightarrow A\times_{M}B$,
the DVB isomorphism $I\colon{{D}}^{{v}}\simeq{{D}}^{\textnormal{f}{{v}}{{h}}}$
is the identity on the bundle $C^{*}\rightarrow M$, and the core bundle
$B^{*}\rightarrow M$, but it is minus the identity on the bundle $A\rightarrow
M$. Moreover, there is no canonical DVB isomorphism which is the identity on
all the building bundles. Nevertheless, $\sigma^{{{v}}}$ gives a vector bundle
isomorphism from $D^{*}_{B}$ to its dual over $C^{*}$ covering
$\operatorname{id}_{C^{*}}$, and so the additional structure of $\sigma$ gives
rise to a non-degenerate bilinear form on $E\rightarrow C^{*}$, where
$E=D^{*}_{B}$. We shall shortly show that the bilinear form is skew-symmetric,
and it induces a DVB isomorphism from
${{E}}={{D}}^{\textnormal{f}{{v}}}=(E;B,C^{*};M)$ to $E^{{h}}$. These
observation lead us to the following definition.

######  Definition 2.11.

A _symplectical double vector bundle_ is a double vector bundle
${{E}}=(E;E_{01}$, $E_{10};M)$ equipped with a linear non-degenerate skew-
symmetric form on $E^{*}_{E_{10}}\rightarrow C^{*}$ that induces a double
vector bundle isomorphism $\beta\colon{{E}}\rightarrow{{E}}^{{h}}$.

A morphism between symplectical DVBs

$\displaystyle\Omega\colon\
{{E}}=(E;E_{01},E_{10};M)\dashrightarrow{{E}}^{\prime}=(E^{\prime};E^{\prime}_{01},E^{\prime}_{10};M^{\prime})$

is a relation of the form $\Omega=\gamma^{{v}}$ for some DVB morphism

$\displaystyle\gamma\colon\
{{E}}^{\prime{{v}}}=({E^{\prime}}^{*}_{E^{\prime}_{01}};E^{\prime}_{01},{E^{\prime}}^{*}_{11};M^{\prime})\rightarrow{{E}}^{{v}}=({E}^{*}_{E_{01}};E_{01},{E}^{*}_{11};M).$

It follows that $\Omega$ is a vector subbundle of $\overline{E}\times
E^{\prime}\rightarrow E_{11}\times E_{11}^{\prime}$. It is required that it is
an isotropic subbundle with respect to the difference of the skew-symmetric
forms on $E^{\prime}$ and $E$ (which we indicated by the bar over $E$).

Evidently, symplectical double vector bundles form a category with we denote
as $\textnormal{{SpVB}}[2]$.

######  Remark 2.12.

We will use the nomenclature _symplectical double vector bundle_ as opposed to
_symplectic double vector_ as to avoid possible confusion with double vector
bundles equipped with symplectic forms (of whatever weight) on their total
spaces. This choice also avoids confusion with $n$-graded symplectic manifolds
as defined and studied in [17].

To highlight the main constructions of this subsection, given any symmetric
double vector bundle $({{D}},\sigma)$ one associates with it a sympletical
double vector bundle $(D_{B}^{*},\langle,\rangle,\sigma^{{{v}}})$, where the
pairing is given by

$\displaystyle\langle\Psi_{1},\Psi_{2}\rangle:=\Psi_{1}(\sigma(d))-\Psi_{2}(d),$

where $\Psi_{1},\Psi_{2}\in D_{B}^{*}$ project to the same point in $C^{*}$.
The point $d\in D$ is any point that projects to $(a_{1},b_{2})\in
A\times_{M}B$, we define $b_{2}:=\pi_{B}^{D_{B}^{*}}(\Psi_{2})\in B$ and
$a_{1}:=\sigma(\pi_{B}^{D_{B}^{*}}(\Psi_{1}))\in A$. This association is
functorial, moreover we have the following theorem.

######  Theorem 2.13.

The category of symplectical double vector bundles is equivalent to the
category of symmetric double vector bundles, and hence is also equivalent to
the category of graded bundles of degree $2$.

We precede the proof with a technical lemma.

######  Lemma 2.14.

Assume
$R\colon{{D}}^{\prime}=(D^{\prime};A^{\prime},B^{\prime};M^{\prime})\dashrightarrow{{D}}=(D;A,B;M)$
is a DVB relation, i.e., $R\subset D^{\prime}\times D$ is a double vector
subbundle. Then
$R^{{v}}\colon{{D}}^{{v}}\dashrightarrow{{{D}}^{\prime}}^{{v}}$, where

$\displaystyle
R^{{v}}:=\big{\\{}(\Phi,\Phi^{\prime})\in{D}^{*}_{A}\times{D^{\prime}}^{*}_{A^{\prime}}:\Phi(d)=\Phi^{\prime}(d^{\prime})\
\text{for all}\ d,d^{\prime}\in D\ \text{such that}\ (d^{\prime},d)\in
R\big{\\}},$

is a DVB relation, as well.

###### Proof.

It amounts to check that $R^{{v}}$ is a vector subbundle of the product of
vector bundles $D^{*}_{A}\rightarrow C^{*}$ and
${D^{\prime}}^{*}_{A^{\prime}}\rightarrow{C^{\prime}}^{*}$. Due to [18,
Theorem 2.3], since $R^{{v}}$ is a submanifold of
$D^{*}_{A}\times{D^{\prime}}^{*}_{A^{\prime}}$ it is enough to show that

$\displaystyle(\lambda\cdot_{C^{*}}\Phi,\lambda\cdot_{{C^{\prime}}^{*}}\Phi^{\prime})\in
R^{{v}}\qquad\text{for any}\quad(\Phi,\Phi^{\prime})\in
R^{{v}},\quad\text{and}\quad\lambda\in{\mathbb{R}}.$

We have
$(\lambda\cdot_{C^{*}}\Phi)(\lambda\cdot_{B}d)=\lambda\cdot\Phi(d)=\lambda\cdot\Phi^{\prime}(d^{\prime})=(\lambda\cdot_{{C^{\prime}}^{*}}\Phi^{\prime})(\lambda\cdot_{B^{\prime}}d^{\prime})$,
hence the result follows as $(d^{\prime},d)\in R$ if and only
$(\lambda\cdot_{B^{\prime}}d^{\prime},\lambda\cdot_{B}d)\in R$. ∎

###### Proof of Theorem 2.13.

Assume $({{D}},\sigma)$ is a symmetric DVB. Then $\sigma^{{v}}$ is DVB
isomorphism and, as we noticed earlier, it defines a bilinear form on
$E\rightarrow C^{*}$, where $E=D^{*}_{B}$. We shall show first that it is
skew-symmetric. Take $\Psi_{1},\Psi_{2}\in D_{B}^{*}$ that project to the same
$\alpha\in C^{*}$. Denote $b_{j}=\pi^{D_{B}^{*}}_{B}(\Psi_{j})\in B$, $j=1,2$,
and set $a_{j}=\sigma(b_{j})\in A$. Thus,

$\displaystyle\alpha(c)=\Psi_{1}(d_{1}{+}_{B}c)-\Psi_{1}(d_{1})=\Psi_{2}(d_{2}{+}_{B}c)-\Psi_{2}(d_{2})$

for any $c\in C_{m}$, $m=\pi^{B}_{M}(b_{1})=\pi^{B}_{M}(b_{1})\in M$, and any
$d_{j}\in D$ such that $\pi^{D}_{B}(d_{j})=b_{j}$, $j=1,2$. The bilinear form
induced by $\sigma$ is given by

$\displaystyle\langle\Psi_{1},\Psi_{2}\rangle_{E}=\langle\langle\sigma^{V}(\Psi_{1}),\Psi_{2}\rangle\rangle=\sigma^{{v}}(\Psi_{1})(d)-\Psi_{2}(d)=\Psi_{1}(\sigma(d))-\Psi_{2}(d),$
(2.15)

where $d$ is any element of $D$ that projects to $(a_{1},b_{2})$ under the
projection $D\rightarrow A\times_{M}B$, and
$\langle\langle\cdot,\cdot\rangle\rangle$ is the pairing (2.14). We have

$\displaystyle\langle\Psi_{2},\Psi_{1}\rangle=\Psi_{2}(\sigma(d^{\prime}))-\Psi_{1}(d^{\prime}),$

where now $d^{\prime}\in D$ is arbitrary that projects to $(a_{2},b_{1})$, so
we may take $d^{\prime}=\sigma(d)$ due to (2.8) and get

$\displaystyle\langle\Psi_{2},\Psi_{1}\rangle=\Psi_{2}(\sigma(\sigma(d)))-\Psi_{1}(\sigma(d))=-\langle\Psi_{1},\Psi_{2}\rangle,$
(2.16)

so $\langle\cdot,\cdot\rangle_{E}$ is skew-symmetric, as we claimed.

