text:
Gabriella Crespi
The style of the design is reminiscent of the designer and/or manufacturer
NMK-547264
Brass, Smoked Glass
Grey, gold
Born near Florence and trained at the Politecnico di Milano, Italian designer-architect Gabriella Crespi (1922-2017) attracted a cult following thanks to her penchant for creating geometric, sculptural forms imbued with an aura of glamour.
Crespi's career spanned from the 1950s to the 1980s. Although she often cited modernist pioneers like Frank Lloyd Wright and Le Corbusier as key influences, her designs for furniture, lighting, and decorative accessories were largely artisanally produced in fine materials—never intended for the mass market. Her her high end collections were the result of expert Italian artisans bringing her sketches and wax prototypes to life.
Among her more famous designs are her Z Table and Z Bar (1972-75); the Tiffany-like Fungo Lamp (1973); the block-shaped Menhir Cabinet (1978); the Yang Yin Desk (1979); and the sculpted Ara Table (1979). Notably, Crespi designed four Plurimo collections, the prototype of which was revealed at an exhibition in Dallas in 1968. The first collection, designed in collaboration with her daughter Elisabetta in 1970, comprised the Magic Cube, 2000, Dama, and Scultura. The second launched in 1976 and included the Ellisse and Cubo Tondo, and the third introduced Blow Up, Eclipse, and Sit & Sip in 1980. The final fourth edition of the Plurimo series was created in 1982, called Punto '83. This final Plurimo was a unique piece, designed specially for the exhibition "I Plurimi di Gabriella Crespi" at Milan's Museum of Science and Technology, organized by art critic Vanni Scheiwiller. It is no surprise therefore, that the Plurimo is often considered Crespi's best work as well as her most well known—the gilded coffee tables with a split tabletop that can be swiveled to serve drinks—are reminiscent of Willy Rizzo's designs: elegant, luxuriant, and conceived with entertainment in mind.
In 1987, Crespi abandoned her high-profile design career to embark on a spiritual journey, which took her to the Indian Himalayas to follow guru Shri Muniraji. She remained in India for almost two decades and documented her experiences in the book Ricerca di Infinito, published in 2007 upon her return to Europe.
In the last decade of her life, Crespi contributed to a number of fashion and design collaborations, such as a limited-edition jewelry collection with Stella McCartney for charity in 2008 and a retrospective exhibition with Sergio Rossi in 2013. In 2015, Galleria Rita Fancsaly in Milan presented New Bronze Age, a series of new limited-edition designs by Crespi.
In 2012, Crespi established an archive of her works, which is managed by her daughter Elisabetta. The Wave Desk—with its undulating curved form in signature Crespi gilt metal—was her last design. She passed away in Milan on February 14th, 2017.
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Mid-Century Modern Italian Chandelier from Stilux, 1950s
Mid-Century Italian Vase, 1960s
Art Deco French Desk Set, 1930s
Mid-Century Modern Blue Glass & Sterling Silver Vanity Box, 1940s
Vintage Italian Square Plexiglas Ice Bucket, 1970s
Italian Space Age Table Lamp, 1960s
Italian Purple Murano Glass Ashtray, 1970s
Mid-Century Modern Italian Coffee Table
Antique Dutch Oak and Brass Candelabra, Set of 2
Mid-Century Modern Italian Jazz Cocktail Set from Decor, 1950s
Italian White and Blue Murano Glass Wall Sconces from Mazzega, 1970s, Set of 2
Art Deco Blue Glass Vase
Vintage Molten Glass Pendant Light, 1960s
Mid-Century Brown & Amber Murano Glass Vase, 1970s
Red Ceramic Jug from Bertoncello, 1980s
Sommerso Murano Glass Bull Sculpture from Seguso, 1970s
Italian Art Deco Floor Lamp, 1930s
Space Age White Wall Sconces by Danilo & Corrado Aroldi for Stilnovo, 1970s, Set of 2
Italian Ceramic Mural Plates from Ceramiche Ariston, 1960s, Set of 2
Italian Brass and Smoked Glass Tableware by Gabriella Crespi, 1960s
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text:
Theory of Economics
Late Capitalism
byErnest Mandel
April 17, 1999|
Late Capitalism is the first major synthesis to have been produced by the contemporary revival of Marxist economics. It represents, in fact, the only systematic attempt so far ever made to combine the general theory of the "laws of motion" of the capitalist mode of production developed by Marx, with the concrete history of capitalism in the twentieth century.
Mandel's book starts with a challenging discussion of the appropriate methods for studying the capitalist economies. He seeks to show why the classical approaches of Luxemburg, Bukharin, Bauer and Grossman failed to accomplish the further development of Marxist theory whose urgency became evident after Marx's death. He then sketches the structure of the world market and the variant types of surplus-profit that have characterized its successive stages. On these foundations, Late Capitalism proceeds to advance an extremely bold schema of the "long waves" of expansion and contraction in the history of capitalism, from the Napoleonic Wars to the present. Mandel criticizes and refines Kondratieff's famous use of the notion.
Mandel's book surveys in turn the main economic characteristics of late capitalism as it has emerged in the contemporary period. The last expansionary long wave, it argues, started with the victory of fascism on the European continent and the advent of the war economies in the US and UK during the 1940s, and produced the record world boom of 1947-72. Mandel discusses the reasons why the dynamic upswing of growth in this period was bound to reach its limits at the turn of the 1970s, and why a long wave of economic stagnation and intensified class struggle has set in today.
Late Capitalism is a landmark in Marxist economic literature. Specifically designed to explain the international recession of the 1970s, it is a central guide to understanding the nature of the world economic crisis today.
Ernest Mandel (1923–95), historian, economist and activist, was a leading figure in the Fourth International from 1945 and was the author of a number of books, including Late Capitalism, Marxist Economic Theory, Long Waves of Capitalist Development, and The Meaning of the Second World War.
Title:Late Capitalism
Product dimensions:618 pages, 8.4 X 5.4 X 1.33 in
Shipping dimensions:618 pages, 8.4 X 5.4 X 1.33 in
Published:April 17, 1999
Publisher:Verso Books

redpajama_set_name: RedPajamaCommonCrawl


text:
Wearing her ghoonghat a few inches higher
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Image and text contributed by Sreenivasan Jain, Mumbai
Some text is paraphrased from a recent Book ? Civil Disobedience, Sreenivasan?s father Late. Shri LC Jain, noted economist and Gandhian.
Chameli Devi Jain and her husband Phool Chand shortly after they were married. Photographer unknown
This image was photographed in Delhi, shortly after my Paternal grandparents Chameli and Phool Chand, got married. She was 14 and he was 16. It was unusual for couples in our family to be photographed, especially holding hands, which turned out to be an indication of the unconventional direction their lives would take. They were both Gandhians and Freedom fighters.
The prestigious Chameli Devi Jain award for Journalists was named after my grandmother . The only visible reminder of her brush with radical politics of the freedom movement was the milky cornea in her right eye, the result of an infection picked up in Lahore Jail where she had spent 4 months in 1943. Otherwise, she was Ammaji: gentle, almost luminous in her white saris, regular with her samaik (Jain prayer), someone who would take great pleasure, on our Sunday visits, to feed us dal chawal (rice and lentils) mixed with her own hands.
My grandmother grew up in a village called Bahadarpur in Alwar, about 4 hours south of Delhi, in a deeply conservative Jain family. The family was locally influential; they were traders in cotton turbans, woven by local Muslim weavers and sold in Indore, Madhya Pradesh. They also were moneylenders. As with much of rural Rajasthan, the women were in purdah. Within two years of their marriage, their first child, my father, LC Jain was born.
Ammaji moved with my grandfather into the family home in the teeming bylanes of Dariba in Chandni Chowk. But he had developed a growing interest in Gandhi and the nationalist movement and soon broke away from the family business to join the Delhi Congress. In 1929, soon after the call for Poorn Swaraj at the Lahore session, he was arrested for the first time.
My grandfather?s stint in jail exposed him to even more radical politics. Along with his Congress membership, he also became part of the revolutionary Hindustan Socialist Republian Association which counted Bhagat Singh and Chandrashekhar Azad as amongst its members. (Azad, in an interview, acknowledged that he received his first revolver from my grandfather). He also became a reporter for the nationalist newspaper at the time, Vir Arjun, whose editor he had met in jail.
In 1932, Gandhi called for a major nationwide satyagraha against foreign goods. It was also the year a bomb was thrown at Lord Lothian, an act in which my grandfather played a role. When he told my grandmother that he was going to jail, she said this time she would go to prison first, by taking part in the swadeshi satyagraha. The household was stunned. Ammaji?s life had revolved around ritual, the kitchen and ghoonghat. Her decision led to the following heated exchange; witnessed by my father, age 7:
Babaji: ?You don?t know anything about jail.?
Ammaji: ?Nor did you when you were first arrested.?
Babaji: ?Who will look after the children ??
Ammaji: ?You will.?
Sensing that things were getting out of hand, my great grandmother, Badi Ammaji locked both of them into a room. But my grandfather apparently fashioned an escape from the window using knotted dhotis and Ammaji, head uncovered, marched with other women pouring out of their homes towards the main bazaar. The crowd had swelled into hundreds. There were cries of ?Mahatma Gandhi ki Jai?. As they began to move around picketing shops selling foreign goods, they were arrested, taken to Delhi Jail, and charged with 4 and half months of rigorous imprisonment.
Her arrest, not surprisingly, outraged the family in Alwar. Umrao Singhji, Ammaji?s father, came to Delhi and had a big argument with my great grandfather, accusing the in-laws of ?ruining our princess?. But Ammaji found an ally in her in-laws, who refused to pay her bail out of respect for her satyagraha. Umrao Singhji then tried to talk his daughter out of it when she was being transferred to Lahore Jail. ?Chameli, apologise, ask for pardon.? But Ammaji asked him not to worry. ?Bolo Bharat Mata ki Jai?, she said, as she was being led away in a rickshaw along with the other prisoners. ?Bharat Mata ki Jai?, responded her father.
She returned from Lahore 4 months later, a minor heroine. But there was also loss. Lakshmi, her daughter, 5 years old, fell from the balcony of the house and died when she was in Lahore jail. And there was the milky cornea ? the loss of an eye. But her world had somewhat widened. She wore her ghoonghat a few inches higher. She gave her Rajasthani ghaghra choli away, and wore only handspun. She spun on the charkha. She would attend meetings with other women on matters of community reform, like widow remarriage and also became more involved in the activities of the local sthanak, the Jain community?s prayer and meditation hall. She had, as it turns out, quietly fashioned her own blend of Jain renunciation and Gandhian abstinence.
In the years that followed, my grandfather retained his engagement with the freedom struggle. He would often go to sit in the family?s property agency in Model Town, but his real passion, which consumed most of his last 30 years was compiling a massive index of freedom fighters, a staggering 11 volume chronicle of the stories of countless ordinary men and women, who took part in protests, bomb conspiracies, went to jail, lived and died. For my grandmother, it was a gradual return to a more conventional domesticity.
But, that single action that morning in 1932 had opened up a world: a young woman from a deeply conservative family, who became the first Jain woman in her neighbourhood to go to jail, who was named on the day of her arrest in the Hindustan Times with all the other satyagrahis, who would return home to other freedoms, even if minor, like a ghoonghat that could be worn a few inches back.
And for that, she would, one day, have an award named after her.
Presented by Anusha S Yadav at the CHHAYA International Conference on Photography of India
Author Shahidul AlamPosted on February 10, 2012 Categories Colonialism, culture, Democracy, Governance, Human rights, Imperialism, Law, Occupation, Photography, politics, Resistance, South AsiaTags freedom, Gandhi, India, jail, Jain, NID, resistance, woman
One thought on "Wearing her ghoonghat a few inches higher"
Nighat Mir says:
What a huge bridge for her to cross!! Admirable.
Sub continental history is replete with such stories: Ordinary women doing extraordinary deeds. So many unsung, uncelebrated. Such as mothers telling their sons to die for the cause, e.g Bi Amma, Maulana Mohammed Ali and Maulana Shaukat Ali's mother.
I salute Chambelli Bai.
Nighat Mir
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text:
\subsection*{Introduction}
\noindent%
With this report we complement the brief communication~\cite{Kiselev:SQS11}
in which we related a class of noncommutative
geometries to associative algebras by
merging the approach of~\cite{Kiselev:KontsevichCyclic} with that
of~\cite{Kiselev:OlverSokolovCMP1998}. We now prove in full detail

the main message in~\cite{Kiselev:SQS11}:
the cyclic\/-\/invariance relation is sufficient for a
construction of differential calculus of variational multivectors over (non)\/commutative
jet spaces. In other words, we remove a much more restrictive hypothesis of the
($\mathbb{Z}_2$-\/graded) commutativity in the models
of mathematical physics.

