# Frölicher structures, diffieties, and a formal KP hierarchy

Jean-Pierre Magnot  Univ. Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers,
France
and
Lycée Jeanne d’Arc
Avenue de Grande Bretagne,
63000 Clermont-Ferrand, France<EMAIL_ADDRESS>,  Enrique G. Reyes
Departamento de Matemática y Ciencia de la Computación, Universidad de
Santiago de Chile (USACH), Casilla 307 Correo 2, Santiago, Chile
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Vladimir Rubtsov  Univ.
Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France  volodya@univ-
angers.fr This paper is dedicated to Alexandre M. Vinogradov, whose
breakthrough insights are at the core of the concepts utilized herein

###### Abstract.

We propose a definition of a diffiety based on the theory of Frölicher
structures. As a consequence, we obtain a natural Vinogradov sequence and,
under the assumption of the existence of a suitable derivation, we can form on
it a Kadomtsev-Petviashvili hierarchy which is well-posed.

###### Key words and phrases:

Diffieties, Frölicher spaces, Kadomtsev-Petviashvili (KP) hierarchy,
Vinogradov sequence

###### 2020 Mathematics Subject Classification:

Research partially supported by the FONDECYT grant #1201894.

##  1\. Introduction

The Erlangen Program of F. Klein and works of S. Lie became a base of the
“Geometric Revolution” in the classical XIX century approach to differential
equations (DEs). The next stage of this revolution was manifested by
applications to DEs of powerful algebro-geometric and algebro-topological
methods developed by the French School (E. and H. Cartan, Ch. Ehresmann, J.
Leray, J.-P. Serre and of course A. Grothendieck). Other very similar or close
(differential-geometric) toolbox was developed in Japan by K. Kodaira and M.
Kuranishi and in US by D. Spencer, H. Goldschmidt, D. Quillen, V. Guillemin
and S. Sternberg.

A. M. Vinogradov (whose scientific start and first twenty years of work
coincided with the so-called “Golden Age” of the Moscow Mathematical School)
was probably the first person who unified successfully two sides of algebro-
geometric instrumentaria in his approach to the study of (non-)linear Partial
Differential Equations ((N)PDEs). He developed and combined ideas of
Grothendieck and the differential-geometric viewpoint of Ehresmann, and
introduced a category whose objects he called diffieties (differential
varieties) to study infinitely prolonged differential equations. He also
proposed a theory for the study of diffieties, what has became known as the
secondary calculus, see [45]. From a technical point of view, the main
ingredients of secondary calculus are an appropriate mixture of commutative
and homological algebra with differential geometry. The concept of a diffiety
plays the same role in the theory of PDEs that affine algebraic varieties do
in the theory of algebraic equations. Various natural characteristics of a
diffiety and, consequently, of the corresponding system of PDEs, are expressed
in terms of this calculus and vice versa.

A.M. Vinogradov himself says the following about the appearance of the notion
of a diffiety (see [46]):

> … it was understood that various natural differential operators and
> constructions that are necessary for the study of a system of PDEs of order
> $k$ do not live necessarily on the $k–$th order jet space, but involve jet
> spaces of any order. This is equivalent to say that a conceptually complete
> theory of PDEs is possible only on infinite order jet spaces. A logical
> consequence of this fact is that objects of the category of partial
> differential equations are diffieties, which duly formalize the vague idea
> of the “space of all solutions” of a PDE. Diffieties are a kind of infinite
> dimensional manifolds, and the specific differential calculus on them,
> called secondary calculus, is a native language to deal with PDEs and
> especially with NPDEs.

Roughly speaking, for each differential equation $\mathcal{E}$, the basic
geometric object $\mathcal{E}^{\infty}$, called “diffiety”, is introduced.
This is the infinitely prolonged equation of the PDE $\mathcal{E}$ or, the
space of (pointwise!) formal solutions of $\mathcal{E}$, and it is endowed
with the Cartan distribution $\mathcal{C}$, which is integrable. The dimension
of this distribution is equal to the number of independent variables of the
given PDE, while local solutions of the system correspond to “integral
submanifolds” of $\mathcal{C}$. This structure is induced by infinite
prolongations of PDE, and it is based on them. The vector fields on
$\mathcal{E}^{\infty}$ which preserve $\mathcal{C}$ are called
$\mathcal{C}-$fields, and the infinitesimal symmetries of $\mathcal{E}$ are
defined to be the equivalence classes of $\mathcal{C}-$fields modulo vector
fields that are tangent to $\mathcal{C}.$

From an algebraic point of view, diffieties are spectra of scalar differential
invariants (see examples in [45]). One can think about them as about infinite-
dimensional varieties enabled with a finite dimensional involutive Cartan
distribution. By means of this distribution, the de Rham algebra on a diffiety
can be used to define a spectral sequence (so-called Vinogradov’s
$\mathcal{C}-$ spectral sequence) that encodes information on the equation at
hand, see [1, 25, 45]. The $\mathcal{C}-$ spectral sequence corresponding to
the “empty” equation, (that is, $\mathcal{E}^{\infty}=\mathcal{J}^{\infty}$)
is the so-called variational bicomplex associated to the bundle $E$, see [1].

Let us give a more precise and computable definition of a diffiety. Let
$\mathcal{E}^{\infty}$ be the infinite prolongation of a differential equation
$\mathcal{E}$ (i.e., $\mathcal{E}$ is a submanifold of the $k-$th jet
fibration $\mathcal{J}^{k}(\pi)$ of a fibration $\pi:E\mapsto M$).

A diffiety
$\mathcal{O}=(\mathcal{O},C^{\infty}(\mathcal{O}),\mathcal{C}(\mathcal{O}))$
is a triple consisting of a manifold, a function field on it, and a finite-
dimensional distribution, that is locally of the form
$(\mathcal{E}^{\infty},C^{\infty}(\mathcal{E}^{\infty}),\mathcal{C}(\mathcal{E}))$,
where $\mathcal{C}(\mathcal{E}))$ is the restriction of the Cartan
distribution on $J^{\infty}E$ to $\mathcal{E}^{\infty}$. The dimension of
$\mathcal{C}(\mathcal{E})$ is called the dimension of $\mathcal{O}$. A smooth
map $F:\mathcal{O}_{1}\mapsto\mathcal{O}_{2}$ is called a morphism of
diffieties if

$dF_{x}({\mathcal{C}}_{x}({\mathcal{O}}_{1}))\subset{\mathcal{C}}_{F(x)}({\mathcal{O}}_{2}),\quad
x\in{\mathcal{O}}_{1}.$

We note at this point that the notion of a diffiety appears to be very useful
also in conceptual definitions of non-local symmetries for PDEs, see [24, 25].

