In this Section we present the experimental validation of our W2WNet. So far,
there is no agreed upon benchmark protocol to evaluate learning methods in the
way they handle measurement and labelling noise. Therefore, we started from
two well-known public datasets, the MNIST mnist and the CIFAR10 cifar, and we
artificially corrupted such datasets in a controlled way. By doing so, we
tried to replicate different types of real-world noisy scenarios:

  1. [label=()]

  2. 1.

Labelling noise (labels from a different classification task).
In text processing, handwritten character classifications are typical
mainstream tasks for CNNs. The MNIST dataset, that is made of 60000 black and
white images of handwritten digits (0 to 9), was corrupted by adding a
controlled percentage of alien samples randomly extracted from the EMNIST
dataset [emnist], which contains handwritten alphabetical characters. Hence,
the resulting corrupted dataset, referred to as _Sp-MNIST_ , contains either
digits (that are still the majority of the images) and letters, all with a
white foreground and black background. By doing so, we simulate a real-world
situation where a pre-processing pipeline may produce spurious samples to a
downstream classifier that was specifically trained on digit classification,
due to text parsing errors. This scenario is similar to any other instances of
data corruption, where the spurious samples share the same characteristics of
the meaningful ones in terms of color range and encoding, but belong to
different classifications tasks (in this case, digits and alphabets).

  3. 2.

Labelling and measurement noise (labels from the same classification task).
As anticipated in Section LABEL:sec:intro, in natural image classification,
datasets may be corrupted by both labelling and measurement noise.
Mislabelling may sometimes occur due to errors during the automatic collection
of a large amount of annotations from the Internet (for example, by extracting
tags from the surrounding texts or keywords from search engines). On the other
hand, measurement errors can always occur because of problems with acquisition
and storage of the images. To simulate such scenarios, we exploited the
CIFAR10 dataset, which consists of 50000 32x32 RGB images of 10 classes of
natural objects. As regards to labelling, the dataset was artificially
corrupted by two different types of noise patterns: symmetric and pair. In the
former, original labels are randomly flipped to another label. In the latter,
labels are systematically flipped to the subsequent one. Both the patterns are
well know in literature, as they are experienced in several image
classification tasks [kohler2019uncertainty]. As regards to measurement noise,
we picked a random pool of images from CIFAR10 and applied three different
types of transformations: (i) blurring, via a median filter with kernel size
11; (ii) random cropping; (iii) random scaling. Even in this case, such image
degradation is widely reported by literature, and known to be troublesome for
CNN learning in many classification tasks [dodge2016understanding]. As a
result of our artificial corruptions, in the final dataset, referred to as
_Sp-CIFAR10_ , a known subset of images are either given a wrong label (which,
differently from the previous case, belongs to the same classification task of
the original dataset), or altered in terms of image definition, scale and
dynamic range.

To push the capabilities of our methodology to its limits, for both the above-
mentioned settings, we introduced increasing amount of spurious samples
(respectively, 10, 20 and 30% of the size of the original dataset). A full
characterization of the obtained validation datasets is reported in Table 1.
In this table, each dataset is referred to as $[Sp]-name-[N]$, where the $Sp$
prefix indicates the presence of spurious samples, $name$ is the acronym of
the original dataset and $N$ is the percentage of spurious samples with
respect to the total size of the corresponding original dataset.

Table : Validation benchmarks: number of images 2*Dataset | Train | Test
---|---|---
| Meaningful | Spurious | Meaningful | Spurious
MNIST | 60000 | - | 10000 | -
$Sp-MNIST-10$ | 60000 | 6000 | 10000 | 1000
$Sp-MNIST-20$ | 60000 | 12000 | 10000 | 2000
$Sp-MNIST-30$ | 60000 | 18000 | 10000 | 3000
$CIFAR10$ | 50000 | - | 10000 | -
$Sp-CIFAR10-10$ | 50000 | 5000 | 10000 | 1000
$Sp-CIFAR10-20$ | 50000 | 10000 | 10000 | 2000
$Sp-CIFAR10-30$ | 50000 | 15000 | 10000 | 3000

##  Data cleansing capability

As a matter of principle, our _W2WNet_ should follow three fundamentals: (i)
if spurious samples are present, it should remove as many as possible (i.e.
high sensitivity); (ii) while removing spurious samples, it should remove as
little meaningful ones as possible, as they might be essential for the
training of the model (i.e. high specificity); (iii) it should be able to
handle datasets that do not contain any spurious samples (that is the ideal
case), and leave them untouched. To assess all the mentioned specifications,
we trained and tested our _W2WNet_ both on the corrupted datasets (i.e. the
ones with the $Sp$ prefix in Table 1) as well as on the corresponding original
ones, in the exact configuration of their reference papers. In Figure 1 we
show the results of our experiments. Bars show the average number of images
removed per dataset, separately for the training and for the test phase. In
the former case, _removed_ images means that the model tagged them as spurious
and hence removed them from the training set. In the latter case, the trained
model tagged them as spurious at inference time, by providing a low-confidence
prediction.