We
formulate the definition of the
Schouten bracket (\cite{Kiselev:Nijenhuis} and~\cite{Kiselev:Schouten},
c.f.~\cite{Kiselev:CattaneoFelderCMP2000}) and establish its main properties
in a way that does not appeal to the commutativity in the rings of smooth functions
on the manifolds at hand. Instead, we consider the spaces of infinite jets
$J^\infty(M^n_{\text{\textup{nC}}}\to{{\EuScript A}})$ of mappings from (non)\/commutative (non)\/graded
ma\-ni\-folds~$M^n_{\text{\textup{nC}}}$ to the factors~${{\EuScript A}}$ of free associative $\Bbbk$-\/algebras
(e.g., $\Bbbk=\mathbb{R}$ or~$\mathbb{C}$)

over the relation of equivalence under cyclic permutations of letters in the words.
Such setup~\cite{Kiselev:SQS11} is a proper noncommutative generalization
of the standard jet bundle geometry for commutative manifolds~\cite{Kiselev:Olver};
this di\-rec\-ti\-on of research was pioneered in~\cite{Kiselev:OlverSokolovCMP1998}
for purely commutative domains~$M^n$ and for the targets~${{\EuScript A}}$ as specified above.
If, at the end of the day, the target algebra~${\EuScript A}$ is proclaimed
(graded-)\/commutative,
one restores the stan\-dard, Gel'fand\/--\/Dorfman calculus of variational
multivectors~\cite{Kiselev:Dorfman,Kiselev:GelfandDorfman1981}.
(We compare several definitions of the variational Schouten bracket in the
commutative setup and verify their equivalence in~\cite{Kiselev:RingersProtaras2012}.)
Alternatively, under the shrinking of
the source manifold~$M^n$
to a point (or by po\-stu\-la\-ting that the only admissible mappings~$M^n\to{{\EuScript A}}$ are
constants), %

we return to the formal noncommutative
geometry of~\cite{Kiselev:KontsevichCyclic}.

We adopt the conventions of~\cite{Kiselev:SQS11} and~\cite{Kiselev:Praha11}
and follow
the notation in the papers~\cite{Kiselev:SQS11,Kiselev:Galli10,Kiselev:RingersProtaras2012},
which contain extra references.

This note
is organized
as follows. We first formalize the language~${{\EuScript A}}$ of cyclic words and describe the multiplicative structures in~it.
Next, we consider the fully
noncommutative jet spaces $J^{\infty}(M^n_{{\text{\textup{nC}}}}\to{{\EuScript A}}}%{\mathcal{A})$ and rephrase the standard notions
of differential calculus in this
setup. The noncommutative variational Schouten bracket~$[\![\,,\,]\!]$
is then defined as the odd Poisson bracket on the noncommutative
superspace which is cotangent to~$J^{\infty}(M^n_{{\text{\textup{nC}}}}\to{{\EuScript A}}}%{\mathcal{A})$,
c.f.~\cite{Kiselev:CattaneoFelderCMP2000} and~\cite{Kiselev:OlverSokolovCMP1998}.
Relating the odd
evolutionary vector fields~$\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}}$ to the operations
$[\![\boldsymbol{\xi},\cdot]\!]$, where $\boldsymbol{\xi}$~is a noncommutative variational
multivector~\cite{Kiselev:Koszul23},

we
affirm
the shifted\/-\/graded skew\/-\/symmetry of~$[\![\,,\,]\!]$ and

directly verify the Jacobi identity, which stems from the
Leibniz rule for the derivations $\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}}$
acting on the bracket $[\![\boldsymbol{\eta},\boldsymbol{\omega}]\!]$ of multivectors;
in turn, that rule holds by virtue of the conceptual equality
$\bigl[\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}},\smash{\boldsymbol{Q}^{\boldsymbol{\eta}}}
\bigr]=\smash{\boldsymbol{Q}^{[\![\boldsymbol{\xi},\boldsymbol{\eta}]\!]}}$.
Having developed the calculus of noncommutative variational $k$-\/vectors
with arbitrary $k\in\BBN\cup\{0\}$, we focus of the Poisson formalism of
bi\/-\/vectors~${\boldsymbol{\mathcal{P}}}$ satisfying the classical master\/-\/equation
$[\![{\boldsymbol{\mathcal{P}}},{\boldsymbol{\mathcal{P}}}]\!]=0$ and rephrase the concept in terms of noncommutative
Hamiltonian operators. In particular, we confirm the substitution principle
that allows us to operate with
identities in total derivatives
(primarily, Jacobi's identity) for variational (co)\/vectors which
depend in an arbitrary way not only on the base coordinates but also on the jet fibre variables.
Finally, we associate
homological evolutionary vector fields~$\boldsymbol{Q}$ and bi\/-\/vectors~${\boldsymbol{\mathcal{P}}}$
over the jet space~$J^{\infty}(M^n_{{\text{\textup{nC}}}}\to{{\EuScript A}}}%{\mathcal{A})$
with skew\/-\/adjoint noncommutative Noether total differential operators with involutive images and then we derive
a
convenient criterion which tells us
whether a given differential operator of this type is Hamiltonian.

Our reasoning is canonical in a sense that it projects down to the commutative setup without modifications. Hence we conclude that --~even in a purely commutative world~--
the determining properties of the Schouten bracket
do not refer to the commutativity assumption.
The crucial observation is that, due to the Leibniz rule, the derivations of~${{\EuScript A}}$
do not feel any cyclic permutations of factors in the products.
Because the construction of differential calculus of smooth functions and multivectors
in the cyclic\/-\/invariant setup is now accomplished,
it is the requirement of invariance under arbitrary shifts
of the words around the circles
--~but not the requirement of a graded commutativity under permutations of adjacent
letters~-- that appears as the weakest but the most natural hypothesis
in the
geometry of fundamental interactions at Planck scale.

\section{The language of cyclic words}
\noindent%
Let $\text{\textsf{Free}}\,(a^1,\ldots,a^m)$ be a free associative algebra
over~$\mathbb{R}}\newcommand{\BBC}{\mathbb{C}$ with
$m$~generators $a^1,\dots,a^m$; in the meanwhile,
let this algebra be not graded.
By definition, the multiplication $\cdot$ in $\text{\textsf{Free}}\,(a^1,\dots,a^m)$ obeys the identity
$(a_1\cdot a_2)\cdot a_3=a_1\cdot(a_2\cdot a_3)$
for the concatenation of any words $a_1,\ a_2$ and $a_3$ written in the alphabet $a^1,\dots,a^m$.
For the sake of clarity,
we discard from consideration here
all other relations (if any) in the algebras at hand, leaving them free, except for the following rule:
We postulate that all words $a\in\text{\textsf{Free}}\,(a^1,\dots,a^m)$ of length $\lambda(a)$ determine the equivalence classes under
the \textsl{cyclic} permutations ${\mathsf{t}}$ of the letters~\cite{Kiselev:KontsevichCyclic}:
\[
a\sim{\mathsf{t}}(a)\sim\ldots\sim{\mathsf{t}}^{\lambda(a)-1}(a)
\sim\frac{1}{\lambda(a)}\sum_{i=1}^{\lambda(a)}{\mathsf{t}}^{i-1}(a).
\]
Otherwise speaking, for any words $a_1,a_2\in\text{\textsf{Free}}\,(a^1,\dots,a^m)$ we set
\[
a_1\cdot a_2\sim a_2\cdot a_1
\]
(no $\mathbb{Z}}\newcommand{\BBE}{\mathbb{E}_2$-\/grading in the free associative algebra\,!) and denote
\[
{{\EuScript A}}}%{\mathcal{A}=\text{\textsf{Free}}\,(a^1,\dots,a^m)/\sim\ .
\]
If a word consists of just one letter, its equivalence under the cyclic permutations is trivial. If both words
$a_1$ and $a_2$ are some generators of $\text{\textsf{Free}}\,(a^1,\dots,a^m)$, then the equivalence encodes the commutativity. However,
at the length three the true noncommutativity starts and it then does not retract to the commutative setup.
For instance, for any $a,b,c\in\text{\textsf{Free}}\,(a^1,\dots,a^m)$ we have that
\[
aabcc\sim abcca\sim bccaa\sim ccaab\sim caabc\quad\nsim abacc.
\]

\begin{rem}
Let us recall that the idea
of a cyclic permutation itself implicitly refers to the transcription of symbols
around the circles~$\mathbb{S}^1$.
By convention, the oriented circles $\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1$ which carry the letters from the
alphabet of ${{\EuScript A}}}%{\mathcal{A}$ also carry the marked
point which is denoted by $\pmb{\infty}$. Thus, each cyclic-invariant word is a necklace
such that the point~$\pmb{\infty}$
is its lock; the
symbols are thread on the circle and are always read
from the infinity on their left to the
infinity on their right.
Elements of the vector space ${{\EuScript A}}}%{\mathcal{A}$ are formal
sums over $\mathbb{R}}\newcommand{\BBC}{\mathbb{C}$ of such words.
The non-graded letters $a^1,\dots,a^m$ can freely pass through the point $\pmb{\infty}$
without contribution to the sign of the coefficient in front of a word.
\end{rem}

The cyclic invariance for the classes in ${{\EuScript A}}}%{\mathcal{A}$ is fundamental.
It establishes a uniform geometric approach to the
following multiplicative structures and notions of differential calculus over the
ring~${{\EuScript A}}$:
\begin{itemize}
\item multiplication of words and word\/-\/valued functions,
\item termwise action of derivations, including
\item commutation of vector fields, and also
\item contraction of vectors with covectors, or
\item the Schouten bracket of multivectors and
\item the Poisson bracket of Hamiltonian functionals.
\end{itemize}
Indeed, all these operations amount to the familiar detach\/-\/and\/-\/join construction
of a pair of topological pants $\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1\times\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1\to\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1$.
We hence conclude that noncommutative differential calculus \textsl{is}
topology. Let us support this claim.

The natural multiplication $\times$ on the space ${{\EuScript A}}}%{\mathcal{A}$ of cyclic words is as follows:
\begin{equation}\label{EqRosary}
a_1\times a_2=
\frac1{\lambda(a_1)\lambda(a_2)}\sum_{i=1}^{\lambda(a_1)}\sum_{j=1}^{\lambda(a_2)}{\mathsf{t}}^{i-1}(a_1)\cdot{\mathsf{t}}^{j-1}(a_2),
\qquad a_1,a_2\in{{\EuScript A}}}%{\mathcal{A}.
\end{equation}
Namely, by taking the sum over all possible positions of the
mark~$\pmb{\infty}$ between the letters in each of the words,
we detach the locks at $\pmb{\infty}$ and, preserving the orientation, join the loose ends of the first string that carries
the word $a_1$ with the two ends of the other string for $a_2$.
We thus endow the space ${{\EuScript A}}}%{\mathcal{A}$ with the structure of an
algebra over the ground field $\mathbb{R}}\newcommand{\BBC}{\mathbb{C}$. The normalization by $(\lambda(a_1)\lambda(a_2))^{-1}$ matches the setup
with the purely commutative world.

\begin{example}
Consider the free algebra $\text{\textsf{Free}}\,(a^1,\dots,a^4)$ and let the two words be $a^1a^2$ and $a^3a^4$. Then we have that
$$a^1a^2\times a^3a^4=\tfrac14(a^1a^2a^3a^4+a^2a^1a^3a^4+a^3a^1a^2a^4+a^3a^2a^1a^4)=a^3a^4\times a^1a^2.$$
\end{example}

\begin{lemma}\label{Kiselev:LCommutNotAssoc}
The multiplication $\times$ in ${{\EuScript A}}}%{\mathcal{A}$ is commutative but in general not associative.
\end{lemma}

The proof is trivial.

\section{Noncommutative jet spaces}\label{SecNCJets}
\noindent%
Let us incorporate the space\/-\/time $M^n$ into the model

(and thus, allow for a possibility to encounter the derivatives
${\boldsymbol{a}}_{\sigma}=\frac{d^{|\sigma|}}{d{\boldsymbol{x}}^{\sigma}}({\boldsymbol{a}})$
of the $m$ pure states ${\boldsymbol{a}}=(a^1,\ldots,a^m)$), which motivates the introduction of the noncommutative jet bundles $J^{\infty}(M^n\to{{\EuScript A}}}%{\mathcal{A})$.

Consider the set of smooth maps from an oriented $n$-\/dimensional real manifold $M^n$ to the algebra ${{\EuScript A}}}%{\mathcal{A}$, which is a
vector space of suitable (possibly, infinite) dimension. Constructing the infinite jet space
$J^{\infty}(M^n\to{{\EuScript A}}}%{\mathcal{A})$
in a standard way~\cite{Kiselev:Olver},
we enlarge the alphabet $a^1,\dots,a^m$
of ${{\EuScript A}}}%{\mathcal{A}$ by the jet variables
${\boldsymbol{a}}_{\sigma}$ of all finite orders $|\sigma|$, here $a^j_{\varnothing}\equiv a^j$
for $1\le j\le m$, and by the base variables ${\boldsymbol{x}}$ (but we expect the theory to be invariant w.r.t.\ translations along~$M^n$).
An expansive notation for a given mapping ${\boldsymbol{a}}={\boldsymbol{s}}({\boldsymbol{x}})$ would be
\[
{\boldsymbol{a}}=s^1({\boldsymbol{x}})\cdot a^1+\ldots+s^m({\boldsymbol{x}})\cdot a^m,
\]
where at each point~${\boldsymbol{x}}\in M^n$ and for each~$j$ we have that
$s^j({\boldsymbol{x}})\in\mathbb{R}}\newcommand{\BBC}{\mathbb{C}$. Likewise, let $\sigma=(\sigma_1,\ldots,\sigma_n)$
be a multi\/-\/index; the respective component of the infinite jet
$j_\infty({\boldsymbol{s}})$ is
\[
{\boldsymbol{a}}_\sigma=\frac{\partial^{|\sigma|}}{\partial{\boldsymbol{x}}^\sigma}(s^1)({\boldsymbol{x}})\cdot a^1+
\ldots+\frac{\partial^{|\sigma|}}{\partial{\boldsymbol{x}}^\sigma}(s^m)({\boldsymbol{x}})\cdot a^m.
\]

We denote by
$C^\infty\bigl(J^\infty(M^n\to{{\EuScript A}})\bigr)$
the $C^{\infty}(M^n)$-algebra of cyclic\/-\/invariant words written in such
alphabet. For example,
\[
a^1a^2_{xx}a^1_x\sim a^2_{xx}a^1_xa^1\sim a^1_xa^1a^2_{xx}\quad\nsim a^1a^1_xa^2_{xx}\in C^\infty\bigl(J^\infty(M^n\to{{\EuScript A}})\bigr).