Assume now that
$\phi\colon({{D}}^{\prime},\sigma^{\prime})\rightarrow({{D}},\sigma)$ is a
morphism in $\textnormal{{SymmVB}}[2]$. The graph of $\phi^{{v}}$ is not only
a vector subbundle of $D_{B}^{*}\times{D^{\prime}}_{B^{\prime}}^{*}$ but is
also a double vector subbundle of ${{D}}^{{v}}\times{{{D}}^{\prime}}^{{v}}$.
In particular, it is a vector subbundle of the product of the vector bundles
$E\rightarrow C^{*}$ and $E^{\prime}\rightarrow C^{{}^{\prime}*}$. We need to
show that it is an isotropic subbundle. Assume
$(\Psi_{j},\Psi_{j}^{\prime})\in\operatorname{graph}\phi^{{v}}$, $j=1,2$, that
project to $(b_{j},b_{j}^{\prime})\in B\times B^{\prime}$ are in the same
fiber over $C^{*}\times{C^{\prime}}^{*}$. Then we can calculate

$\displaystyle\langle(\Psi_{1},\Psi_{1}^{\prime}),(\Psi_{2},\Psi_{2}^{\prime})\rangle_{E\times\bar{E^{\prime}}}=\langle\Psi_{1},\Psi_{2}\rangle_{E}-\langle\Psi^{\prime}_{1},\Psi^{\prime}_{2}\rangle_{E}^{\prime}$
$\displaystyle\hphantom{\langle(\Psi_{1},\Psi_{1}^{\prime}),(\Psi_{2},\Psi_{2}^{\prime})\rangle_{E\times\bar{E^{\prime}}}}{}=\Psi_{1}(\sigma(d))-\Psi_{2}(d)-\Psi^{\prime}_{1}(\sigma^{\prime}(d^{\prime}))+\Psi^{\prime}_{2}(d^{\prime})$
$\displaystyle\hphantom{\langle(\Psi_{1},\Psi_{1}^{\prime}),(\Psi_{2},\Psi_{2}^{\prime})\rangle_{E\times\bar{E^{\prime}}}}{}=\Psi_{1}(\sigma(\phi(d^{\prime})))-\Psi^{\prime}_{1}(\sigma^{\prime}(d^{\prime}))+\Psi^{\prime}_{2}(d^{\prime})-\Psi_{2}(\phi(d^{\prime}))=0,$

as $\sigma\circ\phi=\phi\circ\sigma^{\prime}$ and
$\Psi_{j}^{\prime}=\Psi_{j}\circ\phi$.

Conversely, assume that the graph of $\phi^{{v}}$ is an isotropic subbundle.
Then, the calculation above shows that for any $d^{\prime}\in D^{\prime}$ and
$\Psi_{1}\in D^{*}_{B}$ we have

$\displaystyle\Psi_{1}(\sigma(\phi(d^{\prime})))=\Psi^{\prime}_{1}(\sigma^{\prime}(d^{\prime}))=\Psi_{1}(\phi(\sigma^{\prime}(d^{\prime}))),$

so $\sigma\circ\phi=\phi\circ\sigma^{\prime}$. Thus we have established a
functor $\textnormal{{SymmVB}}[2]\rightarrow\textnormal{{SpVB}}[2]$.

Given a symplectical DVB, say ${{E}}=(E;E_{01},E_{10};M)$, whose skew-
symmetric form is defined by a DVB morphism
$\beta\colon{{E}}\rightarrow{{E}}^{{h}}$, we come back to $({{D}},\sigma)$ in
an obvious way, by setting $\sigma:=(I^{-1}\circ\beta)^{V}$ which is a DVB
morphism ${{D}}\rightarrow{{D}}^{\textnormal{f}}$ satisfying (2.8) and (2.9).
Indeed, $\sigma\colon D_{A}\rightarrow D_{B}$ is characterized by the property

$\displaystyle\beta(\Phi(d))=\Phi(\sigma(d)),$ (2.17)

for any $\Phi\in D_{B}^{*}$ that projects to $b:=\pi^{D}_{B}(d)$. Since $\Phi$
and $\beta(\Phi)$ lies over the same functional $\alpha\in C^{*}$, we find
from (2.17) that $\alpha(c)=\alpha(\sigma(c))$, hence (2.9). To see (2.8),
take $\Psi_{1},\Psi_{2}$, so that

$\displaystyle\langle\Psi_{1},\Psi_{2}\rangle_{E}=\langle\langle\beta(\Psi_{1}),\Psi_{2}\rangle\rangle=\beta(\Psi_{1})(d)-\Psi_{2}(d),$

where $d\in D$ is chosen so that it projects to $(a_{1},b_{2})\in
A\times_{M}B$. By skew-symmetry of the bilinear form,
$\beta(\Psi_{1})(d)-\Psi_{2}(d)$ coincides with
$-\beta(\Psi_{2})(d^{\prime})+\Psi_{1}(d^{\prime})$, where now $d^{\prime}$
should project to $(a_{2},b_{1})$, so we may take $d^{\prime}:=\sigma(d)$.
Using (2.17), we may cancel $\Psi_{1}(d^{\prime})=\beta(\Psi_{1})(d)$ and get
$\Psi_{2}(d)=\beta(\Psi_{2})(d^{\prime})$, so
$\Psi_{2}(d)=\beta(\Psi_{2})(d^{\prime})=\Psi_{2}(d^{\prime\prime})$, where
$d^{\prime\prime}=\sigma(d^{\prime})=\sigma(\sigma(d))$. Since $\Psi_{2}$ is
arbitrary, we get (2.8), this completes our proof. ∎

######  Remark 2.15.

Note that the conditions (2.8) and (2.9) on $\sigma$ are symmetric with
respect to the legs $D\rightarrow A$ and $D\rightarrow B$, whereas the skew-
symmetric form on $E$ is not so. A question arise: what is a connection
between $\sigma^{{v}}$ and $\sigma^{{h}}$? The later defines a skew-symmetric
form on $E^{*}:={{D}}^{fH}=D^{*}_{A}$ which is the dual VB to $E$ over
$C^{*}$. This suggests the answer: the skew-metrics on $E$ and $E^{*}$ defined
by $\sigma^{{v}}$ and $\sigma^{{h}}$, respectively, define the same
identification $E\simeq E^{*}$:

$\displaystyle(\Phi,\Psi)\in\sigma^{{v}}\Leftrightarrow(\Psi,\Phi)\in\sigma^{{h}}.$

Indeed, the former means that $\Phi(\sigma(d))=\Psi(d)$ for any $d\in D$,
while the latter is $\Psi(\sigma(d^{\prime}))=\Phi(d^{\prime})$ which is the
same as $\sigma(\sigma(d))=d$.

######  Remark 2.16.

Note that our observation provides a much simpler and shorter proof of the
equivalence of the categories of metric DVBs and the category of $N$-manifolds
of degree $2$ given in [22], as well. All we need to do is to consider,
instead of $\textnormal{{SymmVB}}[2]$, a similar category of DVBs $(D;A,B;M)$
equipped with a morphism $\sigma\colon{{D}}\rightarrow{{D}}^{\textnormal{f}}$
which satisfies (2.8) and $\sigma|_{C}=-\operatorname{id}_{C}$ instead of
(2.9). Then, the formula (2.15) changes to

$\displaystyle\langle\Psi_{1},\Psi_{2}\rangle_{E}=\langle\langle\sigma^{{{v}}}(\widetilde{\Psi_{1}}),\Psi_{2}\rangle\rangle=\sigma^{{v}}(\widetilde{\Psi_{1}})(d)-\Psi_{2}(d)=-\Psi_{1}(\sigma(d))-\Psi_{2}(d).$
(2.18)

Since $\sigma^{{v}}$ is minus the identity on $C^{*}$, we had to compose
$\sigma^{{v}}$ with an DVB automorphism of
${{D}}^{\textnormal{f}{{v}}}=(D^{*}_{B};B,C^{*};M)$ which is minus the
identity on the core. It is simply given by
$\Psi\mapsto\widetilde{\Psi}=\underset{C^{*}}{-}\Psi$, $\Psi\in D^{*}_{B}$. It
is straightforward to modify (2.16) to see that the form given by (2.18) is
symmetric and that DVB morphisms intertwining $\sigma$’s correspond to
isotropic subbundles. A local form of $\sigma$ satisfying (2.8) and
$\sigma|_{C}=-\operatorname{id}_{C}$ is $\sigma^{\ast}y^{a}_{01}=y^{a}_{10}$,
$\sigma^{\ast}y^{a}_{10}=y^{a}_{01}$,
$\sigma^{\ast}z^{i}=-z^{i}+y^{a}_{01}y^{b}_{10}\sigma^{i}_{ab}$ with
_symmetric_ $\sigma^{i}_{ab}$: $\sigma^{i}_{ab}=\sigma^{i}_{ba}$. Thus we
obtain an equivalence of categories of metric and skew-symmetric DVBs,
skewSymmVB[2], which differs from SymmVB[2] only in that we impose
$\sigma_{C}=-\operatorname{id}_{C}$ instead of (2.8). Then we can consider the
linearisations of $N$-manifolds in a similar manner as graded bundles,
allowing the side bundles to be purely odd and prove that the linearisations
of $N$-manifolds of degree $2$ are exactly skew-symmetric DVBs. We postpone
details until Section 3.