[width=0.5]figures/removal_rates-crop.pdf

Figure : _W2WNet_ removal rates in the validation datasets. Error bars
represent standard deviation of values among different classes.

The first plot of Figure 1 reports the percentage of spurious samples which
were correctly identified and removed from the corrupted datasets (i.e. the
sensitivity of the model). The last two plots report the number of meaningful
samples mistakenly tagged as spurious, respectively on the corrupted datasets
and on the original ones. As mentioned earlier, the lower these numbers, the
higher the specificity of the model. As it can be gathered from the first
plot, _W2WNet_ was able to remove at least 30% and at best 70% of the spurious
images, when considering both the training and the test sets. Apart from the
training of the $Sp-CIFAR10$, where it is possible to see a decreasing trend
of the bars, the performance was quite stable at increasing number of spurious
samples in the datasets. The relation between the sensitivity on the training
and test sets was different for the two applications: higher on the training
than on the test set for the $Sp-CIFAR10$ datasets, the opposite for the $Sp-
MNIST$ ones. As it can be gathered from the second plot, _W2WNet_ proved to be
reasonably specific in the corrupted datasets, removing as little as 17% of
meaningful samples in the worst case ($SP-CIFAR-30$) and almost 0% in the best
case ($SP-MNIST$). Finally, by looking at the last plot, the number of
meaningful images that were on average mistaken as spurious in the original
datasets were 5 and 10%, respectively in MINST and CIFAR10. A more thorough
analysis revealed that in both cases these samples are very ambiguous images,
that a human observer can hardly ascribe to any of the training categories
(see Figure 1). Hence, we believe that tagging such images as spurious is
totally reasonable, and more importantly, it does not have a negative impact
on the training, as will be showed later on. Overall, _W2WNet_ is reasonably
sensitive and specific in the identification of spurious samples, and the
reliability of the uncertainty measure, associated with the final prediction,
is proved by our results.

[width=1]figures/difficult_images-crop.pdf

Figure : Examples of images tagged as _spurious_ , respectively from MNIST (a)
and CIFAR datasets (b).

##  Classification performance

At last, to assess the effectiveness of our solution in terms of positive
impact on the classification performance, we compared _W2WNet_ against a
canonical deterministic counterpart on all the datasets reported in Table 1.
For this purpose we exploited a deterministic DenseNet121 model, as it is also
the backbone of our _W2WNet_ architecture, and hence it is totally equivalent
to our model in terms of depth and classification potential. For the training
of the deterministic CNNs, we followed the same procedure described in Section
LABEL:sec:method, with the only difference of having set the MC dropout rate
to zero. The learning rate was set to 0.1 and 0.01, respectively for the
datasets derived from MNIST and CIFAR10. As already anticipated in Section
LABEL:sec:intro, to the best of our knowledge, there is no published
literature on deep learning methods addressing _measurement_ and _labelling_
noise coexisting together. Nonetheless, to better contextualize our
validation, besides our approach and its deterministic counterpart, we also
provide results obtained by representative algorithms facing either
_measurement_ or _labelling_ noise. For the former category, we tested the
methodology by Roy and colleagues [roy2018effects], which leverages on a not
trainable low-pass filter-like CNN layer to reduce the impact of image
degradation on the classification performance. For the latter, we put into
effect the work by Kohler et al., in the configuration made up of a single MC
droput-based classifier with 25 forward passes [kohler2019uncertainty]. For a
fair comparison, both the methods were implemented using a DenseNet121 model
as the backbone. The results of our experiments are reported in Figure 1,
where we show the mean classification accuracy obtained by the four models
(our _W2WNet_ , a deterministic DenseNet121, ad the two literature data
cleansing approaches). As it can be observed from the plot, for all the
approaches, the mean classification accuracy decreases at increasing number of
spurious samples affecting the dataset (from 10 to 30 %, see also Table 1).
This is absolutely consistent with previous literature
[kohler2019uncertainty]. When considering the corrupted datasets, our _W2WNet_
outperforms the deterministic DenseNet121 of a value between 5% and 10%. In
addition, _W2WNet_ overcomes both the baseline literature solutions, which
both behave similarly to DenseNet121. This is not surprising, as both methods
are specifically tailored to address one type of noise solely. By a lesser
margin, the accuracy of our _W2WNet_ was the highest even in the non-corrupted
datasets.