\]

The domain
of mappings to the algebra~${{\EuScript A}}$ can also be a noncommutative manifold (for example, this hypothesis will
simplify the proof of the substitution principle, see section~\ref{SecSubst}). Anticipating a concrete application of the geometry under study in
theoretical physics,
we employ a proper formulation of

the spectral approach $M^n\leftrightarrow C^{\infty}(M^n)$
to smooth manifolds, in the frames of which we interpret certain algebras as the
(non)\/commutative analogs of the rings of smooth functions on $M^n$
(here, also on the $m$-\/dimensional fibres~$N^m$ of the bundles $\pi\colon E^{m+n}\to M^n$).

Namely, we first consider the \textsl{tangent} bundle $TM^n\to M^n$ and denote by ${\vec{\mathbf{x}}}=({\vec{\mathrm{x}}}_1,\ldots,{\vec{\mathrm{x}}}_n)$ a field of $n$-\/tuples of vectors which span the tangent space $T_{{\boldsymbol{x}}}M^n$ at each point~${\boldsymbol{x}}\in M^n$. We let the elements~${\vec{\mathrm{x}}}_i$ together with their inverses
${\vec{\mathrm{x}}}_i^{\,-1}=1/{\vec{\mathrm{x}}}_i=(-{\vec{\mathrm{x}}}_i)$ be the $2n$~generators of (a sheaf over~$M^n$ of) unital $C^\infty(M^n)$-\/algebras with relations including
${\vec{\mathrm{x}}}_i{\vec{\mathrm{x}}}_i^{\,-1}={\vec{\mathrm{x}}}_i^{\,-1}{\vec{\mathrm{x}}}_i=1+\langle\text{energy}\rangle$, where
$\langle\text{energy}\rangle$ yields a zero\/-\/length cyclic word, but possibly also including many other relations of geometric and physical origin such as, for example, ${\vec{\mathrm{x}}}_i{\vec{\mathrm{x}}}_j{\vec{\mathrm{x}}}_i^{\,-1}{\vec{\mathrm{x}}}_j^{\,-1}{\vec{\mathrm{x}}}_k=
R_{ijk}^\ell({\boldsymbol{x}}){\vec{\mathrm{x}}}_\ell$. Because we shall address the physical aspects of this model
in a separate paper,

let us now not consider those extra relations and the arising problem of synonyms.
Nevertheless, we notice that the cyclic words
$\pmb{\infty}{\vec{\mathrm{x}}}_{i_1}^{\,\pm1}\cdot\ldots\cdot{\vec{\mathrm{x}}}_{i_\ell}^{\,\pm1}\pmb{\infty}$
encode the consecutive displacements, that is, paths (in particular, \textsl{knots}) along the Planck\/-\/scale grid inside the fibre of the tangent bundle over each point~${\boldsymbol{x}}\in M^n$ of space\/-\/time.

Next, we endow each unital algebra generated by ${\vec{\mathbf{x}}}$ and ${\vec{\mathbf{x}}}^{-1}$ with $n$~commuting left derivations ${\vec{\mathfrak{d}}}^i=\vec{\partial}/\partial{\vec{\mathrm{x}}}_i$
so that ${\vec{\mathfrak{d}}}^i({\vec{\mathrm{x}}}_j)=\delta_j^{\,i}$ and ${\vec{\mathfrak{d}}}^i(x^k)\equiv0$.
Gl
ing such pointwise\/-\/defined algebras over~${\boldsymbol{x}}\in M^n$, we obtain
the $C^\infty(M^n)$-\/algebra $C^\infty(M^n_{{\text{\textup{nC}}}})$; it inherits from the ring $C^\infty(M^n)$ the $n$~commuting left derivations $\vec{\partial}_i=\vec{\partial}/\partial x^i$ such that
$\vec{\partial}_i(x^j)=\delta_{i}^{\,j}$ and $\vec{\partial}_i({\vec{\mathrm{x}}}_k)\equiv0$.

Let us regard the new, noncommutative algebra as the ring of smooth functions on the noncommutative manifold which we denote by~$M^n_{{\text{\textup{nC}}}}$
and which we take for the base of a fully noncommutative vector bundle~$\pi^{{\text{\textup{nC}}}}$.

The sections $\boldsymbol{s}\in\Gamma(\pi^{{\text{\textup{nC}}}})$

become
$m$-\/tuples of open strings,
\[
\boldsymbol{a}=\bigl(a^j=\sum\nolimits_{\vec{\imath}}
s^j\bigl({\boldsymbol{x}};\vec{\imath}\,\bigr)\cdot
{\vec{\mathrm{x}}}_{i_1}^{\,\pm1}\cdot\ldots\cdot{\vec{\mathrm{x}}}_{i_\ell}^{\,\pm1},
\qquad 1\leq j\leq m\bigr),
\]
where the miltiplication~$\cdot$
in~$C^\infty(M^n_{\text{nC}})$ is associative but not commutative if
$\ell>0$; by default, $\ell=0$ means
(the image under inclusion of) an element from~$C^\infty(M^n)$.

Finally, we extend the alphabet of ${{\EuScript A}}}%{\mathcal{A}\simeq C^\infty(N^m_{\text{nC}})$ to the set of generators in $C^\infty\bigl(J^\infty(M^n_{\text{nC}}\to{{\EuScript A}})\bigr)$ by incorporating the base elements~$x^1$,\ $\ldots$,\ $x^n$ and~${\vec{\mathrm{x}}}_1$,\ $\ldots$,\ ${\vec{\mathrm{x}}}_n$ and also introducing
the symbols~$\vec{D}^{\tau}(a^j_\sigma)$ for
$1\le j\le m$ and all multi\/-\/indexes $\sigma$ and~$\tau$,
here $\vec{D}^{\varnothing}(a^j_\varnothing)\equiv a^j$.
The definition
\begin{align*}
(\boldsymbol{x},{\vec{\mathbf{x}}})\pinner\,
\bigl(j_\infty(\boldsymbol{s})^*\bigl(\overrightarrow{{\mathrm d}}/{\mathrm d} x^i(f)\bigr)\bigr)&=
\vec{\partial}/\partial x^i\bigl((\boldsymbol{x},{\vec{\mathbf{x}}})\pinner\,
j_\infty(\boldsymbol{s})^*(f)\bigr),\\
(\boldsymbol{x},{\vec{\mathbf{x}}})\pinner\,
\bigl(j_\infty(\boldsymbol{s})^*\bigl(\overrightarrow{D^i}(f)\bigr)\bigr)&=
\vec{\partial}/\partial{\vec{\mathrm{x}}}_i
\bigl((\boldsymbol{x},{\vec{\mathbf{x}}})\pinner\,
j_\infty(\boldsymbol{s})^*(f)\bigr)
\end{align*}
of the total derivatives $\overrightarrow{{\mathrm d}}/{\mathrm d} x^i=\widehat{\vec{\partial}_i}$
and $\overrightarrow{D^i}=\widehat{{\vec{\mathfrak{d}}}^i}$ for a differential function~$f$
on~$J^\infty(M^n_{\text{nC}}\to{{\EuScript A}})$ is reproduced literally
in the fully noncommutative picture, see~\eqref{Kiselev:EqDefTD} below.

We emphasize that the chain of inclusions
$C^\infty(M^n)\hookrightarrow C^\infty(M^n_{\text{nC}})
\hookrightarrow C^\infty\bigl(J^\infty(M^n_{\text{nC}}\to{{\EuScript A}})\bigr)$ is the source of
words of length zero both in~$C^\infty(M^n_{\text{nC}})$, see previous paragraph, and
in the fibre algebras~${{\EuScript A}}$ which now contain the necklaces with strings
of~${\vec{\mathbf{x}}}$'s
and the symbol~$\boldsymbol{\infty}$ at the lock.\footnote{The presence of zero\/-\/length
cyclic words in~$C^\infty\bigl(J^\infty(M^n_{\text{nC}}\to{{\EuScript A}})\bigr)$ is manifest even
if the base of~$\pi^{\text{nC}}$ is commutative; this is a generalization of the standard
fact from a purely commutative jet bundle geometry~\cite{Kiselev:Olver}:
the ring~$C^\infty(M^n)$ plays the r\^o\-le of the ground field in differential calculus
over~$J^\infty(\pi)$, though known conceptual difficulties appear at exactly this point,
c.f.~\cite{Kiselev:Galli10}.}
We note that the multiplication~$\times$ of cyclic words of zero length, i.e., a restriction
of~$\times$ from~$C^\infty\bigl(J^\infty(M^n_{\text{nC}}\to{{\EuScript A}})\bigr)$ to the image
under inclusion of the space~$C^\infty(M^n_{\text{nC}})
$ of open strings remains associative but not commutative, in contrast with
Lemma~\ref{Kiselev:LCommutNotAssoc}. Indeed, a structural difference between the
variables~${\vec{\mathbf{x}}}$ and $\vec{D}^\tau(\boldsymbol{a}_\sigma)$ is that the former never
pass through the marked point~$\boldsymbol{\infty}$ on~$\mathbb{S}^1$: the intermediate order
from the first~${\vec{\mathrm{x}}}_{i_1}$ to the last~${\vec{\mathrm{x}}}_{i_\ell}$ is not broken. The following convention
is thus important: under the multiplication~$\times$ those open strings ${\vec{\mathrm{x}}}_{i_1}^{\,\pm1}\cdot\ldots\cdot{\vec{\mathrm{x}}}_{i_\ell}^{\,\pm1}$ are pasted immediately after (measured counterclockwise) the lock
at~$\boldsymbol{\infty}$ of the second factor but strictly before any other symbols of the
second word (i.e., preceding a letter~${\vec{\mathrm{x}}}_i$ or generator~$a^j$, if any).

We now upgrade necessary definitions
to noncommutative geometry of the jet space $J^{\infty}(\pi^{\text{nC}})$; we emphasize that all the constructions and the statements which we establish in what follows for the new, wider setup
obviously remain true under the stronger, more restrictive assumptions of the (graded-)commutativity.

The total derivatives w.r.t.\ $x^i$ and ${\vec{\mathrm{x}}}_i$, $1\leqslant i\leqslant n$, on $J^\infty(\pi^{\text{nC}})$ are expressed by the respective

formulas
\begin{equation}\label{Kiselev:EqDefTD}
\begin{aligned}
\overrightarrow{\mathrm{d}}/\mathrm{d}x^i&=\overrightarrow{\partial}/\partial x^i +\sum_{|\sigma|,|\tau|\geqslant0}
\vec{D}^\tau(\boldsymbol{a}_{\sigma+1_i}) \overrightarrow{\partial}/\partial\vec{D}^\tau(\boldsymbol{a}_\sigma),\\
\overrightarrow{D}^i&=\overrightarrow{\partial}/\partial{\vec{\mathrm{x}}}_i
+\sum_{|\sigma|,|\tau|\geqslant0}
\vec{D}^{\tau+1_i}(\boldsymbol{a}_{\sigma}) \overrightarrow{\partial}/\partial\vec{D}^\tau(\boldsymbol{a}_\sigma).
\end{aligned}
\end{equation}
The evolutionary derivations
\[
\vec{\partial}^{(\boldsymbol{a})}_\varphi
=
\sum\limits_{|\sigma|,|\tau|\geqslant0}
\bigl(\smash{\overrightarrow{\mathrm{d}}^{|\sigma|}}/\mathrm{d}\boldsymbol{x}^\sigma \circ \overrightarrow{D^\tau}\bigr)
(\varphi)\cdot\smash{\overrightarrow{\partial}}/\partial
\vec{D}^\tau(\boldsymbol{a}_\sigma)
\]
act from the left by the
Leibniz rule.\footnote{By construction,
noncommutative evolutionary derivations determine
flat deformations of sheafs of the algebras ${{\EuScript A}}}%{\mathcal{A}$ over the base manifolds $M^n_{{\text{\textup{nC}}}}$.}

Denote by $\overline{\Lambda}^n(\pi^{\text{nC}})$ the space
of horizontal forms of the
highest ($n$-th) degree and by $\overline{H}^n(\pi^{\text{nC}})$ the respective
cohomology
w.r.t.\ the horizontal differential $\overline{\mathrm{d}}

=\sum\limits_{i=1}^n \mathrm{d}x^i\cdot \overrightarrow{\mathrm{d}}/\mathrm{d}x^i$;
the Cartan differential on $J^\infty(\pi^{\text{nC}})$ is $\mathrm{d}_{\mathcal{C}}= \mathrm{d}_{\text{dR}}-\overline{\mathrm{d}}$;
by convention, the volume form~${\mathrm d}{\boldsymbol{x}}$ plays the r\^o\-le of the marked point on the
necklaces
which carry the highest horizontal forms
$h\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)\,{\mathrm d}{\boldsymbol{x}}\in\overline{\Lambda}^n(\pi^{\text{\textup{nC}}})$.
Denote by $\langle\,,\,\rangle$ the $\overline{\Lambda}^n(\pi^{\text{nC}})$-\/valued coupling
between the spaces of variational covectors $\boldsymbol{p}$ and evolutionary vectors
$\smash{\vec{\partial}^{\,(\boldsymbol{a})}_\varphi}$.
By default, we pass to the cohomology and,

using the integration by parts,
normalize the (non)\/commutative covectors as follows, $\boldsymbol{p}\bigl(
(\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}]\bigr)=
\smash{\sum\nolimits_{j=1}^m} \langle\text{word}\rangle\cdot\mathrm{d}_{\mathcal{C}}a^
$.