######  Example 2.17.

Consider a vector bundle $\tau\colon E\rightarrow M$ equipped with a skew-
symmetric form $\omega\colon E\times_{M}E\rightarrow{\mathbb{R}}$ and take its
tangent lift

$\displaystyle\omega_{{{T}}E}:=\operatorname{d}_{{{T}}}\omega\colon\
{{T}}E\times_{{{T}}M}{{T}}E\rightarrow{\mathbb{R}}.$

Then, ${{E}}=({{T}}E;E,{{T}}M;M)$ is a symplectical DVB. What is the
associated DVB morphism $\sigma$ which goes from
${{D}}=({{T}}^{*}E;E,E^{*};M)$ to its flip ${{D}}^{\textnormal{f}}$? The form
$\omega$ gives identifications

$\displaystyle\tilde{\omega}\colon\ E\rightarrow
E^{*},\qquad\omega^{\prime}\colon\ {{T}}^{*}E\rightarrow{{T}}^{*}E^{*}.$

It is a matter of simple calculation to see that $\sigma$ is obtained by
composing $\omega^{\prime}$ with the canonical diffeomorphism
$\tilde{\mathcal{R}}\colon{{T}}^{*}E^{*}\rightarrow{{T}}^{*}E$ (see [13, 27,
33]) composed with an automorphism of ${{T}}^{*}E$ which is minus the identity
on side bundle $E\rightarrow M$, and on the core bundle ${{T}}^{*}M$, and so
it is the identity on the side bundle $E^{*}\rightarrow M$, so that the
composition $\sigma=\tilde{R}\circ\omega^{\prime}$ is identity on the core
bundle. In standard local coordinates $(x^{a},y^{i},p_{a},p_{i})$ on
${{T}}^{*}E$, $\sigma\colon{{T}}^{*}E\rightarrow{{T}}^{*}E$ reads as

$\displaystyle\sigma^{*}(x^{a})=x^{a},\qquad\sigma^{*}(y^{j})=p_{i}\omega^{ij}(x),\qquad\sigma^{*}(p_{a})=p_{a}+y^{k}\omega_{kj}(x)\frac{\partial\omega^{ji}(x)}{\partial
x^{a}}p_{i},$ $\displaystyle\omega^{*}(p_{j})=y^{i}\omega_{ij}(x),$

where $(\omega_{ij})$ is the matrix of the form $\omega$ and $(\omega^{ij})$
is its inverse. Due to skew-symmetry of $\omega$ the morphism $\sigma$ really
squares to the identity. We may similarly take a symmetric form on $E$ instead
of $\omega$, define a metric DVB structure on ${{T}}E$, and find an analogues
presentation of the associated morphism $\sigma$.

###  2.4 Lie algebroid structures

A further observation here is that a symmetric DVB $({{D}},\sigma)$ gives rise
to a Lie algebroid structure on the leg $D\rightarrow A$. To see this, take a
graded bundle $F_{2}$ of degree $2$ and notice that the vertical subbundle
${{V}}F_{2}$ is a Lie sub-algebroid of the tangent Lie algebroid ${{T}}F_{2}$,
and it induces a Lie algebroid structure on ${\mathbf{L}}(F_{2})\rightarrow
F_{1}$. Indeed, by very definition of ${\mathbf{l}}(F_{2})$, a local section
of the later can be naturally identified with a section $X$ of
${{{V}}F_{2}}\rightarrow F_{2}$ which is invariant with respect to the
canonical action of the model vector bundle on the affine fibration
$\tau^{2}_{1}\colon F_{2}\rightarrow F_{1}$. In standard local coordinates
$\big{(}x^{A},y^{a},z^{i}\big{)}$ on $F_{2}$ it means that the vector field
$X$ has the form

$\displaystyle X=f^{a}(x,y)\partial_{y^{a}}+f^{i}(x,y)\partial_{z^{i}},$

i.e., the coefficient functions $f^{a}$, $f^{i}$ are constant on the fibers of
the fibration $\tau^{2}_{1}$. We see immediately that such vector fields are
closed with respect to the Lie bracket of vector fields. The anchor map
$\rho\colon{\mathbf{l}}(F_{2})\rightarrow{{T}}F_{1}$ is simply
${{T}}\tau^{2}_{1}|_{{\mathbf{l}}(F_{2})}$, so
$\rho(X)=f^{a}(x,y)\partial_{y^{a}}$. Staring with an arbitrary symmetric DVB
$({{D}},\sigma)$, we take as $F_{2}$ the diagonalisation of ${{D}}$ defined by
$\sigma$, and then transfer the obtained algebroid structure on the
linearisation of $F_{2}$ to the vector bundle $D\rightarrow A$ by means of the
isomorphism (2.11)
$I\colon({{D}},\sigma)\rightarrow{\mathbf{L}}(F_{2})\simeq{\mathbf{l}}(F_{2})$.
A simple calculation shows that

$\displaystyle[e_{a},e_{b}]=\sigma_{ab}^{i}(x)f_{i},\qquad[e_{a},f_{i}]=0=[f_{i},f_{j}],\qquad\rho(e_{a})=\partial_{y_{01}^{a}},\qquad\rho(f_{i})=0,$

where $(e_{a},f_{i})$ is a local basis of sections of $\pi^{D}_{A}\colon
D\rightarrow A$ dual to $(y_{10}^{a},z_{11}^{i})$, and we use the local
presentation of $\sigma$ as in (2.7). It follows that the linear Poisson
tensor on the dual bundle $D^{*}_{A}$, in standard fiber coordinates
$(p_{a}^{10},p_{i}^{11})$ dual to $(y_{10}^{a},z_{11}^{i})$, is given by

$\displaystyle\Lambda=\frac{1}{2}\sigma_{ab}^{i}p_{i}^{11}\partial_{p_{a}^{10}}\wedge\partial_{p_{b}^{10}}+\partial_{p_{a}^{10}}\wedge\partial_{y_{01}^{a}}.$
(2.19)

With respect to the standard gradation on the DVB $(D^{*}_{A};A,C^{*};M)$, the
weights of $p_{a}^{10}$, $p_{i}^{11}$ are equal $(1,1)$ and $(1,0)$,
respectively. Therefore, the Poisson tensor $\Lambda$ is homogeneous of bi-
weight $(-1,-2)$.

Thus, we are lead to the following.

######  Proposition 2.18.

Any symmetric double vector bundle $({{D}},\sigma)$ is canonically equipped
with the structure of a Lie algebroid on its leg $D\rightarrow A$ such that
the associated Poisson structure on the dual double vector bundle $D^{*}_{A}$
is homogeneous of bi-weight $(-1,-2)$.

By symmetry, we obtain a Lie algebroid on the leg $D\rightarrow B$, as well.
However, both structure of Lie algebroid on the legs of $D$ do not constitute
a structure of a double Lie algebroid on ${{D}}$ since simply they are of bi-
weights $(-1,-2)$ and $(-2,-1)$, not $(-1,-1)$. Note also, that we do not have
the structure of a weighted Lie algebroid (and so not a
$\mathcal{VB}$-algebroid) as defined in [6].

######  Proposition 2.19.

The Poisson structure on $D^{*}_{B}$ associated with a symmetric double vector
bundle $({{D}},\sigma)$ can be identified with the Poisson structure on the
dual of the linearisation ${\mathbf{l}}^{*}(F_{2})$, which itself is the
reduction of the canonical Poisson structure on ${{T}}^{*}F_{2}$.

###### Proof.

The dual of the linearisation ${\mathbf{l}}^{*}(F_{2})$ can be seen as the
reduction of the cotangent bundle ${{T}}^{*}F_{2}$ (cf. [6, Proposition 3.19])
and the latter, of course, comes with a canonical symplectic structure. Let us
employ local homogeneous coordinates on ${\mathbf{l}}^{*}(F_{2})$ naturally
inherited from canonical (Darboux) coordinates on ${{T}}^{*}F_{2}$

$\displaystyle\Big{(}\underbrace{x^{A}}_{(0,0)},\,\underbrace{y^{a,(1,0)}}_{(1,0)},\,\underbrace{\pi_{b}^{(0,1)}}_{(1,1)},\,\underbrace{\pi_{i}^{(1,1)}}_{(0,1)}\Big{)}.$

In these canonical coordinates, the Poisson structure inherited from the
reduction of the canonical Poisson structure is

$\displaystyle\Lambda_{{\mathbf{l}}}=\partial_{\pi_{a}^{(0,1)}}\wedge\partial_{y^{a,(1,0)}},$

which by inspection is of bi-weight $(-2,-1)$. One should also note that,
although we have defined the Poisson structure here in local coordinates, this
definition is invariant.