In spite of its “universality”, “naturality” and “conceptuality” (the three
most popular adjectives used by A. M. Vinogradov) these important and useful
(and, usually) infinite-dimensional (in a standard sense) objects, may look
“rather esoteric” at a first glance (to quote Toru Tsujishita in his MR
reference to the paper [44], but in the op. cit. he immediately stresses that,
after the paper [44], this notion “now becomes open to everybody.”). We
believe that esoteric they are not: there are some not sufficiently known
curious links between the notion of a diffiety and other not-so-well-known
concepts like Souriau diffeology and Frölicher space structure. In fact, we
hope to show in this paper that Frölicher spaces are a very natural arena for
the development of the theory of diffieties. We can rephrase the definition
given above fully rigorously, and we can rather straightforwardly set up the
Vinogradov sequence.

Frölicher spaces were first described by A. Frölicher in a serie of works in
the 1980’s (see e.g. [18, 19]) and most results of this period are gathered in
his book with A. Kriegl [20]. The terminology “Frölicher space” can be found
for the first time, to our knowledge, in papers by Cherenack (see e.g. [10]),
and Kriegl and Michor confirmed that these spaces were due to A. Frölicher in
[26]. This setting is designed to obtain a safe differential calculus and a
safe differential geometry, and the presence of charts is not assumed.
Smoothness depends directly on tests on a space of functions and on a space of
paths, that are both assumed to be spaces of smooth maps, under mild
conditions for coherence. Refined in [29, 47] by means of the notion of
diffeology due to J.-M. Souriau (see e.g. [32]), the technical possibilities
in this setting have increased drastically these last few years.
Considerations on Frölicher spaces can be found in various contexts, see e.g.
[2, 3, 4, 5, 7, 8, 14, 29, 31, 32, 33, 40].

In this work, written for a proceedings volume in memory of the scientific
work of Alexandre Vinogradov, we intend to propose a formal description of the
interplay of Frölicher spaces and diffieties, as stated in the penultimate
paragraph. We are aware that our formulations may not fully fill all the
refinements on diffieties that have been produced till now (for example the
theory of non-local symmetries and coverings, see [24]), but this note intends
to be a primary base for adaptations and discussions. We organize our paper as
follows: Section 2 is an introduction to the theory of Frölicher spaces and
some basic geometric structures on them (tangent vectors, differential forms,
etc.); Section 3 is our proposal for a definition of a diffiety via Frölicher
spaces; Section 4 is both an application of the theory and a prelude to
further work (we provide some details in the following paragraph); section 5
contains a final remark.

On Section 4: we have explored the initial value problem of the Kadomtsev-
Petviashvili (KP) hierarchy and related non-linear KP equations in several
papers, see [16, 17, 30, 32, 34]. In particular, in [34] we have presented
several curious equations that correspond to “KP flows” posed on particular
algebras equipped with derivations, and in [35] we have applied our
constructions to algebras arising from the elementary diffiety
$(\mathcal{E}^{\infty},C^{\infty}(\mathcal{E}^{\infty}),\mathcal{C}(\mathcal{E}))$.
Can we apply them to general diffieties as defined in Section 3? In Section 4
we show that the answer is affirmative, provided that the diffiety is equipped
with a derivation.

##  2\. Preliminaries on Frölicher spaces

###  2.1. Frölicher spaces and their diffeology

######  Definition 2.1.

$\bullet$ A Frölicher space is a triple $(X,\mathcal{F},\mathcal{P})$ such
that

\- $\mathcal{P}$ is a set of paths $\mathbb{R}\rightarrow X$,

\- A function $f:X\rightarrow\mathbb{R}$ is in $\mathcal{F}$ if and only if
for any $c\in\mathcal{P}$, $f\circ c\in C^{\infty}(\mathbb{R},\mathbb{R})$;

\- A path $c:\mathbb{R}\rightarrow X$ is in $\mathcal{P}$ (i.e. is a contour)
if and only if for any $f\in\mathcal{F}$, $f\circ c\in
C^{\infty}(\mathbb{R},\mathbb{R})$.

$\bullet$ Let $(X,\mathcal{F},\mathcal{P})$ et
$(X^{\prime},\mathcal{F}^{\prime},\mathcal{P}^{\prime})$ be two Frölicher
spaces, a map $f:X\rightarrow X^{\prime}$ is differentiable (=smooth) if and
only if one of the following equivalent conditions is fulfilled:

  * •

$\mathcal{F}^{\prime}\circ f\circ\mathcal{P}\subset
C^{\infty}(\mathbb{R},\mathbb{R})$

  * •

$f\circ\mathcal{P}\subset\mathcal{P}^{\prime}$

  * •

$\mathcal{F}^{\prime}\circ f\subset\mathcal{F}$

Any family of maps $\mathcal{F}_{g}$ from $X$ to $\mathbb{R}$ generates a
Frölicher structure $(X,\mathcal{F},\mathcal{P})$, setting [26]:

\- $\mathcal{P}=\\{c:\mathbb{R}\rightarrow X\hbox{ such that
}\mathcal{F}_{g}\circ c\subset C^{\infty}(\mathbb{R},\mathbb{R})\\}$

\- $\mathcal{F}=\\{f:X\rightarrow\mathbb{R}\hbox{ such that
}f\circ\mathcal{P}\subset C^{\infty}(\mathbb{R},\mathbb{R})\\}.$

We easily see that $\mathcal{F}_{g}\subset\mathcal{F}$. This notion will be
useful in the sequel to describe in a simple way a Frölicher structure. A
Frölicher space carries a natural topology, the pull-back topology of
$\mathbb{R}$ via $\mathcal{F}$. In the case of a finite dimensional
differentiable manifold, the underlying topology of the Frölicher structure is
the same as the manifold topology. In the infinite dimensional case, these two
topologies differ, in general. We have to mention here that the space of
smooth functions $\mathcal{F}$ is a vector space, which has a natural
Frölicher structure called functional Frölicher structure.

Associated to the notion of Frölicher structure, we find the notion of
(reflexive) diffeology.

######  Definition 2.2.

Let $X$ be a set.

$\bullet$ A p-parametrization of dimension $p$ on $X$ is a map from an open
subset $O$ of $\mathbb{R}^{p}$ to $X$.