We note that the derivatives $\vec{\partial}/\partial a^j_{\sigma}$ and the differentials $\mathrm{d}_{\mathcal{C}}\newcommand{\tcC}{\smash{\widetilde{\mathcal{C}}}}a^j_{\sigma}$ play the r\^o\-le of the
marked points $\pmb{\infty}$ in the cyclic words $\vec{\partial}^{\,({\boldsymbol{a}})}_{\varphi}$ and ${\boldsymbol{p}}$, respectively. The coupling
$\langle{\boldsymbol{p}},\vec{\partial}_{\varphi}^{\,({\boldsymbol{a}})}\rangle$
of the two oriented circles $\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1$ carrying ${\boldsymbol{p}}$ and $\vec{\partial}_{\varphi}^{\,({\boldsymbol{a}})}$ goes along
the usual lines of detaching the locks at the marked points and then joining the loose ends, preserving the orientation.
(Again, this constitutes the pair of topological pants $\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1\times\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1\to\mathbb{S}}\newcommand{\BBT}{\mathbb{T}^1$.)

Let $A\colon\boldsymbol{p}\mapsto\vec{\partial}_{A({\boldsymbol{p}})}^{\,(a)}$ be a Noether noncommutative linear matrix operator
in total derivatives,
containing --apart from the total derivatives-- the operators $(a\cdot)$ and $(\cdot a)$ of
left-{} and right\/-\/multiplication
by open words $a\in C^\infty\bigl(J^\infty(\pi^{{\text{\textup{nC}}}})\bigr)$

that are always read from left to right. The
adjoint operator $\overrightarrow{A}^\dagger$ is defined from the equality
$\langle\boldsymbol{p}_1,A(\boldsymbol{p}_2)\rangle=
\langle\boldsymbol{p}_2,\overrightarrow{A}^\dagger(\boldsymbol{p}_1)\rangle$, in which we first
integrate by parts and then transport
the even covector $\boldsymbol{p}_2$ around the circle.
Let us remark that the operators are ``measured from the comma'' in the coupling $\langle\,,\,\rangle$: namely, the
\textsl{right} multiplication in the counterclockwise\/-\/acting operator $\overrightarrow{A}^{\dagger}$ corresponds to the
\textsl{left} multiplication in the clockwise\/-\/acting
$\overleftarrow{A}^{\dagger}$, and \textrm{vice versa}.

\section{Noncommutative variational multivectors}
\noindent%
The covectors $\boldsymbol{p}\bigl((\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}]\bigr)$ were even.
We reverse their parity, $\Pi\colon\boldsymbol{p}\mapsto
\boldsymbol{b}\bigl((\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}]\bigr)$,
preserving the topology of the bundles but postulating that all
objects are at most
polynomial in fi\-ni\-te\-ly many derivatives of $\boldsymbol{b}$.
Next, we consider the noncommutative variational cotangent superspace
$\overline{J^\infty}\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)=
J^\infty(\Pi\widehat{\pi}^{\text{nC}})\mathbin{{\times}_{M^n}} J^\infty(\pi^{\text{nC}})$,
see~\cite{Kiselev:Olver} and~\cite{Kiselev:Praha11,Kiselev:RingersProtaras2012}.
In effect, we declare that $\boldsymbol{b}$,\ $\boldsymbol{b}_{\boldsymbol{x}}$,\ %
$\boldsymbol{b}_{\boldsymbol{x}\boldsymbol{x}}$,\ $\ldots$,\ $\vec{D}^\varrho(\boldsymbol{b}_{\varsigma})$ are
the extra, odd jet variables on top of the old, even~$\vec{D}^\tau(\boldsymbol{a}_{\sigma})$'s.
The total derivatives $\overrightarrow{\mathrm{d}}/\mathrm{d}x^i$ obviously lift onto
$\overline{J^\infty}\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$ as well as
$\overline{\mathrm{d}}$ that yields the cohomology
$\overline{H}^n\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)=
 \overline{\Lambda}^n\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)\bigr/
  (\text{im}\,\overline{\mathrm{d}})$.
The two components of the evolutionary vector fields $\boldsymbol{Q}=
{\vec{\partial}^{\,(\boldsymbol{a})}_{\varphi^{\boldsymbol{a}}}+
\vec{\partial}^{\,(\boldsymbol{b})}_{\varphi^{\boldsymbol{b}}}}$ now begin with
$\dot{\boldsymbol{a}}=
\varphi^{\boldsymbol{a}}\bigl((\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}],[\boldsymbol{b}]\bigr)$ and
$\smash{\dot{\boldsymbol{b}}}=
\smash{\varphi^{\boldsymbol{b}}}\bigl((\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}],[\boldsymbol{b}]\bigr)$.

The definition of noncommutative variational $k$-\/vectors, their evaluation on
$k$ co\-vec\-tors, the definition of the noncommutative variational Schouten bracket, and its
inductive calculation are
two pairs of distinct concepts.

We would like to define a noncommutative
$k$-\/vector~$\boldsymbol{\xi}$, $k\geqslant0$,
as a cohomology class in $\overline{H}^n\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$
whose density is $k$-\/linear in the odd $\boldsymbol{b}$'s or their derivatives.
This is inconsistent because for every $k$ a given $k$-vector can be in fact cohomologically trivial. There are two
ways out: either by a direct inspection of the values of
multivectors on all $k$-tuples of covectors
(the evaluation
is defined in formula~\eqref{EvalkVector} below, but this option is inconvenient) or by using
the normalization of $\boldsymbol{\xi}$.

\begin{define}\label{DefkVector}
A noncommutative $k$-\/vector~$\boldsymbol{\xi}$
is the horizontal cohomology class of the element
\[
\boldsymbol{\xi}=\langle\boldsymbol{b},A(\boldsymbol{b},\ldots,\boldsymbol{b})\rangle/k!\ ,
\]
where the noncommutative total
differential operator $A$ depends on $(k-1)$ odd entries and may
have $\boldsymbol{a}$-\/dependent coefficients.
\end{define}

Integrating by parts and pushing the
letters of the word~$\boldsymbol{\xi}$ along the circle, we infer that
$\langle\boldsymbol{b}_1,A(\boldsymbol{b}_2,\ldots,\boldsymbol{b}_k)\rangle
\smash{{\bigr|}_{\boldsymbol{b}_i\mathrel{{:}{=}}\boldsymbol{b}}}=
\langle\boldsymbol{b}_2,A^\dagger_\circlearrowright(\boldsymbol{b}_3,\ldots,
\boldsymbol{b}_k,\boldsymbol{b}_1)\rangle
\smash{{\bigr|}_{\boldsymbol{b}_i\mathrel{{:}{=}}\boldsymbol{b}}}$;
note that, each time an odd variable $\boldsymbol{b}_\tau$ reaches a marked point
$\boldsymbol{\infty}$ on the circle, it counts the $k-1$
other odd variables whom it overtakes and reports the sign $(-)^{k-1}$
(in particular, $A^\dagger_\circlearrowright=A^\dagger=-A$ if~$k=2$).

\begin{define}
The value of the $k$-\/vector $\boldsymbol{\xi}$ on $k$ arbitrary
covectors $\boldsymbol{p}_i$ is
\begin{equation}\label{EvalkVector}
\boldsymbol{\xi}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_k)=
\frac1{k!}\sum\limits_{s\in S_k}(-)^{|s|}\langle\boldsymbol{p}_{s(1)},
A(\boldsymbol{p}_{s(2)},\ldots,\boldsymbol{p}_{s(k)})\rangl
\ .
\end{equation}
We emphasize that
we shuffle the arguments but never swap their slots,
which are built into the cyclic word~$\boldsymbol{\xi}$.
\end{define}

\section{Noncommutative Schouten bracket}
\noindent%
The commutative concatenation~\eqref{EqRosary} of the \textsl{densities} of two multivectors
provides an ill-defined product in $\overline{H}^n\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$.
The genuine multiplication in the algebra $\overline{H}^n\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$
is the odd Poisson bracket (the \textsl{antibracket}).
We
fix the Dirac ordering
$\delta\boldsymbol{a}\wedge\delta\boldsymbol{b}$ over
each $\boldsymbol{x}$ in $\overline{J^\infty}\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)
\to M^n$;

note that $\delta\boldsymbol{a}$ is a covector and $\delta\boldsymbol{b}$
is an odd vector
so that their coupling equals $+1\cdot\mathrm{d}\boldsymbol{x}$.

\begin{define}\label{DefSchouten}
The noncommutative variational Schouten bracket of two multivectors $\boldsymbol{\xi}$
and $\boldsymbol{\eta}$ is
\begin{equation}\label{EqSchouten}
[\![\boldsymbol{\xi},\boldsymbol{\eta}]\!]=
\langle\overrightarrow{{\delta}\boldsymbol{\xi}}\wedge
\overleftarrow{{\delta}\boldsymbol{\eta}}\rangle.
\end{equation}
In coordinates,\ this yields
\[
[\![\boldsymbol{\xi},\boldsymbol{\eta}]\!]=
\bigl[\overrightarrow{{\delta}\boldsymbol{\xi}}/\delta\boldsymbol{a}\,\cdot
\overleftarrow{{\delta}\boldsymbol{\eta}}/\delta\boldsymbol{b}\,-
\overrightarrow{{\delta}\boldsymbol{\xi}}/\delta\boldsymbol{b}\,\cdot
\overleftarrow{{\delta}\boldsymbol{\eta}}/\delta\boldsymbol{a}\bigr],
\]
where
\begin{enumerate}
\item
all the derivatives are thrown off the variations $\delta\boldsymbol{a}$ and
$\delta\boldsymbol{b}$ via the integration by parts, then
\item
the letters $\boldsymbol{a}_\sigma$, $\boldsymbol{b}_\tau$, $\delta\boldsymbol{a}$, and
$\delta\boldsymbol{b}$, which are thread on the two circles
$\delta\boldsymbol{\xi}$ and $\delta\boldsymbol{\eta}$,
spin
along these rosaries
so that the variations $\delta\boldsymbol{a}$ and $\delta\boldsymbol{b}$ match
in all possible combinations, and finally,
\item
the variations $\delta\boldsymbol{a}$ and $\delta\boldsymbol{b}$ detach
from the circles and couple,
while the loose ends of the two remaining open strings
join
and form the new circle.
\end{enumerate}
\end{define}

\begin{lemma}\label{LSchoutenSkew}
The Schouten bracket is shifted\/-\/graded skew\/-\/symmetric\textup{:}
if $\boldsymbol{\xi}$ is a $k$-\/vector and $\boldsymbol{\eta}$
an $\ell$-\/vector, then $[\![\boldsymbol{\xi},\boldsymbol{\eta}]\!]=
-(-)^{(k-1)(\ell-1)}[\![\boldsymbol{\eta},\boldsymbol{\xi}]\!]$.
\end{lemma}