The isomorphism $I\colon{{D}}\rightarrow{\mathbf{l}}(F_{2})$ gives rise to a
dual isomorphism $\hat{I}\colon{\mathbf{l}}^{*}(F_{2})\rightarrow D^{*}_{B}$
which we write in local coordinates as

$\displaystyle\hat{I}^{*}y^{a}_{10}=y^{a,(1,0)},\qquad\hat{I}^{*}p_{a}^{01}=\pi_{a}^{(0,1)}+\frac{1}{2!}y^{b,(1,0)}\sigma_{ba}^{i}(x)\pi^{(1,1)}_{i},\qquad\hat{I}^{*}p_{i}^{11}=\pi_{i}^{(1,1)}.$

Thus we have

$\displaystyle\hat{I}_{*}\Lambda_{{\mathbf{l}}}=\frac{1}{2}\sigma_{ab}^{i}p_{i}^{11}\partial_{p_{a}^{01}}\wedge\partial_{p_{b}^{01}}+\partial_{p_{a}^{01}}\wedge\partial_{y_{10}^{a}},$

which, by comparison with (2.19), is the desired Poisson structure. ∎

Because we are dealing with symmetric DVBs, application of the flip functor
means that we can also interpret the Poisson structure on $D^{*}_{A}$ as being
inherited from the reduction of the canonical Poisson structure on
${{T}}^{*}F_{2}$.

###  2.5 The equivalence between graded bundles of degree $\boldsymbol{k}$
and symmetric $\boldsymbol{k}$-fold vector bundles

Before giving careful statements and proofs let us sketch the basic idea. From
[18] we know that any graded bundle $F_{k}$ can canonically be embedded in
${{T}}^{k}F_{k}$, and that ${{T}}^{k}F_{k}$ can be canonically embedded in
${{T}}^{(k)}F_{k}$. _Via_ Proposition 2.3 and Corollary 2.6, we see that the
full linearisation of a graded bundle ${\mathbf{L}}(F_{k})$ can be considered
as a substructure of the iterated tangent bundle ${{T}}^{(k)}F_{k}$. The
latter has a number of canonical diffeomorphisms that can be enumerated by
elements of the symmetric group $\mathbb{S}_{k}$. Thus the full linearisation
of a graded bundle of degree $k$ is equipped with a number of canonical
$k$-fold vector bundle morphisms
$\sigma_{g}\colon{\mathbf{L}}(F_{k})\rightarrow{\mathbf{L}}(F_{k})^{g}$, where
$g\in\mathbb{S}_{k}$, and for any $k$-fold graded bundle
${{D}}=(D;\Delta_{1},\ldots,\Delta_{k})$ we define

$\displaystyle{{D}}^{g}=\big{(}D;\Delta_{g(1)},\ldots,\Delta_{g(k)}\big{)}.$

Note that $({{D}}^{g_{1}})^{g_{2}}={{D}}^{g_{1}g_{2}}$. A quick glimpse at the
properties of $\sigma$’s lead us to the following definition of a category
$\textnormal{{SymmVB}}[k]$, which will turn out to be equivalent with the
category of graded bundles of degree $k$ (Theorem 2.24 below).

######  Definition 2.20.

Objects of the category $\textnormal{{SymmVB}}[k]$ are $k$-fold vector bundles
${{D}}=(D;\Delta_{1},\ldots$, $\Delta_{k})$ equipped with $k$-fold vector
bundle isomorphisms $\sigma_{g}\colon{{D}}\rightarrow{{D}}^{g}$, where
$g\in\mathbb{S}_{k}$, such that for each $g_{1},g_{2}\in\mathbb{S}_{k}$ the
composition

$\displaystyle{{D}}\xrightarrow{\sigma_{g_{2}}}{{D}}^{g_{2}}\xrightarrow{(\sigma_{g_{1}})^{g_{2}}}{{D}}^{g_{1}g_{2}}=({{D}}^{g_{1}})^{g_{2}}$
(2.20)

coincides with $\sigma_{g_{1}g_{2}}\colon{{D}}\rightarrow{{D}}^{g_{1}g_{2}}$
and

$\displaystyle\sigma_{(i,j)}|_{C_{ij}}=\operatorname{id}_{C_{ij}},$ (2.21)

for any $1\leq i<j\leq k$, where $C_{ij}$ denotes the core bundle of the DVB
$(D;\Delta_{i},\Delta_{j})$, and $(i,j)\in\mathbb{S}_{k}$ is the transposition
swapping $i$ and $j$. The above $(\sigma_{g_{1}})^{g_{2}}$ coincides with
$\sigma_{g_{1}}$ as a map $D\rightarrow D$ but we consider it here as a
$k$-fold vector bundle morphism ${{D}}^{g_{2}}\rightarrow{{D}}^{g_{1}g_{2}}$.
Note that the assignment ${{D}}\rightsquigarrow{{D}}^{g_{2}}$ is a functor
between the categories of $k$-fold vector bundles, and it takes the morphism
$\sigma_{g_{1}}$ to the morphism $(\sigma_{g_{1}})^{g_{2}}$. A morphism in
$\textnormal{{SymmVB}}[k]$ from $({{D}},(\sigma_{g})_{g\in\mathbb{S}_{k}})$ to
$({{D}}^{\prime},({\sigma^{\prime}}_{g})_{g\in\mathbb{S}_{k}})$ is a $k$-fold
vector bundle morphism $\phi\colon{{D}}\rightarrow{{D}}^{\prime}$ such that

$\displaystyle\sigma^{\prime}_{g}\circ\phi=\phi^{g}\circ\sigma_{g}\colon\
{{D}}\rightarrow{{{D}}^{\prime}}^{g}$ (2.22)

for any $g\in\mathbb{S}_{k}$.

######  Remark 2.21.

One can give a definition of symmetric $k$-fold vector bundle without setting
the order of Euler vector fields, and thus justifying the name “symmetric”.
This is done by setting $T=\\{t\colon\underline{k}\to W\\}$, where
$\underline{k}=\\{1,\ldots,k\\}$, $W=\\{\Delta_{1},\ldots,\Delta_{k}\\}$ and
considering the groupoid $\mathcal{G}\rightrightarrows T$ of all $k$-fold
isomorphisms ${{D}}^{t}\to{{D}}^{t^{\prime}}$, where $t,t^{\prime}\in T$ and
${{D}}^{t}=(D;\Delta_{t(1)},\ldots,\Delta_{t(k)})$. The symmetric group
$\mathbb{S}_{k}$ acts canonically on $T$ and also on $\mathcal{G}$. One can
easily verify that giving a groupoid morphism from the pair groupoid $T\times
T\rightrightarrows T$ to $\mathcal{G}\rightrightarrows T$, covering
$\operatorname{id}_{T}$, correspond with equipping ${{D}}$ with a family of
DVB morphisms $\sigma_{g}$ as in Definition 2.20 satisfying the condition
(2.20), but not necessary (2.21).

######  Definition 2.22.

Let $({{D}},(\sigma_{g})_{g\in\mathbb{S}_{k}})$ be a symmetric $k$-fold vector
bundle, its _diagonalisation_ $F({{D}})\subset D$ is the graded bundle of
degree $k$ constructed as the submanifold of $D$ consisting of all
$\mathbb{S}_{k}$ invariant elements and the weight vector field is given by
the total weight vector field $\Delta=\Delta_{1}+\cdots+\Delta_{k}$ restricted
to the said submanifold.

It will be shortly seen that $\Delta$ is indeed tangent to $F({{D}})$ and the
assignment ${{D}}\to F({{D}})$ is functorial. Moreover, the full linearisation
functor and the diagonalisation functor are mutual inverses.

######  Example 2.23.

The pair $({{T}}^{(k)}M,(\kappa_{g})_{g\in\mathbb{S}_{k}})$ is a symmetric
$k$-fold vector bundle whose diagonalisation is clearly ${{T}}^{k}M$.

######  Theorem 2.24.

The full linearisation and the diagonalisation set up an equivalence of
categories between $\textnormal{{GrB}}[k]$ and the category
$\textnormal{{SymmVB}}[k]$.