$\bullet$ A diffeology on $X$ is a set $\mathcal{D}$ of parametrizations on
$X$ such that:

  * •

For each $p\in\mathbb{N}$, any constant map $\mathbb{R}^{p}\rightarrow X$ is
in $\mathcal{D}$;

  * •

For each arbitrary set of indexes $I$ and family $\\{f_{i}:O_{i}\rightarrow
X\\}_{i\in I}$ of compatible maps that extend to a map $f:\bigcup_{i\in
I}O_{i}\rightarrow X$, if $\\{f_{i}:O_{i}\rightarrow X\\}_{i\in
I}\subset\mathcal{D}$, then $f\in\mathcal{P}$.

  * •

For each $f\in\mathcal{D}$, $f:O\subset\mathbb{R}^{p}\rightarrow X$, and
$g:O^{\prime}\subset\mathbb{R}^{q}\rightarrow O$, in which $g$ is a smooth map
(in the usual sense) from an open set $O^{\prime}\subset\mathbb{R}^{q}$ to
$O$, we have $f\circ g\in\mathcal{P}$.

If $\mathcal{D}$ is a diffeology on $X$, then $(X,\mathcal{D})$ is called a
diffeological space and, if $(X,\mathcal{D})$ and
$(X^{\prime},\mathcal{D}^{\prime})$ are two diffeological spaces, a map
$f:X\rightarrow X^{\prime}$ is smooth if and only if
$f\circ\mathcal{D}\subset\mathcal{D}^{\prime}$.

The notion of a diffeological space is due to J.M. Souriau, see [41]; a
previous but not fully equivalent notion is due to Chen, see [9] and [42] for
a very careful comparison. A comprehensive exposition of basic concepts can be
found in [23]. The category of diffeological spaces is very large, and it
contains many different pathological examples even if it enables a very easy-
to-use framework for infinite dimensional objects. Therefore, the category of
Frölicher spaces as a subcategory of this category (see [29, 47, 42, 33]) with
less technical problems may be useful. The first steps of the comparison were
published in [29]; the reader can also see [30, 32, 47] for extended
expositions. In particular, it is explained in [32] that Diffeological,
Frölicher and Gateaux smoothness are the same notion if we restrict ourselves
to a Fréchet context, in a sense that we explain here. For this, we first need
to analyze how we generate a Frölicher or a diffeological space, that is, how
we implement a Frölicher or a diffeological structure on a given set $X.$

If $(X,\mathcal{F},\mathcal{C})$ is a Frölicher space, we define a natural
diffeology on $X$ by using the following family of maps $f$ defined on open
domains $D(f)$ of Euclidean spaces, see [29]:

$\mathcal{D}_{\infty}(\mathcal{F})=\coprod_{p\in\mathbb{N}^{*}}\\{\,f:D(f)\rightarrow
X\,|\,D(f)\hbox{ is open in }\mathbb{R}^{p}\hbox{ and }\mathcal{F}\circ f\in
C^{\infty}(D(f),\mathbb{R})\\}\;.$

Now we can easily show the following:

######  Proposition 2.1.

[29] Let$(X,\mathcal{F},\mathcal{P})$ and
$(X^{\prime},\mathcal{F}^{\prime},\mathcal{P}^{\prime})$ be two Frölicher
spaces. A map $f:X\rightarrow X^{\prime}$ is smooth in the sense of Frölicher
if and only if it is smooth for the underlying diffeologies
$\mathcal{D}_{\infty}(\mathcal{F})$ and
$\mathcal{D}_{\infty}(\mathcal{F}^{\prime}).$

Thus, Proposition 2.1 and the foregoing remarks imply that the following
implications hold:

smooth manifold | $\Rightarrow$ | Frölicher space | $\Rightarrow$ | diffeological space
---|---|---|---|---

The reader is referred to the Ph.D. thesis [47] for a deeper analysis of these
implications. A more complete and up-to-date exposition, based on infinite
dimensional examples, is actually under preparation in [22], but let us make
some short precisions for the reader. We follow [18, Section 3] closely: as a
consequence of Boman’s theorem, see [26], the triple
$(\mathbb{R}^{n},C^{\infty}(\mathbb{R}^{n},\mathbb{R}),C^{\infty}(\mathbb{R},\mathbb{R}^{n}))$
is a Frölicher structure on $\mathbb{R}^{n}$. More generally, if $V$ is a
finite-dimensional paracompact smooth manifold,
$(V,C^{\infty}(V,\mathbb{R}),C^{\infty}(\mathbb{R},V))$ is a Frölicher
structure on $V$ and the smooth (in the standard sense) maps between two such
manifolds are precisely the smooth maps in the Frölicher sense. We can replace
$\mathbb{R}^{n}$ for a Fréchet space $E$ in our first example, and we can
replace $V$ for a Fréchet manifold $V^{\prime}$ in our second example,
provided that $V^{\prime}$ is a paracompact topological space and it is
modelled on a Fréchet space $E^{\prime}$ that satisfies the following
property: for each neighbourhood $U$ of $0\in E^{\prime}$ there exists a
smooth function $f:U\rightarrow\mathbb{R}$ such that $f(0)=1$ and $f(x)=0$ for
$x\not\in U$.

###  2.2. Tangent spaces, differential forms, diffeomorphisms and all that

On Frölicher spaces, we can define internal tangent cones ${}^{i}T_{x}X,$ see
e.g. [30]. We mention that, diffeologically, internal tangent spaces are
defined in a different way, see e.g. [11], but the approach that we present
here is

  * •

coherent with the definition of the kinematic tangent space of a manifold in
the $c^{\infty}$ category studied in [26]

  * •

and chosen as an interesting definition for tangent space in an applied
setting [21, 30, 32].

For each $x\in X,$ we consider

$C_{x}=\\{c\in C^{\infty}(\mathbb{R},X)|c(0)=x\\}$

and take the equivalence relation $\mathcal{R}$ given by

$c\mathcal{R}c^{\prime}\Leftrightarrow\forall f\in
C^{\infty}(X,\mathbb{R}),\quad\partial_{t}(f\circ
c)|_{t=0}=\partial_{t}(f\circ c^{\prime})|_{t=0}\;.$

The internal tangent cone at $x$ is the quotient

${}^{i}T_{x}X=C_{x}/\mathcal{R}.$

If $X=\partial_{t}c(t)|_{t=0}\in{}^{i}T_{X},$ we define the simplified
notation

$Df(X)=\partial_{t}(f\circ c)|_{t=0}\;.$

Under these constructions, we can define the total space of internal tangent
cones

${}^{i}TX=\coprod_{x\in X}{}^{i}T_{x}X$

with canonical projection $\pi:u\in{}^{i}T_{x}X\mapsto x.$ A Frölicher
structure can be defined on it. We note that in general, the fibration $\pi$
on ${}^{i}TX$ may have non-isomorphic fibers that are all cones but not vector
spaces (an example appears in [33]).