\begin{proof}
It is obvious that the brackets $[\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!]$ and $[\![{\boldsymbol{\eta}},{\boldsymbol{\xi}}]\!]$ contain the same summands which can differ only
by signs; now it is our task to calculate these factors and show that the same signs appear at all the summands
simultaneously. Let us compare the terms
\[
\frac{\!\overrightarrow{{\delta}{\boldsymbol{\xi}}}}{\delta{\boldsymbol{a}}}\langle\delta{\boldsymbol{a}},
\delta{\boldsymbol{b}}\rangle\frac{\!\overleftarrow{{\delta}{\boldsymbol{\eta}}}}{\delta{\boldsymbol{b}}}
\text{\qquad and \quad}
-\frac{\!\overrightarrow{{\delta}{\boldsymbol{\eta}}}}{\delta{\boldsymbol{b}}}\langle\delta{\boldsymbol{a}},
\delta{\boldsymbol{b}}\rangle\frac{\!\overleftarrow{{\delta}{\boldsymbol{\xi}}}}{\delta{\boldsymbol{a}}}
\]
containing the variation of a $k$-vector ${\boldsymbol{\xi}}$ with respect to the even variables ${\boldsymbol{a}}$ and the variation of an
$\ell$-vector ${\boldsymbol{\eta}}$ with respect to the odd entries ${\boldsymbol{b}}$. We first note that the left-to-right transportation of the
differential $\delta{\boldsymbol{b}}$ along the queue of $\ell-1$ odd elements ${\boldsymbol{b}}_{\tau}$ in the variation of ${\boldsymbol{\eta}}$ produces the
sign $(-)^{\ell-1}$. The variations disappear in the coupling $\langle\delta{\boldsymbol{a}},\delta{\boldsymbol{b}}\rangle=+1$,
so there remains to
carry, via the infinity $\boldsymbol{\infty}$, the object $\overrightarrow{\delta{\boldsymbol{\eta}}}/\delta{\boldsymbol{b}}$ of parity $(-)^{\ell-1}$ around
the object $\overleftarrow{\delta{\boldsymbol{\xi}}}/\delta{\boldsymbol{a}}$, which is
$k$-linear in the odd variables after the variation with
respect to the even ${\boldsymbol{a}}$. This yields the sign $(-)^{(\ell-1)\cdot k}$. Therefore, the overall difference in the sign
between the terms $\overrightarrow{\delta{\boldsymbol{\xi}}}/\delta{\boldsymbol{a}}\cdot\!\overleftarrow{\delta{\boldsymbol{\eta}}}/\delta{\boldsymbol{b}}$ and
$\overrightarrow{\delta{\boldsymbol{\eta}}}/\delta{\boldsymbol{b}}\cdot\!\overleftarrow{\delta{\boldsymbol{\xi}}}/\delta{\boldsymbol{a}}$ equals
\[
-(-)^{\ell-1}\cdot(-)^{(\ell-1)\cdot k}=-(-)^{(k+1)(\ell-1)}=-(-)^{(k-1)(\ell-1)}.
\]
Due to the $(k\leftrightarrow\ell)$-symmetry of the exponent, the same sign factor matches the other pair of summands
in the Schouten bracket, namely,
$\overrightarrow{\delta{\boldsymbol{\xi}}}/\delta{\boldsymbol{b}}\cdot\!\overleftarrow{\delta{\boldsymbol{\eta}}}/\delta{\boldsymbol{a}}$ and
$\overrightarrow{\delta{\boldsymbol{\eta}}}/\delta{\boldsymbol{a}}\cdot\!\overleftarrow{\delta{\boldsymbol{\xi}}}/\delta{\boldsymbol{b}}$.
The proof is complete.
\end{proof}

\enlargethispage{0.7\baselineskip}
Define the evolutionary vector field
$\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}}$ on
$\overline{J^\infty}\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$
by the rule
\begin{equation}\label{EqRuleQ}
\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}}(\boldsymbol{\eta})=
[\![\boldsymbol{\xi},\boldsymbol{\eta}]\!],
\end{equation}
whence $\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}}=
-\vec{\partial}^{\,(\boldsymbol{a})}_{\overrightarrow{\delta\boldsymbol{\xi}}/\delta\boldsymbol{b}}
+\vec{\partial}^{\,(\boldsymbol{b})}_{\overrightarrow{\delta\boldsymbol{\xi}}/\delta\boldsymbol{a}}$.
The normalization $\boldsymbol{\xi}=\bigl\langle\boldsymbol{b},
A(\boldsymbol{b},\ldots,\boldsymbol{b})\bigr\rangle/k!$ determines
\[
\smash{\boldsymbol{Q}^{\boldsymbol{\xi}}}=-(-)^{k-1}\tfrac{1}{(k-1)!}
\vec{\partial}^{\,(\boldsymbol{a})}_{A(\boldsymbol{b},\ldots,\boldsymbol{b})}
+(-)^{k-1}\tfrac{1}{k!}\smash{\vec{\partial}^{\,(\boldsymbol{b})}_{
\overrightarrow{\ell}^{\,(\boldsymbol{a})\,\dagger}_{A(\boldsymbol{b}_2,\ldots,\boldsymbol{b}_k)}
 (\boldsymbol{b}_1)}} \smash{{\bigr|}_{\boldsymbol{b}_i\mathrel{{:}{=}}\boldsymbol{b}}},
\]
where for all (non)\/commutative
objects~$\varphi$ we denote by ${\vec{\ell}^{\,(\boldsymbol{a})\,\dagger}_\varphi}$ the adjoint to the linearization\textup{,} which is
$\vec{\ell}^{\,(\boldsymbol{a})}_\varphi(\delta\boldsymbol{a})=
\tfrac{\mathrm{d}}{\mathrm{d}\varepsilon}{\bigr|}_{\varepsilon=0}
\varphi\bigl((\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}+\varepsilon\,\overrightarrow{\delta{\boldsymbol{a}}}]\bigr)$.

\begin{example}[see~\cite{Kiselev:Praha11,Kiselev:Galli10}]
We have that
\begin{equation}\label{QPoisson}
\boldsymbol{Q}^{\frac{1}{2}\langle\boldsymbol{b},A(\boldsymbol{b})\rangle}=
\partial^{(\boldsymbol{a})}_{A(\boldsymbol{b})}-\tfrac{1}{2}
\partial^{(\boldsymbol{b})}_{\overrightarrow{\ell}^{\,(\boldsymbol{a})\,\dagger}_{A(\boldsymbol{b})} (\boldsymbol{b})}.
\end{equation}
\end{example}

\begin{proposition}\label{Prop1}
The equality
\begin{equation}\label{EqRougeEtNoir}
\bigl[\boldsymbol{Q}^{\boldsymbol{\xi}},
\boldsymbol{Q}^{\boldsymbol{\eta}}\bigr]=\boldsymbol{Q}^{[\![\boldsymbol{\xi},
\boldsymbol{\eta}]\!]}
\end{equation}
correlates the graded commutator of
graded evolutionary vector fields with

the the noncommutative variational Schouten bracket
of two multivectors.
\end{proposition}

\begin{proof
To verify Eq.~\eqref{EqRougeEtNoir}, it suffices to inspect the composition of the variations
\[
\overrightarrow\delta\!([\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!])=\frac{\!\overrightarrow\delta\!([\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!])}{\delta{\boldsymbol{a}}}\cdot\delta{\boldsymbol{a}}+
\frac{\!\overrightarrow\delta\!([\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!])}{\delta{\boldsymbol{b}}}\cdot\delta{\boldsymbol{b}}.
\]
They determine the two terms in ${\boldsymbol{Q}}^{[\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!]}$ and, moreover, contain the variations $\delta{\boldsymbol{a}}$ and $\delta{\boldsymbol{b}}$
standing at the right of the variational derivatives (which may themselves bear the total derivatives at their left due
to the integration by parts).

Suppose that initially the $k$-vector ${\boldsymbol{\xi}}$ was written in red ink and ${\boldsymbol{\eta}}$ in black. Let us also agree that the colour
of the ink is preserved in all the formulas which involve ${\boldsymbol{\xi}}$ and ${\boldsymbol{\eta}}$ (say, the red vector field ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$
inserts, by the Leibniz rule, the red sub-words in the black word ${\boldsymbol{\eta}}$).
It is readily seen that some
variations $\delta{\boldsymbol{a}}$ or $\delta{\boldsymbol{b}}$ in $\overrightarrow\delta\!([\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!])$
are red and some are black; for indeed, they stem from the argument $[\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!]$ by the Leibniz rule,
whereas some letters in the argument are red and some are black.

We now study the superposition of the two Leibniz rules:
one for the variation $\overrightarrow\delta$, which drags the differentials to the \textsl{right}, and the other rule
for the evolutionary fields ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ and ${\boldsymbol{Q}}^{{\boldsymbol{\eta}}}$, which act on the arguments from the \textsl{left}.
Consider first the application of ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ to (without loss of generality) the ${\boldsymbol{a}}$-component
$\overrightarrow{\delta{\boldsymbol{\eta}}}/\delta{\boldsymbol{b}}$ of the generating section for ${\boldsymbol{Q}}^{{\boldsymbol{\eta}}}$ in the left-hand side
of~\eqref{EqRougeEtNoir}. The field ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ is evolutionary, so it dives under all the total derivatives and acts,
by the Leibniz rule, on the letters of the argument. That argument is ``almost'' ${\boldsymbol{\eta}}$ from which, by a yet another
Leibniz rule for the Cartan differential, the black variation $\delta_{{\boldsymbol{\eta}}}{\boldsymbol{b}}$ of one ${\boldsymbol{b}}$ is taken out and transported
to the right. Consequently, the graded derivation ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ never overtakes that odd object $\delta_{{\boldsymbol{\eta}}}{\boldsymbol{b}}$, hence
no extra sign is produced (for the even variation $\delta_{{\boldsymbol{\eta}}}{\boldsymbol{a}}$ in $\overrightarrow{\delta{\boldsymbol{\eta}}}$, no signs would
appear at all, but such permutations also never occur).
The red field ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ acts on the black coefficient of $\delta_{{\boldsymbol{\eta}}}{\boldsymbol{b}}$: on one hand, it differentiates~---
one after another~--- all letters in all the terms. (Note that, in particular, by the Leibniz rule it replaces the black
variables ${\boldsymbol{b}}_{\tau}$ by their velocities, except for the variables which turned into the variations $\delta_{{\boldsymbol{\eta}}}{\boldsymbol{b}}$.
But such elements are duly processed in the other summands of the Leibniz formula, in which other odd letters ${\boldsymbol{b}}_{\tau}$
yield the variations.) On the other hand, none of the red ${\boldsymbol{b}}$'s from ${\boldsymbol{\xi}}$ in ${\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ shows up in the form
of the differential at the right. To collect the red variations $\delta_{{\boldsymbol{\xi}}}{\boldsymbol{b}}$, which contribute to the rest of the
${\boldsymbol{a}}$-component of the generating section of the evolutionary field
\begin{equation}\label{EqGradedCommutator}
[{\boldsymbol{Q}}^{{\boldsymbol{\xi}}},{\boldsymbol{Q}}^{{\boldsymbol{\eta}}}]={\boldsymbol{Q}}^{{\boldsymbol{\xi}}}\circ{\boldsymbol{Q}}^{{\boldsymbol{\eta}}}-(-)^{(k-1)(\ell-1)}{\boldsymbol{Q}}^{{\boldsymbol{\eta}}}
\circ{\boldsymbol{Q}}^{{\boldsymbol{\xi}}},
\end{equation}
we repeat the above reasoning for the term ${\boldsymbol{Q}}^{{\boldsymbol{\eta}}}\circ{\boldsymbol{Q}}^{{\boldsymbol{\xi}}}$ in the graded commutator
({\textrm{c.\,f.}}\ Lemma~\ref{LSchoutenSkew}).

It only remains to notice that the variations $\delta{\boldsymbol{b}}$ of all the letters ${\boldsymbol{b}}$ (red or black) are now properly counted.
Comparing the object
\[
{\boldsymbol{Q}}^{{\boldsymbol{\xi}}}(\overrightarrow{\delta{\boldsymbol{\eta}}}/\delta_{{\boldsymbol{\eta}}}{\boldsymbol{b}})-
(-)^{(k-1)(\ell-1)}{\boldsymbol{Q}}^{{\boldsymbol{\eta}}}(\overrightarrow{\delta{\boldsymbol{\xi}}}/\delta_{{\boldsymbol{\xi}}}{\boldsymbol{b}})
\]
with
\[
\overrightarrow\delta\!({\boldsymbol{Q}}^{{\boldsymbol{\xi}}}({\boldsymbol{\eta}}))/\delta_{{\boldsymbol{\eta}}}{\boldsymbol{b}}-
(-)^{(k-1)(\ell-1)}\overrightarrow\delta\!({\boldsymbol{Q}}^{{\boldsymbol{\eta}}}({\boldsymbol{\xi}}))/\delta_{{\boldsymbol{\xi}}}{\boldsymbol{b}}=
\!\overrightarrow\delta\!([\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!])/\delta{\boldsymbol{b}},
\]
we conclude that the two expressions coincide. The same holds for the variation of $[\![{\boldsymbol{\xi}},{\boldsymbol{\eta}}]\!]$ with respect
to ${\boldsymbol{a}}$, which is also composed by the sum of red variations $\delta_{{\boldsymbol{\xi}}}{\boldsymbol{a}}$ and black $\delta_{{\boldsymbol{\eta}}}{\boldsymbol{a}}$.
This implies equality~\eqref{EqRougeEtNoir}.
\end{proof}

Finally, let us act by the graded evolutionary vector field that appears in both sides of
equality~\eqref{EqRougeEtNoir} upon a test multivector~$\boldsymbol{\omega}$
and then use definition~\eqref{EqGradedCommutator} of the graded commutator
(which captures the signs) and formula~\eqref{EqRuleQ}, which reinstates the Schouten brackets.
We thus prove the other fundamental property of the bracket~$[\![\,,\,]\!]$.