###### Proof.

Consider any graded coordinate system
$\big{(}x^{A},y^{i}_{\varepsilon}\big{)}$ on ${{D}}$, where $1\leq i\leq
N_{\varepsilon}$, and the weight of $y^{i}_{\varepsilon}$ is
$\varepsilon\in\\{0,1\\}^{k}$, $\varepsilon\neq 0^{k}$. Since $\sigma_{g}$ are
diffeomorphisms of $D$ that mixes coordinates of the same total weight, the
numbers $N_{\varepsilon}$, $N_{\varepsilon^{\prime}}$ of coordinates of
weights $\varepsilon$ and $\varepsilon^{\prime}$ coincides if
$|\varepsilon|=|\varepsilon^{\prime}|$. Let
$\prescript{g}{}{Y}^{i}_{\varepsilon}$ denote the same function on $D$ as
$y^{i}_{\varepsilon}$, but considered as a coordinate function on ${{D}}^{g}$,
hence its weight is
$\varepsilon.g=(\varepsilon_{g(1)},\ldots,\varepsilon_{g(k)})$
($(\varepsilon,g)\mapsto\varepsilon.g$ is a right action of $\mathbb{S}_{k}$
on $\\{0,1\\}^{k}$, and also on $\\{0,1\\}^{k}\setminus\\{0\\}^{k}$). Define

$\displaystyle
z^{i}_{\varepsilon}:=\frac{1}{k!}\sum_{g\in\mathbb{S}_{k}}\sigma^{*}_{g}\big{(}\prescript{g}{}{Y}^{i}_{\varepsilon.g^{-1}}\big{)}.$

Note that the weight of $\prescript{g}{}{Y}^{i}_{\varepsilon.g^{-1}}$ is
$\varepsilon.g^{-1}.g=\varepsilon$, so $z^{i}_{\varepsilon}$ has weight
$\varepsilon$ on ${{D}}$. Moreover, due to (2.21),

$\displaystyle\sigma_{g}^{*}\big{(}\prescript{g}{}{Y}^{i}_{\varepsilon}\big{)}=y^{i}_{\varepsilon.g}+\text{lower
terms}$

holds for any transposition $g=(i,j)$, and so for any $g\in\mathbb{S}_{k}$,
thanks to (2.20), where _lower terms_ means the sum of products of at least
two functions of non-zero weights that sum up to $\varepsilon.g$, hence they
have to be of lower weight than $\varepsilon.g$ with respect to the (partial)
product order on $\\{0,1\\}^{k}$. It follows that

$\displaystyle z^{i}_{\varepsilon}=y^{i}_{\varepsilon}+\sum_{{j\geq
2,\,i_{1},\ldots,i_{j},}\atop{\varepsilon^{1}+\cdots+\varepsilon^{j}=\varepsilon}}C^{i;\varepsilon^{1},\ldots,\varepsilon^{j}}_{i_{1},\ldots,i_{j}}y^{i_{1}}_{\varepsilon^{1}}\cdots
y^{i_{j}}_{\varepsilon^{j}},$

where the coefficients
$C^{i;\varepsilon^{1},\ldots,\varepsilon^{j}}_{i_{1},\ldots,i_{j}}$ are
functions of $\big{(}x^{A}\big{)}$, so
$\big{(}x^{A},z^{i}_{\varepsilon}\big{)}$ form a new graded coordinate system
for ${{D}}$. Denote the corresponding coordinates on ${{D}}^{g}$ by
$\prescript{g}{}{Z}^{i}_{\varepsilon}$. We shall show that

$\displaystyle\sigma_{h}^{*}\big{(}\prescript{h}{}{Z}^{i}_{\varepsilon}\big{)}=z^{i}_{\varepsilon.h}.$
(2.23)

Both sides of (2.23) are of weight $\varepsilon.h$, while abstracting from
weights of coordinates we get

$\displaystyle\sigma_{h}^{*}(z^{i}_{\varepsilon})=\frac{1}{k!}\sum_{g\in\mathbb{S}_{k}}\sigma_{h}^{*}\sigma_{g}^{*}\big{(}y^{i}_{\varepsilon.g^{-1}}\big{)}=\frac{1}{k!}\sum_{g\in\mathbb{S}_{k}}\sigma_{gh}^{*}\big{(}y^{i}_{\varepsilon.g^{-1}}\big{)}=z^{i}_{\varepsilon.h},$

what justifies our claim.

Thus the morphisms $\sigma_{h}\colon{{D}}\rightarrow{{D}}^{h}$ in new
coordinates look extremely simple. The graded bundle $F_{k}\subset D$
associated with $({{D}},(\sigma_{g})_{g\in\mathbb{S}_{k}})$ is given by

$\displaystyle F_{k}=\\{q\in D\colon\sigma_{g}(q)=q\ \text{for any}\
g\in\mathbb{S}_{k}\\}=\big{\\{}\big{(}z^{i}_{\varepsilon}\big{)}\colon
z^{i}_{\varepsilon}=z^{i}_{\varepsilon^{\prime}}\ \text{if}\
|\varepsilon|=|\varepsilon^{\prime}|\big{\\}},$

and thus we may define without ambiguity graded coordinates on $F_{k}$ by
$z^{i}_{w}:=z^{i}_{\varepsilon}|_{F_{k}}$, where $w=|\varepsilon|$. Therefore,
$F_{k}$ is indeed a graded bundle of degree $k$ and the diagonalisation is a
functor $\textnormal{{SymmVB}}[k]\to\textnormal{{GrB}}[k]$. Indeed, if
$\psi\colon{{D}}\to{{D}}^{\prime}$ is a morphism in
$\textnormal{{SymmVB}}[k]$, then for any $g\in\mathbb{S}_{k}$, and $q\in
F_{k}$ holds $\sigma_{g}(\psi(q))=\psi(\sigma_{g}(q))=\psi(q)$ and $\psi$
preserves the total weight given by $\Delta=\Delta_{1}+\cdots+\Delta_{k}$,
hence $\psi|_{F_{k}}$ is a morphism in $\textnormal{{GrB}}[k]$ between
diagonalisations of ${{D}}$ and ${{D}}^{\prime}$.

Consider the full linearisation ${\mathbf{L}}{F_{k}}\subset{{T}}^{(k)}F_{k}$
with coordinates, denoted by $y^{i,(\varepsilon)}$, inherited from
${{T}}^{(k)}F_{k}$. It is clear that
$I\colon{{D}}\rightarrow{\mathbf{L}}(F_{k})$, defined locally by

$\displaystyle I^{*}y^{i,(\varepsilon)}=z^{i}_{\varepsilon}$

is well defined globally and makes the following diagram commutative:

$\textstyle{{{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{g}}$$\scriptstyle{I}$$\textstyle{{{D}}^{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I^{g}}$$\textstyle{{\mathbf{L}}F_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{g}|_{{\mathbf{L}}F_{k}}}$$\textstyle{({\mathbf{L}}F_{k})^{g}.}$

We have described a natural one-to-one correspondence between objects in our
categories. Now we shall point the correspondence of morphisms between these
categories.

It is clear that a morphism of graded bundles $\phi\colon F_{k}\rightarrow
F_{k}^{\prime}$ gives rise to a morphism ${\mathbf{L}}{\phi}\colon$
${\mathbf{L}}(F_{k})\rightarrow{\mathbf{L}}(F_{k}^{\prime})$ satisfying
(2.22). Conversely, assume
$\psi\colon({{D}},(\sigma_{g}))\rightarrow({{D}}^{\prime},(\sigma^{\prime}_{g}))$
is a $k$-fold vector bundle morphism satisfying (2.22). It follows that we may
restrict $\psi$ to $F_{k}\subset D$ and get a graded bundle morphism
$\phi:=\psi|_{F_{k}}\colon F_{k}\rightarrow F_{k}^{\prime}$. Actually, the
morphism $\psi\colon{{D}}\rightarrow{{D}}^{\prime}$ is fully determined by its
restriction to $F_{k}$. To see this write a general form of $\psi$ in the
local coordinates $(z^{i}_{\varepsilon})$ defined above:

$\displaystyle\psi^{*}z^{i^{\prime}}_{\varepsilon}=\sum_{{j\geq
1;\,k_{1},\ldots,k_{j},}\atop{\varepsilon^{1}+\cdots+\varepsilon^{j}=\varepsilon}}Q^{i^{\prime};\varepsilon^{1},\ldots,\varepsilon^{j}}_{k_{1},\ldots,k_{j}}z^{k_{1}}_{\varepsilon^{1}}\cdots
z^{k_{j}}_{\varepsilon^{j}},\qquad
Q^{i^{\prime};\varepsilon^{1},\ldots,\varepsilon^{j}}_{k_{1},\ldots,k_{j}}\in{\mathbb{R}}.$

It follows from (2.22) that the latter expression has to coincide with

$\displaystyle\sum_{{j\geq
1;\,k_{1},\ldots,k_{j},}\atop{\varepsilon^{1}+\cdots+\varepsilon^{j}=\varepsilon}}Q^{i^{\prime};\varepsilon^{1},\ldots,\varepsilon^{j}}_{k_{1},\ldots,k_{j}}z^{k_{1}}_{\varepsilon^{1}.g}\cdots
z^{k_{j}}_{\varepsilon^{j}.g}$

for any $g\in\mathbb{S}_{k}$, so the coefficients $Q^{\ldots}_{\ldots}$ must
be symmetric in indices $k_{a},k_{b}$ if
$|\varepsilon^{a}|=|\varepsilon^{b}|$. Note that the formula for
$\psi|_{F_{k}}$ involves only symmetric part of $Q^{\ldots}_{\ldots}$. This
completes our proof. ∎