Differential forms on $X$ and a de Rham differential also exist in this
setting. It is easier to understand their construction under the diffeology
viewpoint:

######  Definition 2.3.

[41] Let $(X,\mathcal{D})$ be a diffeological space and let $V$ be a vector
space equipped with a differentiable structure. A $V-$valued $n-$differential
form $\alpha$ on $X$ (noted $\alpha\in\Omega^{n}(X,V))$ is a map

$\alpha:\\{p:O_{p}\rightarrow
X\\}\in\mathcal{D}\mapsto\alpha_{p}\in\Omega^{n}(O_{p};V)$

such that

$\bullet$ Let $x\in X.$ $\forall p,p^{\prime}\in\mathcal{D}$ such that $x\in
Im(p)\cap Im(p^{\prime})$, the forms $\alpha_{p}$ and $\alpha_{p^{\prime}}$
are of the same order $n.$

$\bullet$ Moreover, let $y\in O_{p}$ and $y^{\prime}\in O_{p^{\prime}}.$ If
$(X_{1},...,X_{n})$ are $n$ germs of paths in $Im(p)\cap Im(p^{\prime}),$ if
there exists two systems of $n-$vectors $(Y_{1},...,Y_{n})\in(T_{y}O_{p})^{n}$
and
$(Y^{\prime}_{1},...,Y^{\prime}_{n})\in(T_{y^{\prime}}O_{p^{\prime}})^{n},$ if
$p_{*}(Y_{1},...,Y_{n})=p^{\prime}_{*}(Y^{\prime}_{1},...,Y^{\prime}_{n})=(X_{1},...,X_{n}),$

$\alpha_{p}(Y_{1},...,Y_{n})=\alpha_{p^{\prime}}(Y^{\prime}_{1},...,Y^{\prime}_{n}).$

We note by

$\Omega(X;V)=\oplus_{n\in\mathbb{N}}\Omega^{n}(X,V)$

the set of $V-$valued differential forms.

We feel we need to make two remarks:

$\bullet$ If do not exist $n$ linearly independent vectors $(Y_{1},...,Y_{n})$
defined as in the last point of the definition, $\alpha_{p}=0$ at $y.$

$\bullet$ Let $(\alpha,p,p^{\prime})\in\Omega(X,V)\times\mathcal{D}^{2}.$ If
there exists $g\in C^{\infty}(D(p);D(p^{\prime}))$ (in the usual sense) such
that $p^{\prime}\circ g=p,$ then $\alpha_{p}=g^{*}\alpha_{p^{\prime}}.$

######  Proposition 2.2.

The set $\mathcal{P}(\Omega^{n}(X,V))$ made of maps $q:x\mapsto\alpha(x)$ from
an open subset $O_{q}$ of a finite dimensional vector space to
$\Omega^{n}(X,V)$ such that for each $p\in\mathcal{P},$

$\\{x\mapsto\alpha_{p}(x)\\}\in C^{\infty}(O_{q},\Omega^{n}(O_{p},V)),$

is a diffeology on $\Omega^{n}(X,V).$

Working with plots of the diffeology, we can define the wedge product and the
exterior differential of differential forms, which are (diffeologically)
smooth and have the same properties as in the standard case.

We need to introduce one last construction. Again following [33], we let
$Diff(X)$ be the group of diffeomorphisms of the Frölicher space $X$. This is
a well-defined object, it can be equipped with a diffeology, and its internal
tangent space at the identity is a diffeological vector space that defines
(see [33, Section 2]) a restricted tangent space on $X$ called diff-tangent
space and noted by ${}^{d}TX$. The space ${}^{d}TX$ is a Frölicher subspace of
${}^{i}TX$ and it is also a Frölicher vector bundle. Hereafter we assume that
$Diff(X)$ is a Frölicher Lie group, that is, the tangent space at the identity
has a natural diffeology for which the Lie bracket is smooth. This condition
is not automatically fulfilled, and the study [28], completed in the paper
[27], shows that this happens only on a “restricted” class of examples.
However it is only through this setting that one can, to our knowledge, define
a Lie bracket on vector fields that are identified as elements of the Lie
algebra of $Diff(X).$

##  3\. Diffieties and their Frölicher structure

We recall from [45, p. 7, 191] and Section 1, that a diffiety is, classically,
a (finite or infinite dimensional) manifold ${\mathcal{O}}$ that is locally of
the form $\mathcal{E}^{\infty}$, where $\mathcal{E}^{\infty}$ denotes an
infinitely prolonged equation that fibers over an $n$-dimensional manifold of
“independent variables”. We refer the reader to [25] for information on
equation manifolds. This definition means, in particular, that ${\mathcal{O}}$
must be equipped with an $n$-dimensional involutive distribution that
coincides (locally) with the Cartan distribution of $\mathcal{E}^{\infty}$.

We now propose a very general definition of a diffiety that does not use
topology explicitly. Let $\mathcal{O}$ be a set, and let
$(\mathcal{O},\mathcal{F}({\mathcal{O}}),\mathcal{P}(\mathcal{O}))$ be a
Frölicher structure. Since $\mathcal{P}(\mathcal{O})$ is generated by
$\mathcal{F}(\mathcal{O}),$ we remark that it is redundant to mention it.

######  Definition 3.1.

We consider a Frölicher space
$(\mathcal{O},\mathcal{F}({\mathcal{O}}),\mathcal{P}(\mathcal{O}))$. A Cartan
distribution $\mathcal{C}(\mathcal{O})$ on $\mathcal{O}$ is a finite-
dimensional vector subbundle of ${}^{d}TX$ that is involutive, that is, there
is a Lie subalgebra $\mathcal{H}$ of the Lie algebra of $Diff(\mathcal{O})$
such that for all $p\in\mathcal{O}$,

$\mathcal{C}(\mathcal{O})_{p}=\\{X(p):X\in\mathcal{H}\\}\;.$

A diffiety is a triple
$(\mathcal{O},\mathcal{F}({\mathcal{O}}),\mathcal{P}(\mathcal{O}))$.