\begin{cor}
The Leibniz rule
$\boldsymbol{Q}^{\boldsymbol{\xi}}\bigl([\![\boldsymbol{\eta},\boldsymbol{\omega}]\!]\bigr)
=[\![\boldsymbol{Q}^{\boldsymbol{\xi}}(\boldsymbol{\eta}),\boldsymbol{\omega}]\!]
+(-)^{(k-1)(\ell-1)}[\![\boldsymbol{\eta},
\boldsymbol{Q}^{\boldsymbol{\xi}}(\boldsymbol{\omega})]\!]$, where
$\boldsymbol{\omega}\in\overline{H}^n\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$,
is the Jacobi identity
\begin{equation}\label{JacobiSchouten}
[\![\boldsymbol{\xi},[\![\boldsymbol{\eta},\boldsymbol{\omega}]\!]
 ]\!]=[\![ [\![\boldsymbol{\xi},\boldsymbol{\eta}]\!],\boldsymbol{\omega}]\!]
+(-)^{(k-1)(\ell-1)}[\![\boldsymbol{\eta},[\![\boldsymbol{\xi},\boldsymbol{\omega}]\!]
]\!]
\end{equation}
for the noncommutative variational Schouten bracket~$[\![\,,\,]\!]$.
\end{cor}

\section{Noncommutative Poisson formalism}
\noindent%
The differential calculus on noncommutative jet spaces which we have reported here is fully
capable to serve the noncommutative \textsl{Poisson} formalism (but not only: other
domains of its applicability are outlined in~\cite{Kiselev:Praha11}).
To endow the space $\overline H^n(\pi^{{\text{\textup{nC}}}})$ of the functionals for the jet space $J^{\infty}(\pi^{{\text{\textup{nC}}}})$ with the
variational Poisson algebra structure, let us notice first that

each skew\/-\/adjoint noncommutative linear total differential operator
$A\colon\boldsymbol{p}\mapsto\vec{\partial}^{\,(\boldsymbol{a})}_{A({\boldsymbol{p}})}$ yields the bi\/-\/vector
$\boldsymbol{\mathcal{P}}\newcommand{\cR}{\mathcal{R}}=\tfrac{1}{2}\langle\boldsymbol{b},A(\boldsymbol{b})\rangle$.
Let $\mathcal{H}_1$,\ $\mathcal{H}_2$,\ $\mathcal{H}_3$ be zero\/-\/vectors, \textrm{i.\,e.},
$\mathcal{H}_i=\int h_i\bigl((\boldsymbol{x},{\vec{\mathbf{x}}}),[\boldsymbol{a}]\bigr)\,\mathrm{d}\boldsymbol{x}\in\overline{H}{}^n(\pi^{\text{\textup{nC}}})$.
By definition, put
\begin{equation}\label{PBr}
\{\mathcal{H}_i,\mathcal{H}_j\}_{{\boldsymbol{\mathcal{P}}}}
\mathrel{{:}{=}}{\boldsymbol{\mathcal{P}}} \bigl(\smash{\overrightarrow{\delta\mathcal{H}_i}}
/\delta\boldsymbol{a},
\smash{\overrightarrow{\delta\mathcal{H}_j}}/\delta\boldsymbol{a}\bigr),
\end{equation}
which equals
\[
\bigl\langle\overrightarrow{\delta\mathcal{H}_i}/\delta\boldsymbol{a},
 A\bigl(\overrightarrow{\delta\mathcal{H}_j}/\delta\boldsymbol{a}\bigr)\bigr\rangle=
\vec{\partial}^{\,(\boldsymbol{a})}_{A(\overrightarrow{\delta\mathcal{H}_j}/\delta\boldsymbol{a})}
 (\mathcal{H}_i)\pmod{\text{im}\,\overline{\mathrm{d}}}.
\]
The bra\-cket $\{\,,\,\}_{{\boldsymbol{\mathcal{P}}}}$ is
bilinear and skew\/-\/symmetric by construction;
however, the bracket $\{\,,\,\}\colon\overline H^n(\pi^{{\text{\textup{nC}}}})\times\overline H^n(\pi^{{\text{\textup{nC}}}})\to\overline H^n(\pi^{{\text{\textup{nC}}}})$
does not in general restrict as a \textsl{bi}\/-\/derivation
to the horizontal co\-ho\-mo\-lo\-gy with respect
to~$\overline{\mathrm{d}}$.

\begin{define}
Bracket~\eqref{PBr} is \textsl{Poisson} if it
satisfies the Jacobi identity
\begin{equation}\label{JacobiPoisson}
\sum_{\circlearrowright}\bigl\{ \{\mathcal{H}_1,
 \mathcal{H}_2\}_A,\mathcal{H}_3\bigr\}_A=0,
\end{equation}
which also is
\begin{equation*}
\sum_{s\in S_3} (-)^{|s|}
\smash{\vec{\partial}^{\,(\boldsymbol{a})}_{A(\overrightarrow{\delta\mathcal{H}}_{s(3)}/\delta\boldsymbol{a})}}
\bigl(\tfrac{1}{2}\bigl\langle\smash{\overrightarrow{\delta\mathcal{H}}}_{s(1)}/\delta\boldsymbol{a},
A\bigl(\smash{\overrightarrow{\delta\mathcal{H}}}_{s(2)}/\delta\boldsymbol{a}\bigr)\bigr\rangle\bigr)=0;
\end{equation*}
the operator $A$ in ${\boldsymbol{\mathcal{P}}}=\frac12\langle{\boldsymbol{b}},A({\boldsymbol{b}})\rangle$ is then called a \textsl{Hamiltonian operator}.
\end{define}

\begin{lemma}\label{LiftP}
For each evolutionary vector field $\vec{\partial}_{\varphi}^{\,({\boldsymbol{a}})}$, the induced velocity $\dot{\boldsymbol{p}}=
\mathrm{L}_{\vec{\partial}^{\,(\boldsymbol{a})}_\varphi}(\boldsymbol{p})$
of a variational covector~$\boldsymbol{p}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)$
equals
$\dot{\boldsymbol{p}}=\vec{\partial}^{\,(\boldsymbol{a})}_\varphi(\boldsymbol{p})+
(\boldsymbol{p})\smash{\overleftarrow{ \bigl(\ell^{(\boldsymbol{a})\,\dagger}_\varphi
\bigr)}}$\textup{,}
where

$\vec{\partial}^{\,(\boldsymbol{a})}_\varphi$ acts on~${\boldsymbol{p}}$ componentwise.
\end{lemma}

The proof is straightforward: it amounts to a definition of the Lie derivative~$\mathrm{L}$.

\begin{rem}
The tempting notation $\vec{\partial}^{\,(\boldsymbol{a})}_{A(\boldsymbol{b})}(\boldsymbol{\mathcal{P}}\newcommand{\cR}{\mathcal{R}})
\bigl(\smash{\bigotimes^3 \overrightarrow{\delta\mathcal{H}_i}}/\delta\boldsymbol{a}\bigr)=0$
is illegal by Lemma~\ref{LiftP},
which forbids us
to set $\dot{\boldsymbol{p}}\equiv0$ at will
so that the vector field $\vec{\partial}^{\,(\boldsymbol{a})}_{\varphi}$ would be ill\/-\/defined
on $\overline{J^\infty}\bigl(\Pi\widehat{\pi}^{\text{nC}}_\pi\bigr)$.
Nevertheless, the Jacobi identity~\eqref{JacobiPoisson} is equivalent to the classical (non)\-com\-mu\-ta\-tive master\/-\/equation
\begin{equation}\label{ClassMaster}
\smash{\boldsymbol{Q}^{\boldsymbol{\mathcal{P}}\newcommand{\cR}{\mathcal{R}}}}(\boldsymbol{\mathcal{P}}\newcommand{\cR}{\mathcal{R}})=
[\![\boldsymbol{\mathcal{P}}\newcommand{\cR}{\mathcal{R}},\boldsymbol{\mathcal{P}}\newcommand{\cR}{\mathcal{R}}]\!]=0
\end{equation}
upon the noncommutative variational Poisson bi-vector~${\boldsymbol{\mathcal{P}}}$.
\end{rem}

\begin{example}\label{ExSkewIsHam}
Every skew\/-\/adjoint linear total differential operator $A\colon{\boldsymbol{p}}\mapsto\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}})}$
whose coefficients belong to the sub\/-\/ring $C^{\infty}(M^n_{{\text{\textup{nC}}}})$
--in particular, with constant coefficients-- is
a Hamiltonian operator.
\end{example}

Satisfying master\/-\/equation~\eqref{ClassMaster},
each Poisson bi\/-\/vector~$\boldsymbol{\mathcal{P}}$ yields
the Poisson\/--\/Lich\-ne\-ro\-wicz differential~${\boldsymbol{Q}}^{{\boldsymbol{\mathcal{P}}}}$
and the respective cohomology.
The cohomological Lenart\/--\/Magri scheme
(see~\cite{Kiselev:Christoffel} and references therein)
then provides us with a concept
of noncommutative infinite\/-\/dimensional completely integrable systems.

\begin{rem}
One has
to pay due attention to the understanding of noncommutative evolution
equations because of the subtle difference between the equalities of \textsl{words}, so that the velocities are defined up to cyclic permutations, and the equalities of corresponding evolutionary \textsl{vector fields}, which are uniquely defined by the derivatives that are firmly attached to the marked points~$\boldsymbol{\infty}$ on the circles~$\mathbb{S}^1$.
\end{rem}

\begin{rem}
An extension of a given commutative differential equation to the noncommutative ({\textrm{e.\,g.}}, to cyclic\/-\/invariant)
setup is a nontrivial problem if one requires that the richness of the original equation's geometry must be preserved.
Meaningful noncommutative generalizations are known for the Burgers, KdV, modified KdV, the nonlinear Schr\"odinger
equation (NLS), and for several other commutative
integrable systems (e.g., see the seminal paper~\cite{Kiselev:OlverSokolovCMP1998}).
Nevertheless, we stress that this particular
transition between the two worlds, (graded-)\/commutative $\longmapsto$ associative, \textsl{may not}\footnote
Let us recall that
each cyclic word describes the
field $\mathcal{A}\bigl([{\boldsymbol{a}}({\boldsymbol{x}},{\vec{\mathbf{x}}})]\bigr)$ which co\/-\/exists in the states
$\mathcal{A}$,\ ${\mathsf{t}}(\mathcal{A})$,\ $\dots$,\ ${\mathsf{t}}^{\lambda(\mathcal{A})-1}(\mathcal{A})$ at all
points of the space\/-\/time~$M^n$.
The interaction $|\mathcal{A}_1\rangle\otimes|\mathcal{A}_2\rangle\mapsto|\mathcal{A}_1\times\mathcal{A}_2\rangle$
is such that it does not matter
(up to \textsl{left}${}={}$\textsl{right}~mod $O(\hbar)$)
whether $|\mathcal{A}_1\rangle$ scatters on $|\mathcal{A}_2\rangle$ or \textrm{vice versa} because
the multiplication $\times$ in~${{\EuScript A}}$ is
commutative.
(The new word $\mathcal{A}_1\times\mathcal{A}_2$ could then simplify
under a factorization over the extra relations,

which we do not address
here.)
It then becomes a nontrivial task to restore the \textsl{associativity} of the scattering, {i.\,e.}, the independence of the out\/-\/going state from the order of two pairwise collisions.
The deformation quantization technique~\cite{Kiselev:KontsevichFormality}
which extends the multiplication~$\times$ to the power
series~$\star=\times+\hbar\cdot\{\,,\,\}_{\text{Poisson}}+o(\hbar)$
in the Planck constant~$\hbar$
is the key to
solution of the triangle equation
$|1\rangle\times\bigl(|2\rangle\times|3\rangle\bigr)=
 \bigl(|1\rangle\times|2\rangle\bigr)\times|3\rangle$.
Therefore, it is the matching of the two channels in the associative scattering
which has
a quantum origin
}
be appropriately called a ``quantization.''
\end{rem}

\section{Noncommutative substitution principle}\label{SecSubst}
\noindent%
In order to make working the proofs of known convenient reformulations of the Jacobi identity $[\![{\boldsymbol{\mathcal{P}}},{\boldsymbol{\mathcal{P}}}]\!]=0$ (e.g., see the next section), let us establish the substitution principle~\cite[Problem~5.32${}^*$]{Kiselev:Olver} in the fully noncommutative setup.

\begin{rem}
We emphasize that for the substitution principle to hold one does not require that the setup is commutative.
Moreover, the noncommutative substitution principle

hints us why the base of the
bundle~$\pi^{{\text{\textup{nC}}}}$ should itself be a noncommutative manifold: indeed, the proof then becomes particularly transparent, also specifying as the only admissible the class of horizontal bundles over~$J^\infty(\pi^{{\text{\textup{nC}}}})$ whose sections~${\boldsymbol{p}}$ naturally bear the marked points~$\pmb{\infty}$.
(For instance, such is the case of noncommutative variational (co)\/vectors: the derivatives $\vec{\partial}/\partial\vec{D}^\tau({\boldsymbol{a}}_\sigma)$ in a vector and the differentials ${\mathrm d}\vec{D}^\tau({\boldsymbol{a}}_\sigma)$ in a covector play the exclusive r\^ole of
the marked points~$\pmb{\infty}$ on the circles that carry the cyclic words in the formal sum~${\boldsymbol{p}}$. Consequently,
the position of the lock on such necklaces~$\mathbb{S}^1$ unambiguously determines the rules according to which the sections~${\boldsymbol{p}}$ are never split by any other words
in the course of multiplication.)
We note that the assertion is not valid for sections ${\boldsymbol{p}}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)$ of \textsl{unspecified} nature.
\end{rem}