Diagrammatically, we have

$\textstyle{\textnormal{{GrB}}[k]}$$\textstyle{\textnormal{{SymmVB}}[k].}$full
linearisationdiagonalisation

######  Example 2.25.

Following Example 2.5 of the full linearisation of a graded bundle of degree
$3$, we see that the diagonalisation of ${\mathbf{L}}(F_{3})$, which is of
course just $F_{3}$ itself, can be described in natural coordinates as

$\displaystyle y^{a}:=y^{a,(00)}=y^{a,(10)}=y^{(a),(01)},$ $\displaystyle
z^{i}:=2!z^{i,(10)}=2!z^{i,(01)}=2!z^{i,(11)},\qquad
w^{\kappa}:=3!w^{\kappa,(11)},$

and by quick inspect it is clear that we recover the correct transformation
laws for the non-zero weight coordinates. Up to combinatorial factors, one
sets the coordinates of a given total weight equal: in a loose sense we look
at the ‘diagonal submanifold’ of the full linearisation. The generalisation of
this local diagonalisation to higher degree graded bundles is clear.

##  3 $\boldsymbol{N}$-manifolds and their linearisation

###  3.1 The linearisation of $\boldsymbol{N}$-manifolds

Given a supermanifold $\mathcal{M}$, consider its tangent bundle
${{T}}\mathcal{M}$. It is known that there exists a canonical involution
$\kappa_{\mathcal{M}}$ of ${{T}}{{T}}\mathcal{M}$, which in adapted
coordinates looks the same as the classical (purely even) case. Let us denote
local coordinates on $\mathcal{M}$ by $x^{A}$, the _parity_ of $x^{A}$ by
$\widetilde{x}^{A}=\tilde{A}\in{\mathbb{Z}}_{2}$. Then the adapted coordinates
$\big{(}x^{A},\dot{x}^{A}\big{)}$ on ${{T}}\mathcal{M}$ transform in the usual
way

$\displaystyle
x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad\dot{x}^{A^{\prime}}=\dot{x}^{B}\frac{\partial
x^{A^{\prime}}}{\partial x^{B}},$

where $\dot{x}^{A}$ has the same Grassmann parity as $x^{A}$, i.e.,
$\widetilde{\dot{x}}^{A}=\widetilde{x}^{A}=\tilde{A}$. Thus we can employ on
${{T}}{{T}}\mathcal{M}$ standard coordinates $\big{(}x^{A},\dot{x}^{A};\delta
x^{A},\delta\dot{x}^{A}\big{)}$ with additional transformation law of the form

$\displaystyle\delta{x}^{A^{\prime}}=\delta{x}^{B}\frac{\partial
x^{A^{\prime}}}{\partial
x^{B}},\qquad\delta\dot{x}^{A^{\prime}}=\delta{\dot{x}^{B}}\frac{\partial
x^{A^{\prime}}}{\partial x^{B}}+\delta
x^{C}\dot{x}^{B}\,\frac{\partial^{2}x^{A^{\prime}}}{\partial x^{B}\partial
x^{C}},$ (3.1)

hence the usual formula

$\displaystyle\kappa_{\mathcal{M}}^{*}\big{(}\dot{x}^{A}\big{)}=\delta
x^{A},\qquad\kappa_{\mathcal{M}}^{*}\big{(}\delta{x}^{A}\big{)}=\dot{x}^{A},\qquad\kappa_{\mathcal{M}}^{*}\big{(}\delta\dot{x}^{A}\big{)}=\delta\dot{x}^{A}$

works fine for supermanifolds.

Consider an $N$-manifold ${\cal F}$ (of degree $k$ with base $M$) (cf. [31,
32]) on which we employ coordinates $\big{(}x^{A},y^{a}_{w}\big{)}$ as usual,
so the parity of $y^{a}_{w}$ is $w\bmod\,2$ and $1\leq w\leq k$. Then, we take
the vertical subbundle ${{V}}{\cal F}$ of ${{T}}{\cal F}$ which we consider as
a (super) submanifold of ${{T}}{\cal F}$ defined locally by equations
$\dot{x}^{A}=0$, but we change the weight as in the ordinary case. Thus
$\dot{y}^{a}_{w}$ is of weight $w-1$ and the same parity as $y^{a}_{w}$. Then,
we may get rid of coordinates $y^{a}_{k}$ to obtain a local description of the
_linearisation_ ${\mathbf{l}}({\cal F})$ of $N$-manifold ${\cal F}$, exactly
as in the case of (even) graded bundle. As possible bi-weights of our
coordinates on ${\mathbf{l}}({\cal F})$ are $(w_{1},w_{2})$ with
$w_{1}\in\\{0,1\\}$, $0\leq w_{2}\leq k-1$, the linearisation
${\mathbf{l}}({\cal F})$ is a GrL-bundle but in the category of
supermanifolds. We may define the _full linearisation_ functor exactly as in
even case as ${\mathbf{L}}({\cal F}):={\mathbf{l}}^{(k-1)}({\cal F})$.

Consider the case $k=2$ and employ coordinates $\big{(}x^{A},\mathchar
28952\relax^{a},z^{i}\big{)}$ on ${\cal F}$ of degrees $0$, $1$, $2$,
respectively, with transformation law of the form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad\mathchar
28952\relax^{a^{\prime}}=\mathchar
28952\relax^{b}T_{b}^{\>\>\>a^{\prime}}(x),\qquad
z^{i^{\prime}}=z^{j}T_{j}^{\>\>\>i^{\prime}}(x)+\frac{1}{2}\mathchar
28952\relax^{a}\mathchar 28952\relax^{b}T_{ba}^{\>\>\>\>i^{\prime}}(x),$ (3.2)

where $T_{ba}^{i^{\prime}}=-T_{ab}^{i^{\prime}}$, as $\mathchar
28952\relax^{a}\mathchar 28952\relax^{b}=-\mathchar 28952\relax^{b}\mathchar
28952\relax^{a}$. The formula

$\displaystyle i^{*}\big{(}x^{A}\big{)}=x^{A},\qquad i^{*}(\dot{\mathchar
28952\relax}^{a})=\mathchar 28952\relax^{a},\qquad i^{*}(\delta\mathchar
28952\relax^{a})=\dot{\mathchar 28952\relax}^{a},\qquad
i^{*}(\delta\dot{z}^{i})=\dot{z}^{i},$ $\displaystyle
0=i^{*}\big{(}\dot{x}^{A}\big{)}=i^{*}(\delta
x^{A})=i^{*}\big{(}\delta\dot{x}^{A}\big{)}=i^{*}(\mathchar
28952\relax^{a})=i^{*}(\delta\dot{\mathchar
28952\relax}^{a})=i^{*}(z^{i})=i^{*}(\dot{z}^{i})=i^{*}(\delta z^{i}),$

defines a map $i\colon{\mathbf{l}}({\cal F})\rightarrow{{T}}{{T}}{\cal F}$.
Indeed, e.g.,
$i^{*}(\delta\dot{z}^{i^{\prime}})=\dot{z}^{i^{\prime}}=\dot{z}^{j}T_{j}^{\>\>\>i^{\prime}}(x)+\dot{\mathchar
28952\relax}^{a}\mathchar 28952\relax^{b}T_{ba}^{\>\>\>\>i^{\prime}}(x)$,
since using (3.1) we find that
$\delta\dot{z}^{i^{\prime}}=\delta\dot{z}^{j}\frac{\partial
z^{i^{\prime}}}{\partial z^{j}}+\delta\mathchar
28952\relax^{a}\,\dot{\mathchar
28952\relax}^{b}\frac{\partial^{2}z^{i^{\prime}}}{\partial\mathchar
28952\relax^{b}\partial\mathchar 28952\relax^{a}}$ plus other terms on which
$i^{*}$ vanishes, so $i^{*}$ takes $\delta\dot{z}^{i^{\prime}}$ to
$\dot{z}^{j}T_{j}^{\>\>\>i^{\prime}}(x)+\dot{\mathchar
28952\relax}^{a}\mathchar 28952\relax^{b}T_{ba}^{\>\>\>\>i^{\prime}}(x)$, the
latter coincides with $\dot{z}^{i^{\prime}}$, by (3.2). Thus we may obtain
${\mathbf{l}}({\cal F})$ as a substructure of ${{T}}{{T}}{\cal F}$.
Furthermore, the canonical involution on the double tangent bundle of a
supermanifold behaves exactly the same as in the classical case and hence
${\mathbf{l}}({\cal F})$ is equipped with a canonical diffeomorphism
$\sigma={\kappa_{{\cal F}}}|_{{\mathbf{l}}({\cal F})}$.