This definition includes the particular case of
$\mathcal{O}=\mathcal{E}^{\infty}$ if all prolongations of $\mathcal{E}$
satisfy natural conditions as in the standard case of [1, 25]. Indeed, let us
assume that $\mathcal{E}^{\infty}$ is contained in the infinite jet bundle
$J^{\infty}(E)$ for some fiber bundle $E\rightarrow M$. The bundle
$J^{\infty}(E)$ is the inverse limit of the sequence of finite-dimensional jet
bundles $J^{k}(E)$, $k\geq 0$, and we have observed that these manifolds have
natural Frölicher structures. Now, the category of Frölicher spaces and smooth
maps is closed under inverse limits (see e.g., [20]), and therefore we obtain
that $J^{\infty}(E)$ is a Frölicher space. Since
$\mathcal{E}^{\infty}\subseteq J^{\infty}(E)$, we conclude that
$\mathcal{E}^{\infty}$ has a natural Frölicher structure. It follows from the
standard geometry reviewed for example in [1, 25], that functions over
$\mathcal{E}^{\infty}$ that are Frölicher-smooth are precisely the standard
smooth functions on $\mathcal{E}^{\infty}$. Thus, the classical example of a
diffiety (what Vinogradov calls an elementary diffiety in [45, P. 191]) is a
diffiety in our sense.

We now show that within this framework, we can build the Vinogradov sequence
along the lines of [43, 45]: the algebra of differential forms

$\Omega^{*}(\mathcal{O})=\sum_{k\in\mathbb{N}}\Omega^{*}(\mathcal{O})$

is a graded differential algebra for the de Rham differential operator. Let us
define

$C\Omega^{k}(\mathcal{O})=\left\\{\alpha\in\Omega^{k}\,|\,\alpha|_{\mathcal{C}(\mathcal{O})}=0\right\\},$
$\mathcal{C}\Omega(\mathcal{O})=\sum_{k\in\mathbb{N}}\mathcal{C}\Omega^{k}(\mathcal{O})$

and

$\mathcal{C}^{l}\Omega(\mathcal{O})=\mathcal{C}\Omega(\mathcal{O})^{\wedge
l}.$

Since $\mathcal{C}(\mathcal{O})$ is involutive, $\mathcal{C}(\mathcal{O})$ is
a differential ideal, and so is $\mathcal{C}^{l}(\mathcal{O})$ for
$l\in\mathbb{N}^{*}.$ Therefore we can define

$E^{p,q}_{0}=\frac{\mathcal{C}^{p}\Omega^{p+q}(\mathcal{O})}{\mathcal{C}^{p+1}\Omega^{p+q}(\mathcal{O})},$
$E^{p,q}_{r+1}=H^{*}(E^{p,q}_{r})$

and

$\mathcal{C}E(\mathcal{O})=\left\\{E_{r}^{p,q}\right\\}$

equipped with the restriction/extension of the de Rham differential to the
corresponding spaces. This is the spectral sequence we sought.

##  4\. KP hierarchy on a diffiety

In this section we consider the KP hierarchy. As explained in Section 1, we
present this section as an advance of [35], where we study differential
equations posed on elementary diffieties. Here, for the sake of brevity we
simply assume that there is a non-trivial derivation $D$ on
$\mathcal{F}(\mathcal{O})$ that is smooth (in [35] we use explicit
derivations). We denote by $A$ any Frölicher subalgebra of
$\mathcal{F}(\mathcal{O)}$ such that $D\,A\subset A.$

Let $\xi$ be a formal variable not in $A$. The algebra of symbols over $A$ is
the vector space

$\Psi_{\xi}(A)=\left\\{P_{\xi}=\sum_{\nu\in{\bf
Z}}a_{\nu}\,\xi^{\nu}:a_{\nu}\in A\,,\;a_{\nu}=0\mbox{ for }\nu\gg
0\right\\}\;$

equipped with the associative multiplication $\circ\,$, where

(4.1) $P_{\xi}\circ Q_{\xi}=\sum_{k\geq
0}\frac{1}{k!}\,\frac{\partial^{k}P_{\xi}}{\partial\xi^{k}}\,D^{k}Q_{\xi}\;,$

with the prescription that the multiplication on the right hand side is the
standard multiplication of Laurent series in $\xi$ with coefficients in $A$.
The algebra $A$ is included in $\Psi_{\xi}(A)$.

The algebra of formal pseudo-differential operators over $A$ is the vector
space

$\Psi(A)=\left\\{P=\sum_{\nu\in\mathbb{Z}}a_{\nu}\,D^{\nu}:a_{\nu}\in
A\,,\;a_{\nu}=0\mbox{ for }\nu\gg 0\right\\}\;,$

in which a multiplication $\circ$ is defined so that the map

$\sum_{\nu\in\mathbb{Z}}a_{\nu}\,\xi^{\nu}\;\mapsto\;\sum_{\nu\in\mathbb{Z}}a_{\nu}\,D^{\nu}$

from $\Psi_{\xi}(A)$ to $\Psi(A)$ is an algebra homomorphism. The algebra
$\Psi(A)$ is associative; it becomes a Lie algebra over $K$ if we define, as
usual,

(4.2) $[P,Q]=P\circ Q-Q\circ P\;.$

The order of $P\neq 0\in\Psi(A)$ is $N$ if $a_{N}\neq 0$ and $a_{\nu}=0$ for
all $\nu>N$, and order($P$) = $-\infty$ if $P=0$. If $P$ is of order $N$, the
coefficient $a_{N}$ is called the leading term of $P$.

We collect some properties of $\Psi(A)$ for completeness.

######  Lemma 4.1.

  1. (1)

If $P$ and $Q$ are formal pseudo–differential operators over $A$, and the
leading terms of $P$, $Q$ are not divisors of zero in $A$, then ${\rm
order}(P\circ Q)={\rm order}(P)+{\rm order}(Q)$, and ${\rm
order}([P,Q])\leq{\rm order}(P)+{\rm order}(Q)-1$. In particular, if $A$ is an
integral domain, the ring $\Psi(A)$ does not have zero divisors.

  2. (2)

Every non–zero formal pseudo–differential operator $P$ for which its leading
term is invertible has an inverse in $\Psi(A)$.