Let us recall from section~\ref{SecNCJets} that the variables~${\vec{\mathrm{x}}}_i^{\,\pm1}$, $1\leq i\leq n$, are the local generators of associative unital $C^\infty(M^n)$-\/algebras~$C^\infty(M^n_{{\text{\textup{nC}}}})$. This means that we shall not attempt swapping the adjacent letters ${\vec{\mathrm{x}}}_{i_\alpha}$ and ${\vec{\mathrm{x}}}_{i_{(\alpha\mod\ell)+1}}$ in a word $f({\boldsymbol{x}})\cdot{\vec{\mathrm{x}}}_{i_1}^{\,\pm1}\cdot\ldots\cdot{\vec{\mathrm{x}}}_{i_\ell}^{\,\pm1}$.

\begin{theor}[The Substitution Principle]\label{ThSubstPrinciple}
Suppose that a tuple of identities
\[
\boldsymbol{I}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}],
\bigl[{\boldsymbol{p}}({\boldsymbol{x}},{\vec{\mathbf{x}}})\bigr]\bigr)\equiv0
\]
holds on~$J^\infty(\pi^{{\text{\textup{nC}}}})$

for every noncommutative variational \textup{(}co\textup{)}vector ${\boldsymbol{p}}({\boldsymbol{x}},{\vec{\mathbf{x}}})$ whose coefficients belong to the subring $C^\infty(M^n_{{\text{\textup{nC}}}})\hookrightarrow C^\infty(J^\infty(\pi^{{\text{\textup{nC}}}}))$. Then the identities
\[
\boldsymbol{I}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}],\bigl[{\boldsymbol{p}}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)\bigr]\bigr)\equiv0
\]
in total derivatives~\eqref{Kiselev:EqDefTD} with respect to~${\boldsymbol{p}}$ are valid on~$J^\infty(\pi^{{\text{\textup{nC}}}})$ for all \textup{(}local\textup{)} sections~${\boldsymbol{p}}$ depending not only
on~${\boldsymbol{x}}$ and~${\vec{\mathbf{x}}}$ but also endowed with arbitrary, finite differential order dependence on the unknowns~${\boldsymbol{a}}$ and their derivatives.
\end{theor}

\begin{cor}
If, under the assumptions of Theorem~\textup{\ref{ThSubstPrinciple},}
the identities~$\boldsymbol{I}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}],[{\boldsymbol{p}}]\bigr)\equiv0$ in total derivatives with respect to~${\boldsymbol{p}}$ hold on~$J^\infty(\pi^{{\text{\textup{nC}}}})$ for \textsl{exact} variational covectors ${\boldsymbol{p}}=\overrightarrow{\delta\mathcal{H}}/\delta{\boldsymbol{a}}$ which are obtained by variation of arbitrary functionals $\mathcal{H}\in\overline{H}^n(\pi^{{\text{\textup{nC}}}})$, then these identities hold for \textsl{all} covectors ${\boldsymbol{p}}\in\Gamma\bigl((\pi^{{\text{\textup{nC}}}}_\infty)^*
(\widehat{\pi^{{\text{\textup{nC}}}}})\bigr)$, i.e., not necessarily exact.
\end{cor}

\noindent%
Indeed, it is always possible to represent locally an $({\boldsymbol{x}},{\vec{\mathbf{x}}})$-\/dependent section $\sum\nolimits_{j=1}^m p_j({\boldsymbol{x}},{\vec{\mathbf{x}}})\cdot\delta a^j$ as the right variation $\vec{\delta}\mathcal{H}$ of the functional
$\sum\nolimits_{j=1}^m \int p_j({\boldsymbol{x}},{\vec{\mathbf{x}}})\cdot a^j\,{\mathrm d}{\boldsymbol{x}}$ and then apply Theorem~\ref{ThSubstPrinciple}.

\begin{proof}[Proof of Theorem~\textup{\ref{ThSubstPrinciple}}]
For the sake of brevity, let ${\boldsymbol{p}}$ consist of just one word written in the alphabet of~$C^\infty(J^\infty(\pi^{{\text{\textup{nC}}}}))$.
The crucial idea is that the position of the lock~$\pmb{\infty}$ is fixed on the circles which carry the word~${\boldsymbol{p}}$. This means that, whenever one declares an arbitrary differential dependence of~${\boldsymbol{p}}$ on~${\boldsymbol{a}}$, the words~$\boldsymbol{I}$ in principle lengthen but still ${\boldsymbol{p}}$~is never torn in between every occurence of the letters~${\boldsymbol{a}}$ under the multiplication~$\times$. Namely,
during the evaluation of~$\boldsymbol{I}$
the word~${\boldsymbol{p}}$ is unlocked,
the letters~${\vec{\mathbf{x}}}$ and $\vec{D}^\tau({\boldsymbol{a}}_\sigma)$ and the coefficient which depends on~${\boldsymbol{x}}$ are then stretched to an open string (ordered counterclockwise) and finally, this string is pasted into~$\boldsymbol{I}$ without splitting, i.e., the adjacent letters of~${\boldsymbol{p}}$ never become separated by any other symbols. (We note that this scenario is realized irrespectively of a presence or absence of~${\boldsymbol{a}}$'s in the factor~${\boldsymbol{p}}$, which is in contrast with formula~\eqref{EqRosary}.)

The total derivatives~\eqref{Kiselev:EqDefTD}

then work by their definition: under a restriction of~$\boldsymbol{I}$ (hence of~${\boldsymbol{p}}$) onto the jet~$j_\infty({\boldsymbol{s}})$ of a section ${\boldsymbol{a}}={\boldsymbol{s}}({\boldsymbol{x}},{\vec{\mathbf{x}}})$, each symbol~$a^j$ is replaced with the respective sum of open strings~$s^j({\boldsymbol{x}},{\vec{\mathbf{x}}})$ so that left derivations~\eqref{Kiselev:EqDefTD}, either in~$\vec{D}^\tau({\boldsymbol{a}}_\sigma)$ if there is an explicit dependence of~$\boldsymbol{I}$ on~$[{\boldsymbol{a}}]$ or the
derivatives falling on~${\boldsymbol{p}}$, then reduce to the left derivations~$\vec{\partial}_i$ and~${\vec{\mathfrak{d}}}^i$ in~$C^\infty(M^n_{{\text{\textup{nC}}}})$. By the initial assumption of the theorem, its assertion is valid for any strings in the basic alphabet~$({\boldsymbol{x}},{\vec{\mathbf{x}}})$ replacing the entries~${\boldsymbol{p}}$ in~$\boldsymbol{I}$.
(We note that one does not even have to postulate that the sections~${\boldsymbol{a}}={\boldsymbol{s}}({\boldsymbol{x}},{\vec{\mathbf{x}}})$ inserted in the explicit dependence of~$\boldsymbol{I}$ on~$[{\boldsymbol{a}}]$ coincide with the sections now standing for~${\boldsymbol{a}}$ in the implicit dependence $\bigl[{\boldsymbol{p}}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)\bigr]$.) Hence we conclude that the identities $\boldsymbol{I}\equiv0$ hold on~$J^\infty(\pi^{{\text{\textup{nC}}}})$ for the full set of arguments of the (co)\/vec\-tors.
\end{proof}

\begin{rem}
The proof remains literally valid in the case of evolutionary vector fields instead of variational covectors. This is important for the description of variational noncommutative symplectic structures.
However, the proof reveals why this noncommutative extension of the substitution principle does \textsl{not} hold for arbitrary sections
${\boldsymbol{p}}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)$ of generic horizontal bundles over~$J^\infty(\pi^{{\text{\textup{nC}}}})$.
\end{rem}

\section{Noncommutative $QP$-\/structures}
\noindent%
We now prove
a
criterion

which states whether a given differential operator~$A$ is Hamiltonian
and endows the space $\overline H^n(\pi^{{\text{\textup{nC}}}})$ of functionals on $J^{\infty}(\pi^{{\text{\textup{nC}}}})$ with a Poisson bracket.

\begin{theor}\label{ThAlmostCriterion}
Let $A\colon{\boldsymbol{p}}\in P\mapsto\smash{\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}})}}$ be
a skew\/-\/adjoint linear \textup{(}non\textup{)}\/commutative total differential operator with involutive image, meaning that,
due to the Leibniz rule,
\begin{equation}\label{EqOplusB}
\bigl[A({\boldsymbol{p}}_1),A({\boldsymbol{p}}_2)\bigr]

= A\bigl( \vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)}({\boldsymbol{p}}_2) -
\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_2)}({\boldsymbol{p}}_1) + \ib{{\boldsymbol{p}}_1}{{\boldsymbol{p}}_2}{A}\bigr)
\end{equation}
for any variational covectors~${\boldsymbol{p}}_1$,\ ${\boldsymbol{p}}_2\in\widehat{\varkappa(\pi^{{\text{\textup{nC}}}})}$.

Take the bi\/-\/vector
${\boldsymbol{\mathcal{P}}}=\frac12\langle{\boldsymbol{b}},A({\boldsymbol{b}})\rangle$ and
let ${\boldsymbol{Q}}^A=\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{b}})}-\frac12\vec{\partial}^{\,({\boldsymbol{b}})}_{
\ib{{\boldsymbol{b}}}{{\boldsymbol{b}}}{A}}$
be an odd\/-\/parity
vector field\,\footnote{We recall from~\cite{Kiselev:Galli10} that

the odd\/-\/parity vector field~${\boldsymbol{Q}}^A$ is a Lie\/-\/algebroid
differential.}
on the total space $\overline{J^{\infty}}(\Pi\widehat\pi_{\pi})$
of the \textup{(}non\textup{)}\/commutative variational cotangent bundle over~$\pi^{{\text{\textup{nC}}}}$.

Then the operator~$A$ is Hamiltonian if and only if
\[
{\boldsymbol{Q}}^A({\boldsymbol{\mathcal{P}}})=0\in\overline{H}^n(\Pi\widehat{\pi}_\pi).
\]
\end{theor}

\begin{rem}
The equality $A=-A^{\dagger}$ of the mapping
$A\colon P\to\varkappa(\pi^{{\text{\textup{nC}}}})$ to the space of evolutionary vector fields and, respectively,
the mapping $A^{\dagger}\colon\widehat{\varkappa(\pi^{{\text{\textup{nC}}}})}\to\widehat P$ from the $\langle\,,\,\rangle$-\/dual space
$\Hom_{C^\infty(J^\infty(\pi^{{\text{\textup{nC}}}}))}\bigl(\varkappa(\pi^{{\text{\textup{nC}}}}),
\overline{\Lambda}^n(\pi^{{\text{\textup{nC}}}})\bigr)$
of variational covectors uniquely determines the domain $P=\widehat{\varkappa(\pi^{{\text{\textup{nC}}}})}$ of~$A$.
For instance, such are the (non)\/commutative Hamiltonian operators;
moreover, the ``only if'' statement is their well\/-\/known property
(see~\cite{Kiselev:Galli10} and references therein).
\end{rem}

\begin{proof}
We note that if $\ib{\,}{\,}{A}\equiv0$, there is nothing to prove
(see Example~\ref{ExSkewIsHam} on p.~\pageref{ExSkewIsHam}). From now on we suppose that $\ib{\,}{\,}{A}\not\equiv0$.

Using the Leibniz rule, we deduce that
\begin{align*}
0&=
{\boldsymbol{Q}}^A({\boldsymbol{\mathcal{P}}})=\tfrac12\langle-\tfrac12\ib{{\boldsymbol{b}}}{{\boldsymbol{b}}}{A},A({\boldsymbol{b}})\rangle
-\tfrac12\langle{\boldsymbol{b}},\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{b}})}(A)({\boldsymbol{b}})\rangle
+\tfrac12\langle{\boldsymbol{b}},\tfrac12A(\ib{{\boldsymbol{b}}}{{\boldsymbol{b}}}{A})\rangle,\\
\intertext{where we integrate by parts in the first term of the
right\/-\/hand side and obtain}
{}&\cong\tfrac12\cdot
\left
\tfrac12\langle{\boldsymbol{b}},A(\ib{{\boldsymbol{b}}}{{\boldsymbol{b}}}{A})\rangle+\langle{\boldsymbol{b}},\underline{-\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{b}})}(A)({\boldsymbol{b}})
+\tfrac12A(\ib{{\boldsymbol{b}}}{{\boldsymbol{b}}}{A})}\rangle\right).\\
\intertext{The evaluation~\eqref{EvalkVector} of the underlined factor in the second coupling at any two covectors yields zero so that
there only remains}
{}&
\tfrac1
\cdot\langle{\boldsymbol{b}},A(\tfrac12\ib{{\boldsymbol{b}}}{{\boldsymbol{b}}}{A})\rangle.
\end{align*}
By the hypothesis of the theorem, the value of this tri\/-\/vector at any three covectors ${\boldsymbol{p}}_1$, ${\boldsymbol{p}}_2$, and~${\boldsymbol{p}}_3\in\widehat{\varkappa(\pi)}$

is equal to
\begin{equation}\label{Ogryzok}
\frac{1}{2}\cdot
\sum\limits_{\circlearrowright}\langle{\boldsymbol{p}}_1,A(\ib{{\boldsymbol{p}}_2}{{\boldsymbol{p}}_3}{A})\rangle=0\in\overline{H}^n(\pi^{{\text{\textup{nC}}}}).
\end{equation}
We claim that the sum~\eqref{Ogryzok} is zero if and only if
$A$~is a Hamiltonian operator ({\textrm{i.\,e.}}, whenever it
induces the Poisson bracket $\{\,,\,\}_A$ which satisfies the Jacobi identity).