To describe $\sigma$ in terms of ordinary even differential geometry, we shall
consider the _parity reversion functor_ for double vector bundles (in the
category of supermanifolds) as defined in [37].

###  3.2 Parity reversion functor for double vector bundles

All the necessary notions from the definition of a double vector bundle carry
over to the supermanifold setup. Thus a double vector bundle $(D;A,B;M)$
admits local homogeneous coordinates
$\big{(}x^{A};u^{i},w^{\alpha};z^{\mu}\big{)}$ of weights $(0,0)$, $(0,1)$,
$(1,0)$, $(1,1)$, respectively, and of arbitrary parity. Within our notation,
$(u^{i})$, $(w^{\alpha})$ are pullbacks of linear fiber coordinates of the
side bundles $A\rightarrow M$, $B\rightarrow M$, respectively, and coordinate
changes are of the form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad
u^{i^{\prime}}=u^{i}T_{i}^{i^{\prime}},\qquad
w^{\alpha^{\prime}}=w^{\alpha}T_{\alpha}^{\alpha^{\prime}},\qquad
z^{\mu^{\prime}}=z^{\mu}T_{\mu}^{\mu^{\prime}}+u^{i}w^{\alpha}T_{\alpha
i}^{\mu^{\prime}}.$ (3.3)

We apply the parity reversion functor to the vector bundle projection
$D\rightarrow A$ in the first step and get a DVB denoted by $(\Pi_{1}D;A,\Pi
B;M)$. The index $1$ at $\Pi$ indicates that we take into account the first
vector bundle structure on $D$ and $\Pi B\rightarrow M$ is the bundle obtained
from the initial vector bundle $B\rightarrow M$ by changing parity to fiber
coordinates. (This operation in not possible for graded bundles of higher
degree.) Next we apply the parity reversion functor $\Pi_{2}$ to the second
vector bundle structure on the obtained DVB and get a DVB $(\Pi_{21}D;\Pi
A,\Pi B;M)$, where $\Pi_{21}=\Pi_{2}\circ\Pi_{1}$. Following explanation from
[37], we denote local coordinates on $(\Pi_{1}D;A,\Pi B;M)$ and
$(\Pi_{2}\Pi_{1}D;\Pi A,\Pi B;M)$ by
$\big{(}x^{A};u^{i},\eta^{\alpha};\theta^{\mu}\big{)}$ and
$\big{(}x^{A};\mathchar 28952\relax^{i},\eta^{\alpha};z^{\mu}\big{)}$,
respectively, and find that coordinate changes in $\Pi_{21}{{D}}$ have the
form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad\mathchar
28952\relax^{i^{\prime}}=\mathchar
28952\relax^{i}T_{i}^{i^{\prime}},\qquad\eta^{\alpha^{\prime}}=\eta^{\alpha}T_{\alpha}^{\alpha^{\prime}},\qquad
z^{\mu^{\prime}}=z^{\mu}T_{\mu}^{\mu^{\prime}}+(-1)^{\tilde{i}}\mathchar
28952\relax^{i}\eta^{\alpha}T_{\alpha i}^{\mu^{\prime}},$ (3.4)

where $\tilde{i}$ is the parity of $u^{i}$, not $\mathchar 28952\relax^{i}$.
Consider the square $S=\Pi_{21}\Pi_{21}$ of the functor $\Pi_{21}$. Although
all the building bundles of $D$ and $SD$ can be canonically identified, it
follows from (3.4) that coordinates changes on $S{{D}}$ have the form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad
u^{i^{\prime}}=u^{i}T_{i}^{i^{\prime}},\qquad
w^{\alpha^{\prime}}=w^{\alpha}T_{\alpha}^{\alpha^{\prime}},\qquad
z^{\mu^{\prime}}=z^{\mu}T_{\mu}^{\mu^{\prime}}-u^{i}w^{\alpha}T_{\alpha
i}^{\mu^{\prime}},$

resulting from the fact that the parity of $\mathchar 28952\relax^{i}$ is
$\tilde{i}+1\,\bmod\,2$, and $(-1)^{\tilde{i}}(-1)^{\tilde{i}+1}=-1$. Although
there is a canonical DVB isomorphism between $D$ and $SD$, namely given by

$\displaystyle I_{{D}}\colon\ {{D}}\rightarrow
S{{D}},\qquad\\!\\!I_{{{D}}}^{*}(x^{A})=x^{A},\qquad\\!\\!I_{{{D}}}^{*}(u^{i})=u^{i},\qquad\\!\\!I_{{{D}}}^{*}(w^{\alpha})=w^{\alpha},\qquad\\!\\!I_{{{D}}}^{*}(z^{\mu})=-z^{\mu},\\!\\!\\!$
(3.5)

and although there exist many (non-canonical) DVB isomorphisms between ${{D}}$
and $S{{D}}$ inducing the identities on all building vector bundles of ${{D}}$
(i.e., the core and on the side bundles), by referring to the theory of
_statomorphisms_ introduced in [20], there is not, in general, a _canonical_
double vector bundle isomorphism between ${{D}}$ and $S{{D}}$ which induces
the identities on all building vector bundles. This is in full analogy to DVB
isomorphisms between the cotangent bundles ${{T}}^{*}E$ and ${{T}}^{*}E^{*}$
of total spaces of a vector bundle $E\rightarrow M$ and its dual $E^{*}$.

######  Remark 3.1 ([37]).

The functors $\Pi_{12}$ and $\Pi_{21}$ are different in the sense presented in
[20], as they are linked by the functor $S$.

Another observation is that, in general, there is no canonical statomorphism
between $\Pi_{21}{{D}}^{\textnormal{f}}$ and
$(\Pi_{21}{{D}})^{\textnormal{f}}$, but the following holds.

######  Proposition 3.2.

There is a canonical statomorphism between $\Pi_{21}({{D}}^{\textnormal{f}})$
and $S(\Pi_{21}{{D}})^{\textnormal{f}}$.

###### Proof.

Let us write the transformation law (3.3) for coordinates $(z^{\mu})$ on
${{D}}^{\textnormal{f}}$ in the form

$\displaystyle
z^{\mu^{\prime}}=z^{\mu}T_{\mu}^{\mu^{\prime}}+w^{\alpha}u^{i}\hat{T}_{i\alpha}^{\mu^{\prime}},$

where
$\hat{T}_{i\alpha}^{\mu^{\prime}}=(-1)^{\tilde{i}\tilde{\alpha}}T_{\alpha
i}^{\mu}$. Then, according to (3.4), for coordinates $(z^{\mu})$ on
$\Pi_{21}{{D}}^{\textnormal{f}}$ we have

$\displaystyle
z^{\mu^{\prime}}=z^{\mu}T_{\mu}^{\mu^{\prime}}+(-1)^{\alpha}\eta^{\alpha}\mathchar
28952\relax^{i}\hat{T}_{i\alpha}^{\mu^{\prime}}.$

Directly from (3.4) we find that coordinates $(z^{\mu})$ on
$(\Pi_{21}{{D}})^{\textnormal{f}}$ transform as

$\displaystyle
z^{\mu^{\prime}}=z^{\mu}T_{\mu}^{\mu^{\prime}}+\epsilon\mathchar
28952\relax^{i}\eta^{\alpha}T_{\alpha i}^{\mu^{\prime}},$

where $\epsilon=(-1)^{\tilde{i}}(-1)^{(\tilde{i}+1)(\tilde{\alpha}+1)}$.
Hence, the functors $\Pi_{21}\circ\textnormal{f}$ and
$\textnormal{f}\circ\Pi_{21}$ differs by $S$, as
$(-1)^{\tilde{\alpha}}(-1)^{\tilde{i}\tilde{\alpha}}=-\epsilon$, and the
result follows. ∎

###  3.3 $\boldsymbol{N}$-manifolds of degree 2 and metric double vector
bundles

Apply the parity reversion functor $\Pi_{21}$ to the linearisation of an
$N$-manifold ${\cal F}$ of degree $2$ and the morphism
$\sigma\colon{\mathbf{L}}({\cal F})\rightarrow{\mathbf{L}}({\cal
F})^{\textnormal{f}}$ defined by $\kappa_{\cal F}\colon{{T}}{{T}}{\cal
F}\to{{T}}{{T}}{\cal F}$ and the inclusion ${\mathbf{L}}({\cal
F})\subset{{T}}{{T}}{\cal F}$. We shall obtain an ordinary (even) double
vector bundle ${{D}}=(D;A,B;M)=\Pi_{21}{\mathbf{L}}({\cal F})$ with isomorphic
side bundles $A\simeq B\simeq\Pi{\cal F}_{1}$ and a DVB morphism
$\Pi_{21}(\sigma)\colon{{D}}\rightarrow\Pi_{21}({\mathbf{L}}({\cal
F})^{\textnormal{f}})$ which is the identity on all the building bundles as
$\kappa_{\cal F}$ is so. To get a morphism
${{D}}\rightarrow{{D}}^{\textnormal{f}}$ we compose $\Pi_{21}(\sigma)$ with a
DVB isomorphism (3.5)