The following observation is at the basis of all our analysis: the Lie algebra
$\Psi(A)$ admits a left $A$–module direct sum decomposition

(4.3) $\Psi(A)={\mathcal{I}}_{A}\oplus{\mathcal{D}}_{A}\;,$

in which ${\mathcal{D}}_{A}$ is the Lie subalgebra of all differential
operators of order greater or equal to zero, and ${\mathcal{I}}_{A}$ is the
Lie subalgebra of integral operators, that is, the set of all formal pseudo-
differential operators in $\Psi(A)$ of order at most $-1$. It is known since
[37, 38, 39] that there exist formal Lie groups $G(\Psi(A))$,
$G_{+}({\mathcal{D}}_{A})$, and $G_{-}({\mathcal{I}}_{A})$ with Lie algebras
$\Psi(A)$, $\mathcal{D}_{A}$, and $\mathcal{I}_{A}$ respectively, such that

(4.4) $G(\Psi(A))=G_{-}({\mathcal{I}}_{A})\cdot G_{+}({\mathcal{D}}_{A})\,.$

In order to explain what these groups are, we need to specialize the algebra
$A$ considered in the previous subsection. Let $R$ be a fixed commutative
algebra over a field $K$ equipped with a derivation $D$. We take as $A$ the
ring of formal power series over $R$ in an infinite number of variables
$\tau_{1},\tau_{2},\cdots.$ Certainly, it is not necessary to make this
particular choice in the classical algebraic theory of KP (see for instance
[15, 36]) but, unless we equip $A$ with a further structure as in Demidov [12,
13] and specially [17], taking $A$ as a ring of power series is crucial for
the explicit construction of the groups $G(\Psi(A))$,
$G_{+}({\mathcal{D}}_{A})$ and $G_{-}({\mathcal{I}}_{A})$.

######  Remark 4.2.

For the benefit of the reader, we recall the construction of $A$ following
Bourbaki [6, p. 454-457]. We consider a countable set of indices $I,$ and we
take $T$ as the additive monoid of all sequences of natural numbers
$t=(n_{i})_{i\in I}$ such that $n_{i}=0$ except for a finite number of
indices. A formal power series is a function $u$ from $T$ to $R$,
$u=(u_{t})_{t\in T}$. Consider an infinite number of formal variables
$\tau_{i}$, $i\in I$ and set $\tau=(\tau_{1},\tau_{2},\cdots)$. Then, the
formal power series $u$ is also written as $u=\sum_{t\in T}u_{t}\tau^{t}$ in
which $\tau^{t}=\tau_{1}^{n_{1}}\tau_{2}^{n_{2}}\cdots$. We say that $u_{t}\in
R$ is a coefficient and that $u_{t}\tau^{t}$ is a term.
Operations on the ring $A$ are defined in an usual manner: If $u=(u_{t})_{t\in
T},$ $v=(v_{t})_{t\in T},$ then we set

$u+v=(u_{t}+v_{t})_{t\in T}\quad\mbox{ and }\quad uv=w,$

in which $w=(w_{t})_{t\in T}$ and

$w_{t}=\sum_{p,q\in T\atop{p+q=t}}u_{p}v_{q}.$

It is shown in [6, p. 455] that this multiplication is well defined, and that
$A$ equipped with these two operations is a commutative algebra with unit. The
derivation $D$ on $R$ extends to a derivation on $A$ via

$Du=\sum_{t\in T}(Du_{t})\tau^{t}.$

We also define the order of a power series. Let $u\in A,$ $u\neq 0.$ Let us
write $u=\sum_{t\in T}u_{t}\tau^{t}$, and if $t=(n_{i})_{i\in I}$, let us set
$\mid t\mid=\sum n_{i}$. The terms $u_{t}\tau^{t}$ such that $\mid t\mid=p$
are called terms of total degree $p.$ The formal power series $u_{p}$ whose
terms of total degree $p$ are those of $u,$ and whose other terms are zero, is
called the homogeneous part of $u$ of degree $p.$ The series $u_{0},$ the
homogeneous part of $u$ of degree $0$, is identified with an element of $R$
called the constant term of $u.$ For a formal series $u\neq 0,$ the least
integer $p\geq 0$ such that $u_{p}\neq 0$ is called the order of $u,$ and it
is denoted by $ord_{t}(u)$, and we extend this definition to the case $u=0$
setting $ord_{t}(0)=\infty$ (see [6, p. 457]). The following properties hold:
if $u,v$ are formal power series different from zero, then

$ord_{t}(u+v)\geq inf(ord_{t}(u),ord_{t}(v))\;,\quad\mbox{ if }u+v\neq 0,$
$ord_{t}(u+v)=inf(ord_{t}(u),ord_{t}(v))\;,\quad\mbox{ if }ord_{t}(u)\neq
ord_{t}(v),$ $ord_{t}(uv)\geq ord_{t}(u)+ord_{t}(v)\;,\quad\mbox{ if }uv\neq
0.$

Following Mulase, [39], we now define the space

(4.5)
$\widehat{\Psi}(A)=\left\\{P=\sum_{\alpha\in{\mathbb{Z}}}a_{\alpha}\,D^{\alpha}:a_{\alpha}\in
A\mbox{ and }\exists C\in\mathbb{R}^{+},N\in\mathbb{Z}^{+}\mbox{ so that
}ord_{t}(a_{\alpha})>C\,\alpha-N\ \forall\ \alpha\gg 0\right\\}$

and the subspace

(4.6)
$\widehat{\mathcal{D}}_{A}=\left\\{P=\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\,D^{\alpha}:P\in\widehat{\Psi}(A)\mbox{
and }a_{\alpha}=0\mbox{ for }\alpha<0\right\\}\;.$

The definition of order explained in the last remark implies that $A$ and
$\Psi(A)$ are contained in $\widehat{\Psi}(A)$. The operations on
$\widehat{\Psi}(A)$ are natural extensions of the operations on $\Psi(A)$. As
mentioned by Mulase in [39] and proved explicitly in [16, 17], the following
holds:

######  Lemma 4.3.

The space $\widehat{\Psi}(A)$ has a natural algebra structure, and
$\widehat{\mathcal{D}}_{A}$ is a subalgebra of $\widehat{\Psi}(A)$.