Indeed, define a bi\/-\/linear skew\/-\/symmetric bracket $\{\,,\}_A$ for any~$\mathcal{H}_i,\mathcal{H}_j\in\overline{H}^n(\pi)$ by the rule
\[
\{\mathcal{H}_i,\mathcal{H}_j\}_A\mathrel{{:}{=}}\bigl\langle\overrightarrow{\delta\mathcal{H}_i}/\delta{\boldsymbol{a}},
A\bigl(\overrightarrow{\delta\mathcal{H}_j}/\delta{\boldsymbol{a}}\bigr)\bigr\rangle\cong\vec{\partial}^{\,({\boldsymbol{a}})}_{A(\overrightarrow{\delta\mathcal{H}_j}/\delta{\boldsymbol{a}})}(\mathcal{H}_i).
\]
Now let $\mathcal{H}_1$,\ $\mathcal{H}_2$, and~$\mathcal{H}_3\in\overline{H}^n(\pi^{{\text{\textup{nC}}}})$ be three arbitrary functionals and put ${\boldsymbol{p}}_i\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)=\overrightarrow{\delta\mathcal{H}_i}/\delta{\boldsymbol{a}}$ for $i=1,2,3$.
Then the left\/-\/hand side of the Jacobi identity for
the bracket~$\{\,,\,\}_A$ is, up to a nonzero constant factor, equal to
\begin{multline}
\sum_\circlearrowright \vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)} \langle{\boldsymbol{p}}_2,A({\boldsymbol{p}}_3)\rangle\\
{}= \sum_\circlearrowright\left(
 \langle\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)}({\boldsymbol{p}}_2),A({\boldsymbol{p}}_3)\rangle
+\langle{\boldsymbol{p}}_2,\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)}(A)({\boldsymbol{p}}_3)\rangle
+\bigl\langle{\boldsymbol{p}}_2,A\bigl(\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)}({\boldsymbol{p}}_3)\bigr)\bigr\rangle
\right).\label{DivideEt}
\end{multline}
Let us integrate by parts in the last term of the right\/-\/hand side;
at the same time, using the property $\vec{\ell}^{\,({\boldsymbol{a}})}_{{\boldsymbol{p}}_2}=\vec{\ell}^{\,({\boldsymbol{a}})\dagger}_{{\boldsymbol{p}}_2}$ of the exact covector~${\boldsymbol{p}}_2=\overrightarrow{\delta\mathcal{H}_2}/\delta{\boldsymbol{a}}$
(we assume that the topology of the bundle~$\pi^{{\text{\textup{nC}}}}$ matches the hypotheses of
the Helmholtz theorem
\cite{Kiselev:Olver}), we conclude that the first term in the right\/-\/hand side of~\eqref{DivideEt} equals
\begin{multline*}
\langle\vec{\ell}^{\,({\boldsymbol{a}})}_{{\boldsymbol{p}}_2}\bigl(A({\boldsymbol{p}}_1)\bigr),A({\boldsymbol{p}}_3)\rangle
=\langle\vec{\ell}^{\,({\boldsymbol{a}})\dagger}_{{\boldsymbol{p}}_2}\bigl(A({\boldsymbol{p}}_1)\bigr),A({\boldsymbol{p}}_3)\rangle
\cong\langle\vec{\ell}^{\,({\boldsymbol{a}})}_{{\boldsymbol{p}}_2}\bigl(A({\boldsymbol{p}}_3)\bigr),A({\boldsymbol{p}}_1)\rangle
=\\=\langle\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_3)}({\boldsymbol{p}}_2),A({\boldsymbol{p}}_1)\rangle.
\end{multline*}
Continuing the equality from~\eqref{DivideEt}, we have
\begin{align*}
\cong\sum_\circlearrowright&\left(
\langle{\boldsymbol{p}}_2,\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)}(A)({\boldsymbol{p}}_3)\rangle
+\underline{\langle\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_3)}({\boldsymbol{p}}_2),A({\boldsymbol{p}}_1)\rangle
-\langle\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_1)}({\boldsymbol{p}}_3),A({\boldsymbol{p}}_2)\rangle}\right).\\
\intertext{The $3+3$ underlined terms cancel in the sum, whence we obtain the equality}
{}&=\sum_\circlearrowright
\langle{\boldsymbol{p}}_1,\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_3)}(A)({\boldsymbol{p}}_2)\rangle
=\frac{1}{2}\sum_\circlearrowright
\langle{\boldsymbol{p}}_1,\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_3)}(A)({\boldsymbol{p}}_2)
-\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{p}}_2)}(A)({\boldsymbol{p}}_3)\rangle\\
{}&=-\frac{1}{2}\cdot\sum_\circlearrowright
\bigl\langle{\boldsymbol{p}}_1,A\bigl(\ib{{\boldsymbol{p}}_2}{{\boldsymbol{p}}_3}{A}\bigr)\bigr\rangle.
\end{align*}
By~\eqref{Ogryzok}, this expression equals zero.\footnote{Moreover, the substitution principle (see Theorem~\ref{ThSubstPrinciple}) states that if such an expression vanishes for the \textsl{exact} covectors ${\boldsymbol{p}}_i=\overrightarrow{\delta\mathcal{H}_i}/\delta{\boldsymbol{a}}$, then it vanishes for \textsl{all} sections~${\boldsymbol{p}}_i\in\widehat{\varkappa(\pi)}$.}
Therefore, the Jacobi identity for the bracket~$\{\,,\,\}_A$ holds, whence the operator~$A$ is Hamiltonian. The proof is complete.
\end{proof}

\begin{rem}
Theorem~\ref{ThAlmostCriterion} converts the \textsl{quadratic} equation
$[\![{\boldsymbol{\mathcal{P}}},{\boldsymbol{\mathcal{P}}}]\!]=0$ $\Leftrightarrow$ $\smash{({\boldsymbol{Q}}^{{\boldsymbol{\mathcal{P}}}})^2}=0$
to a \textsl{bi\/-\/linear} condition ${\boldsymbol{Q}}^A({\boldsymbol{\mathcal{P}}})=0$
upon the bi\/-\/vector~${\boldsymbol{\mathcal{P}}}$ and
$Q$-\/\textsl{structure} for the infinite jet
superspace~$\overline{J^\infty}(\Pi\widehat{\pi}_\pi)$,\ which is endowed with the odd Poisson bracket~$[\![\,,\,]\!]$,~\textrm{c.\,f.}~\cite{AKZS}.
\end{rem}

\begin{rem}
The odd\/-\/parity evolutionary vector field $\smash{{\boldsymbol{Q}}^A}$ was introduced
\textsl{formally} by postulating that the image of~$A$ is involutive,
whence we deduced~\cite{Kiselev:Galli10} that ${\boldsymbol{Q}}^A$~is a differential.
Unfortunately, it follows from nowhere that such evolutionary
vector field should at the same time
be \textsl{geometric}, \textrm{i,\,e.}, why
the velocities of the va\-ri\-ab\-les~${\boldsymbol{b}}$

evaluated on true sections ${\boldsymbol{b}}=\Pi{\boldsymbol{p}}\bigl(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]\bigr)$ of the noncommutative variational cotangent bundle

over $J^{\infty}(\pi^{{\text{\textup{nC}}}})$ do coincide, up to a necessity
to recover the first two
stan\-dard summands in the right\/-\/hand side of
expansion~\eqref{EqOplusB},
with such sections' velocities
\
\mathrm{L}_{\vec{\partial}^{\,({\boldsymbol{a}})}_{A(\Pi{\boldsymbol{p}}(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}]))}}\bigl(\Pi{\boldsymbol{p}}(({\boldsymbol{x}},{\vec{\mathbf{x}}}),[{\boldsymbol{a}}])\bigr)
\
induced by the evolution
$\vec{\partial}^{\,({\boldsymbol{a}})}_{A({\boldsymbol{b}})}$
on the unknowns~${\boldsymbol{a}}$, c.f.\ Lemma~\ref{LiftP} on p.~\pageref{LiftP}.
\end{rem}

\begin{example
Formal evolutionary vector fields~\eqref{QPoisson}

\textsl{are} geometric for all Hamiltonian operators (whence we deduce that Lemma~\ref{LiftP} calculates the respective
bi\/-\/differential Christoffel symbols $\ib{\,}{\,}{A}$).
\end{example}

On the other hand, if we accept
that
all homological evolutionary vector fields
$\smash{{\boldsymbol{Q}}^A}$
are geometric, then we strengthen Theorem~\ref{ThAlmostCriterion} as follows:
\textit{every skew\/-\/adjoint linear total differential operator with involutive image
is a Hamiltonian operator.}

\subsection*{Conclusion
There has been much debate about a proper construction of differential calculus for a proper choice of noncommutative geometry in models of mathematical physics.

We have shown that the traditional calculus, in the form as we know it from Newton and Leibniz (or Gel'\-fand and Dorfman), itself is noncommutative because it does not appeal to the commutativity assumption. Moreover, we have demonstrated that within the setup of cyclic\/-\/invariant words the calculus becomes a topological theory so that all basic operations amount to the standard pair of pants. We also introduced the fully noncommutative bases
of the bundles~$\pi^{{\text{\textup{nC}}}}$ in such a way that their sections acquire the nature of paths (meaning here \textsl{walks} but not Feynman's trajectories, which are
grasped later by the construction of jet spaces over~$\pi^{{\text{\textup{nC}}}}$);
consequently, the total spaces acquire the nature of \textsl{knot bundles}. We anticipate that our new approach and techniques are applicable to an axiomatic description of the geometry of fundamental interactions in (non)\/commutative space\/-\/time at Planck scale. The
physical sense of
this mathematical picture will be the subject of another paper.

\subsubsection*{Acknowledgements}
The author thanks the Organizing committee of
the workshop `Noncommutative algebraic geometry and its applications to physics' (Lorentz Centre, The Netherlands, 2012)
for partial support and a warm atmosphere during the meet\-ing.
The author is grateful to
J.~W.~van de~Leur, S.~M.~Natanson, and
P.~J.~Olver for stimulating remarks and thanks
Yu.~I.~Manin, V.~N.~Roubtsov, and V.~V.~Sokolov
for helpful discussions at the Max Planck Institute for
Mathematics~(Bonn).

This research was supported in part
by NWO~VENI grant~639.031.623 (Utrecht) and JBI~RUG
project~103511 (Groningen).
A~part of this research was done while the author was visiting at
the $\smash{\text{IH\'ES}}$ (Bures\/-\/sur\/-\/Yvette);
the financial support and hospitality of this institution are gratefully acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Metallica's Kirk Hammett releases first song of upcoming solo album
The legendary Metallica guitarist Kirk Hammett released "High Plains Drifter", the first single of his upcoming debut solo album "Portals". The instrumental record will be out next April 23.
Hear the new Kirk Hammett song "High Plains Drifter":
In an official statement, Hammett talked about the record, saying: "Initially, before I even had the idea for a solo EP, I was inspired by the need to create some sort of soundtrack music to accompany The Kirk Hammett Collection for the first "It's Alive" exhibition at the Peabody Essex Museum in Salem, Massachusetts, 2017. I wanted to conceive music to play on a loop in the background as people walked through the exhibit. I sat down one night with a progression, and before I knew it, all the parts were there for 'Maiden and the Monster.'"
"The initial concept for Portals started with that one song. Following that, I realized I could create different soundtrack moments. These songs are what I call 'Audio Cinematic;' I'm creating sounds and pieces of music for the movies playing in my head. Hopefully they'll create movies in other people's heads in a similar fashion."
Other musicians who were also on the record are drummers Jon Theodore (Kirk's fellow Wedding Band member) and Abraham Laboriel, Greg Fidelman on bass, Emmy-winning arranger Blake Neely, and Bob Rock (Metallica's 'Black Album' producer).
The official press-release of the album explained Hammett inspirations, saying:
"Recorded around the world, from Los Angeles to Paris to Oahu, Portals is the result of Kirk's life's journeys and an invitation to tap into your own inner world. Heavily influenced by classical music, soundtracks, horror movies, and maybe a little Ennio Morricone, Kirk wanted the EP to span the decades and touch upon all the horror he loves. The idea behind the EP was to have all the songs flow into each other. So he relied on a whole cast of his friends to record the four songs."
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