$\displaystyle\Pi_{21}\big{(}{\mathbf{L}}({\cal
F})^{\textnormal{f}}\big{)}\simeq S(\Pi_{21}{\mathbf{L}}({\cal
F}))^{\textnormal{f}}\xrightarrow{\eqref{eqn:iso_D_sD}}(\Pi_{21}{\mathbf{L}}({\cal
F}))^{\textnormal{f}}={{D}}^{\textnormal{f}}.$

Due to properties of above isomorphisms, the obtained DVB morphism
$\hat{\sigma}\colon{{D}}\rightarrow{{D}}^{\textnormal{f}}$ is minus the
identity on the core and the identity on the side bundles. A morphism
$\phi\colon{\cal F}\rightarrow{\cal F}^{\prime}$ between $N$-manifolds ${\cal
F}$ and ${\cal F}^{\prime}$ of degree $2$ is given locally by

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad\mathchar
28952\relax^{a^{\prime}}=\mathchar
28952\relax^{b}Q_{b}^{\>\>\>a^{\prime}}(x),\qquad
z^{i^{\prime}}=z^{j}Q_{j}^{\>\>\>i^{\prime}}(x)+\frac{1}{2}\mathchar
28952\relax^{a}\mathchar 28952\relax^{b}Q_{ba}^{\>\>\>\>i^{\prime}}(x),$

where $Q_{ba}^{\>\>\>\>i^{\prime}}=-Q_{ab}^{\>\>\>\>i^{\prime}}$, and it
results in a DVB morphism
$(\Pi_{21}{\mathbf{L}})(\phi):{{D}}\rightarrow{{D}}^{\prime}$ of the form

$\displaystyle x^{A^{\prime}}=x^{A^{\prime}}(x),\qquad
y^{a^{\prime}}_{01}=y^{b}_{01}Q_{b}^{\>\>\>a^{\prime}}(x),\qquad
y^{a^{\prime}}_{10}=y^{b}_{10}Q_{b}^{\>\>\>a^{\prime}}(x),$ $\displaystyle
z^{i^{\prime}}=z^{j}Q_{j}^{\>\>\>i^{\prime}}(x)-y^{b}_{01}y^{a}_{10}Q_{ab}^{\>\>\>\>i^{\prime}}(x),$

which obviously ‘respects’ additional structures $\hat{\sigma}$,
$\hat{\sigma^{\prime}}$. This lead us to introducing a category
$\textnormal{{skewSymmVB}}[2]$ of _skew-symmetric DVBs_ , whose objects are
ordinary double vector bundles ${{D}}$ equipped with a DVB morphism
$\sigma\colon{{D}}\rightarrow{{D}}$ with only such difference to the category
$\textnormal{{SymmVB}}[2]$ that (2.9) is replaced with
$\sigma_{C}=-\operatorname{id}_{C}$ where $C$ is the core of ${{D}}$. As we
have mentioned earlier in the paper, the category
$\textnormal{{skewSymmVB}}[2]$ just introduced is equivalent to the category
of metric DVBs (cf. [22]). Thus we have proved

######  Proposition 3.3.

The category of metric double vector bundles is equivalent to the category
$\textnormal{{skewSymm}}[2]$, and hence is also equivalent to the category of
$N$-manifolds of degree $2$.

##  4 Superisation of graded bundles

###  4.1 Using the parity shifted full linearisation

It is clear that we can use the total linearisation and the parity reversion
functor in each ‘direction’ to associate with a graded bundle an $N$-manifold
(cf. [17, 37]). Of course this is not a categorical equivalence in general,
but we do have a categorical equivalence when we restrict attention to the
subcategory of $N$-manifolds that arise as the superisation of symmetric
$k$-fold vector bundles.

######  Proposition 4.1.

Let $F_{k}$ be a graded bundle of degree $k$, $\Pi^{(k)}{\mathbf{L}}(F_{k})$
is a $N$-manifold of degree $k$, where
$\Pi^{(k)}:=\Pi_{k}\Pi_{k-1}\cdots\Pi_{1}$ is the total parity reversion
functor. Moreover, this association is functorial.

However, one has to be carefully applying parity reversion functors to
$k$-fold vector bundles. A choice in ordering the coordinates as they appear
in the transformation law or, equivalently, a choice which order to apply the
parity reversion functors is needed. Different choices here will result in
spurious minus sign differences at the level of coordinates and their
transformation rules. We will obtain different functors for each of these
choices, however they will be related via natural transformations. From our
perspective, the fact that one has to make choices and consider natural
transformations here is somewhat unpleasing. For a full discussion consult
[17, 37].

Note that if one restricts attention to vector bundles, then the subtleties
explained above do not appear. The standard notion of a superisation works
_perfectly_ for vector bundles, and this suggests that some more general
notion of a ‘superisation’ is needed for $k$-fold vector bundles.

###  4.2 Using $\boldsymbol{\mathbb{Z}_{2}^{k}}$-supermanifolds

As an alternative to the standard parity reversion functor, one can use the
full linearisation to define a $\mathbb{Z}_{2}^{k}$-supermanifold by declaring
the commutation rules of the coordinates to be defined by the scalar product
of the weights. The reader should consult [9, 10] for the locally ringed space
approach, noting that Molotkov [28] developed his notions in the categorical
framework of the functor of points. We denote the category of
$\mathbb{Z}_{2}^{k}$-supermanifolds with $\textnormal{{SMan}}[k]$.

From [9, Proposition 5.5] we know that the superisation of a $k$-fold vector
bundle, $(k>1)$ is a $\mathbb{Z}_{2}^{k}$-supermanifold. More specifically, in
local coordinates $(y^{a,(\varepsilon)}_{w})_{w=|\varepsilon|}$ on
${\mathbf{L}}(F_{k})$ one can define a $\mathbb{Z}_{2}^{k}$-supermanifold
using the sign rule

$\displaystyle
Y^{a,(\varepsilon)}_{w}Y^{b,(\delta)}_{v}=(-1)^{\langle\varepsilon,\delta\rangle}Y^{b,(\delta)}_{v}Y^{a,(\varepsilon)}_{w},$

where $\langle\varepsilon,\delta\rangle=\sum\mathchar 28962\relax_{i}\mathchar
28942\relax_{i}$ is the standard scalar product on $\mathbb{Z}_{2}^{k}$: to
each $y$ one assigns a corresponding $Y$ of the same multi-weight. The
_Grassmann parity_ of a given coordinate $Y$ is defined as the total weight of
the coordinate, but note that the parity does _not_ determine the commutation
rules.

Let us denote the $\mathbb{Z}_{2}^{k}$-supermanifold build from the full
linearisation of a graded bundle with $\Pi{\mathbf{L}}(F_{k})$. As the changes
of local coordinates on ${\mathbf{L}}(F_{k})$ respect the weight, this
superisation is well-defined. Furthermore, the changes of local coordinates on
$\Pi{\mathbf{L}}(F_{k})$ are exactly the same as for the full linearisation
itself. As we have assumed $F_{k}$ to be a manifold, there are no potential
minus signs that arise via any reordering, since the changes of coordinates
never involve coordinates that do not strictly commute. Evidently, the graded
commutation rules for the local coordinates do not effect the form of the
transformation laws and so the local gluings of coordinate charts. There are
no choices by hand here that manifest themselves as specious minus signs; we
can all agree on $\Pi{\mathbf{L}}(F_{k})$ without any further choices or
conventions.

If we start with a vector bundle in the category of smooth manifolds, say $E$,
then it is easy to see that the full linearisation acts as the identity
functor, i.e., ${\mathbf{L}}(E)\simeq E$. Thus the
$\mathbb{Z}_{2}^{k}$-superisation reduces to the standard
$\mathbb{Z}_{2}$-superisation via the parity reversion functor.

We summarise the main observations and results of this paper in the following
final theorem.

######  Theorem 4.2.

There exists a canonical superisation functor

$\displaystyle\Pi{\mathbf{L}}\colon\
\textnormal{{GrB}}[k]\longrightarrow\textnormal{{SMan}}[k],$

defined as the composition of the full linearisation functor and the canonical
$\mathbb{Z}_{2}^{k}$-superisation functor of $k$-fold vector bundles.

### Acknowledgements

The authors thank the anonymous referees whose comments and suggestions have
served to improve the presentation of this work. Research funded by the Polish
National Science Centre grant under the contract number
DEC-2012/06/A/ST1/00256.

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