######  Definition 4.4.

Let $\mathcal{K}$ be the ideal of $A$ generated by $t_{1},t_{2},\cdots$. If
$P\in\widehat{\Psi}(A)$, we denote by $P|_{t=0}$ the equivalence class
$P\,mod\;\mathcal{K}$, and we identify it with an element of $\Psi(A)$. We
also set $G_{A}=1+\mathcal{I}_{A}$, and we define the spaces

$\widehat{\Psi}(A)^{\times}=\\{P\in\widehat{\Psi}(A):P|_{t=0}\in G_{A}\\}$

and

$\widehat{\mathcal{D}}_{A}^{\times}=\\{P\in\widehat{\mathcal{D}}_{A}:P|_{t=0}=1\\}\;.$

The most important fact about these two spaces is the following result proven
by Mulase in [39], and reviewed in full detail in [16] and [17].

######  Proposition 4.1.

The spaces $\widehat{\Psi}(A)^{\times}$ and
$\widehat{\mathcal{D}}_{A}^{\times}$ are formal Lie groups: each element $P$
in $\widehat{\Psi}(A)^{\times}$ and $\widehat{\mathcal{D}}_{A}^{\times}$ has
an inverse of the form

$P^{-1}=\sum_{n\geq 0}(1-P)^{n}\;.$

Moreover, we have: $G(\Psi(A))=\widehat{\Psi}(A)^{\times}$,
$G_{+}(\mathcal{D}(A))=\widehat{\mathcal{D}}_{A}^{\times}$, and
$G_{-}(\mathcal{I}_{A})=1+{\mathcal{I}}_{A}$.

######  Theorem 4.5.

For any $U\in G({\Psi}(A))$ there exist unique $W\in G_{-}(\mathcal{I}_{A})$
and $Y\in G_{+}({\mathcal{D}}_{A})$ such that

$U=W^{-1}\,Y\;.$

In other words, there exists a unique global factorization of the formal Lie
group $G({\Psi}(A))$ as a product group,

$G({\Psi}(A))=G_{-}(\mathcal{I}_{A})\,G_{+}({\mathcal{D}}_{A})\;.$

This theorem is proven in [39], see also the later papers [16, 17] for full
proofs.

The Kadomtsev-Petviashvili (KP) hierarchy on $\mathcal{O}$ reads

(4.7) $\frac{dL}{dt_{k}}=\left[(L^{k})_{+},L\right]\;,\quad\quad k\geq 1\;,$

with initial condition $L(0)=L_{0}\in\Psi^{1}(R)$. The dependent variable $L$
is chosen to be of the form

$L=D+\sum_{\alpha\leq-1}u_{\alpha}\,D^{\alpha}\in{\Psi}^{1}(A_{t})\;.$

Standard references on (4.7) are [15, 36, 37, 38, 39]. The following result
gives a solution to the Cauchy problem for the KP hierarchy (4.7).

######  Theorem 4.6.

Consider the KP hierarchy with initial condition $L(0)=L_{0}$.

  1. (1)

There exists a pair $(S,Y)\in
G_{-}(\mathcal{I}_{A})\times\widehat{\mathcal{D}}_{A}^{\times}$ such that the
unique solution to Equation $(\ref{lolo})$ with $L(0)=L_{0}$ is

$\displaystyle L(t)=Y\,L_{0}\,Y^{-1}=SL_{0}S^{-1}\;.$

  2. (2)

The pair $(S,Y)$ is uniquely determined by the decomposition problem

$exp\left(\sum_{k\in\mathbb{N}}t_{k}L_{0}^{k}\right)=S^{-1}Y.$

  3. (3)

The solution operator $L$ is smoothly dependent on the variable $t$ and on the
initial value $L_{0}.$ This means that the map

$(L_{0},s)\in(D+\Psi^{-1}(A))\times
T\mapsto\sum_{n\in\mathbb{N}}\left(\sum_{|t|=n}[L(s)]_{t}\right)\in(D+\Psi^{-1}(R))^{\mathbb{N}}$

is smooth, in which $s\in\cup_{n\in\mathbb{N}}\mathbb{K}^{n}$ and this set is
equipped with the structure of locally convex topological space given by the
inductive limit.

The existence of an algebraic decomposition as in Parts (1) and (2) of Theorem
4.6 appears already in Mulase’s seminal papers [37, 39], and a formal solution
to (4.7) as in Part (1) is in [16]. The richness of Theorem 4.6 steams from
the fact that we pose the KP equations (4.7) in the Frölicher algebra
$\Psi(A)$ using an analytically rigorous factorization in Frölicher groups of
the infinite-dimensional Frölicher group $G({\Psi}(A))$. For all details, we
refer to [17, 32].

##  5\. Outlook

As anticipated, we have a more concrete example of Frölicher structure on a
classical elementary diffiety in the work in progress [35]; in [35] we also
investigate analogues to the KP hierarchy and equations very explicitly. The
definition of a diffiety given here, plus Section 5, compels us to study non-
linear equations “of KP type” in full detail in this global setting. We hope
to do so in the near future.

## References

  * [1] Anderson, I.M.; Introduction to the variational bicomplex. _Contemporary Mathematics_ 132 51–73 (1992).
  * [2] Batubenge, A.; Ntumba, P.; On the way to Frölicher Lie groups Quaestionnes mathematicae 28 no1, 73–93 (2005)
  * [3] Batubenge, A.; Tshilombo, H. Topologies and smooth maps on initial and final objects in the category of Frölicher spaces. _Demonstr. Math._ 42, No. 3, 641-655 (2009).
  * [4] Batubenge, A.; A survey on Frölicher spaces. _Quaest. Math._ 38, no. 6, 869-884 (2015).
  * [5] Batubenge, A.; Tshilombo, H.; Finsler metric topology coincides with Frölicher topology. _Balkan J. Geom. Appl._ 22, no. 2, 1-12 (2017)
  * [6] Bourbaki, N.; Algebra I. Chapters 1–3. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998.
  * [7] Canarutto, D.; Frölicher-smooth geometries, quantum jet bundles and BRST symmetry. _J. Geom. Phys._ 88, 113–128 (2015).
  * [8] Canarutto, D.; _Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics_ Oxford University Press (2020)
  * [9] Chen, K.T.; _Iterated Path Integrals_. Bulletin AMS 83 , 831–879 (1977).
  * [10] Cherenack, P.; Applications of Frölicher spaces to cosmology. _Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math._ 41, 63-91 (1998).
  * [11] Christensen, J.D.; Wu, E.; Tangent spaces and tangent bundles for diffeological spaces; Cahiers de Topologie et Géométrie Différentielle, LVII, 3-50 (2016)
  * [12] Demidov, E. E., On the Kadomtsev-Petviashvili hierarchy with a noncommutative timespace, Funct. Anal. Appl., 29, 2, 131-133, (1995)
  * [13] Demidov, E. E., Noncommutative deformation of the Kadomtsev-Petviashvili hierarchy. In “Algebra. 5, Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI)”, Moscow, 1995. (Russian), J. Math. Sci. (New York), 88, 4, 520-536, (1998)
  * [14] Dugmore, B.; Ntumba, P. P. On tangent cones of Frölicher spaces. _Quaest. Math._ 30 no. 1, 67-83 (2007).
  * [15] Dickey, L.A.; Soliton equations and Hamiltonian systems, second edition (2003).
  * [16] Eslami Rad, A.; Reyes, E. G.; The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups J. Geom. Mech. 5 no 3 (2013), 345–363.
  * [17] Eslami Rad, A.; Magnot, J.-P.; Reyes, E. G.; The Cauchy problem of the Kadomtsev- Petviashvili hierarchy with arbitrary coefficient algebra. J. Nonlinear Math. Phys. 24:sup1 (2017), 103–120.
  * [18] Frölicher, A.; Smooth structures. in: _Category theory. Applications to algebra, logic and topology, Proc. int. Conf., Gummersbach 1981, Lect. Notes Math._ 962, 69-81 (1982).
  * [19] Frölicher, A.; Cartesian closed categories and analysis of smooth maps in: _Categories in continuum physics, Lect. Notes in Math._ 1174J 43–51 (1983)
  * [20] Frölicher, A; Kriegl, A; Linear spaces and differentiation theory Wiley series in Pure and Applied Mathematics, Wiley Interscience (1988)
  * [21] Goldammer, N. Welker, K.; Optimization in diffeological spaces. _Proc. Appl. Math. Mech._ 21 no S1 e202100260 (2021)
  * [22] Goldammer, N.; Magnot, J-P.; Welker, K.; On diffeologies in infinite dimensional geometry and shape analysis (preliminary title). In preparation.
  * [23] Iglesias-Zemmour, P.; Diffeology Mathematical Surveys and Monographs 185 (2013).
  * [24] Krasil’shchik, I.S.; Vinogradov, A.M.; Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws and Bäcklund transformations. Acta Appl. Math. 15 (1989), 161–209.
  * [25] I.S. Krasil’shchik, and A.M. Vinogradov (Eds.) _Symmetries and conservation laws for differential equations of mathematical physics_ Translations of Mathematical Monographs Vol. 182, AMS, Providence(1999).
  * [26] Kriegl, A.; Michor, P.W.; The convenient setting for global analysis Math. surveys and monographs 53, American Mathematical society, Providence, USA. (2000)
  * [27] Laubinger, M.; A Lie algebra for Frölicher groups Indag. Math. 21 no 3-4, 156–174 (2011)
  * [28] Leslie, J.; On a Diffeological Group Realization of certain Generalized symmetrizable Kac- Moody Lie Algebras J. Lie Theory 13 (2003), 427-442.
  * [29] Magnot, J-P.; Difféologie sur le fibré d’holonomie d’une connexion en dimension infinie C. R. Math. Acad. Sci., Soc. R. Can. 28, no. 4, 121-128 (2006)
  * [30] Magnot, J-P.; Ambrose-Singer theorem on diffeological bundles and complete integrability of KP equations. Int. J. Geom. Meth. Mod. Phys. 10, no 9 Article ID 1350043 (2013)
  * [31] Magnot, J-P.; Remarks on the geometry and the topology of the loop spaces $H^{s}(S^{1},N),$ for $s\leq 1/2.$ _Int. J. Maps Math._ 2 no. 1, 14–37 (2019).
  * [32] Magnot, J-P.; Reyes, E. G.; Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups. Ann. H. Poincaré 21, No. 6, 1893–1945 (2020).
  * [33] Magnot, J-P.; On the differential geometry of numerical schemes and weak solutions of functional equations. _Nonlinearity_ 33, No. 12, 6835-6867 (2020)
  * [34] Magnot,J.-P.; Reyes, E.G.; Roubtsov, V.N.; On $(t_{2},t_{3})$-Zakharov–Shabat equations of generalized Kadomtsev-Petviashvili hierarchies. _J. Math. Phys._ 63 no. 9, Paper No. 093501, 11 pp. (2022)
  * [35] Magnot, J-P.; Reyes, E.G., Rubtsov, V.N.; Infinite order structures on differential equations in preparation, 2022.
  * [36] Miwa, T.; Jimbo, M.; Date, E.; Solitons, Differential Equations, Symmetries and Infinite Dimensional Algebras (translated by Miles Reid) Cambridge University Press (2000)
  * [37] Mulase, M.; Complete integrability of the Kadomtsev-Petvishvili equation. Advances in Math. 54 57–66 (1984).
  * [38] Mulase, M.; Cohomological structure in soliton equations and Jacobian varieties. J. Diff. Geom. 19 403–430 (1984).
  * [39] Mulase, M.; Solvability of the super KP equation and a generalization of the Birkhoff decomposition. Invent. Math. 92 1–46 (1988).
  * [40] Ntumba, P.; Sikorski and Frölicher CW-complexes compared. _Demonstratio Math._ 38 , no. 1, 207–221 (2005).
  * [41] Souriau, J-M.; un algorithme générateur de structures quantiques Astérisque (hors série) 341-399 (1985)
  * [42] Stacey, A.; Comparative Smootheology. _Theory and Applications of Categories_ 25, no. 4, 64–117 (2011).
  * [43] Vinogradov, A.M.; The $\mathcal{C}-$spectral sequence, Lagrangian formalism and conservation laws _J. Math. Anal. Appl._ 100 1–129 (1984)
  * [44] Vinogradov, A. M.; Local symmetries and conservation laws. _Acta Appl. Math._ 2 , no. 1, 21–78 (1984).
  * [45] Vinogradov, A.M.; Cohomological Analysis of Partial Differential Equations and Secondary Calculus . Translations of Mathematical Monographs 204, AMS, Providence (2001).
  * [46] Vinogradov, A.M.; What are symmetries of PDEs and what are PDEs themselves? Sophus Lie and Felix Klein: the Erlangen program and its impact in mathematics and physics, 137–190, IRMA Lect. Math. Theor. Phys., 23, Eur. Math. Soc., Zürich, (2015).
  * [47] Watts, J.; Diffeologies, differentiable spaces and symplectic geometry PhD thesis, university of Toronto (2012) arXiv:1208.3634