[width=0.8]figures/comparison_accuracies-crop.pdf

Figure : Mean accuracy of _W2WNet_ compared with representative works from
literature. Error bars represent standard deviation of values among different
classes.

##  Real-world case study: histological images classification

Histological image analysis is the gold standard for the diagnosis and gauging
of large number of cancers [ponzio2019dealing]. Typically, when there is a
suspicion of cancer, the patient goes through a biopsy, where a thin layer of
tissue sample is resected, fixed on a slide, and stained (for example, by
Hematoxylin and Eosin). Then, the pathologist analyzes the slide on the
microscope looking for malignancies, which commonly cause alterations of the
normal tissue architecture. The recent diffusion of digital scanners imposed
the transition from standard histological slides to very large born-digital
multi-resolution images called Whole-Slide Images (WSIs, see Figure 1(a)),
whose typical size may be $100,000\times 100,000$ pixels. This is rapidly
changing the workflow of clinical laboratories [wsi]: the traditional visual
evaluation of the samples directly under the microscope is progressively
shifting to Computer-Aided Diagnosis (CAD) systems, encouraging a complete
automatization of downstream image analysis. Recently, researchers have shown
an increased interest in applying DL techniques (most often based on CNNs) to
the automated assessment of the WSIs. Nonetheless, obtaining good quality
training sets for the CNNs is an extremely cumbersome task, involving a number
of steps: (i) manually dividing each WSI into regions of interest (ROIs), that
should be homogeneous in terms of tissue architecture; (ii) manually labelling
ROIs, based on the tissue category (e.g. cancer vs no-cancer, see Figure
1(b)); (iii) cropping ROIs into a regular grid of small tiles, that can be fed
into a CNN together with their corresponding label (the same of the
corresponding ROI, Figure 1(c)). Due to image artifacts, imprecision in the
ROI delineation, or non-homogeneous content of the ROIs, the outcome of this
procedure is typically a dataset which may contain a large amount of spurious
tiles: that is, a significant number of tiles may have a content that is
either too blurred (measurement noise) or unrelated to the label they were
associated to (labelling noise), and then potentially harmful for the training
of the CNN. For example, in Figure 1(e), a number of tiles labeled as _cancer_
contain a prevalence of background glass, which is obviously not meaningful to
the _cancer_ category. This makes it a significant case-study for the
exploitation of our _W2WNet_.

[width=1]figures/roi_cropping-crop.pdf

Figure : Generation of a digital patholohy dataset to train CNNs: typical
automated procedure. (a) Whole Slide Image (WSI). (b) Identification and
labelling of homogeneous Regions of Interest (ROIs). (c) Cropping ROIs into
small tiles, which are all given the same label of the originating ROI. (d)
Meaningful tiles (e) Spurious tiles (that is, tiles whose content is not fully
representative of the given label).

More specifically, in our experiments we refer to the same case study
described in our earlier work [ponzio], focused on Colorectal Cancer (CRC)
categorization. In this case, the classes of interest are three: (i)
Adenocarcinoma (AC), corresponding to recognizable CRC; (ii) Tubulovillous
adenoma (AD), a precursive lesion of CRC, and (iii) Healthy tissue (H). As
detailed in [ponzio], downstream of the automated ROI cropping and labelling
procedure represented in Figure 1, a total number of 19644 non-overlapping
annotated tiles were obtained from 27 different WSIs. After ad-hoc re-
examination of the tiles by a pathologist, 6144 of them were tagged as
spurious, as the prevailing content of such slides (either blood vessels,
adipose cells, background glass or stroma, see figure 1(e)), was not deemed
meaningful to any of the three classes of interest. For training and testing
purposes, the initial cohort of 27 WSIs was randomly split into two disjoint
subsets (18 for training and 9 for testing), roughly balanced with respect to
the classes, and then fed into our _W2WNet_ for data cleansing and
classification. The results of our experiments are shown in Figure 1. As it is
visible from the plot on the left, our framework was able to identify 55% and
58% of the spurious samples from training and test set respectively. The
impact on the classification is shown on the right plot, where we compare the
mean classification accuracy of _W2WNet_ with the one obtained by the
deterministic counterpart, a state-of-the-art DenseNet121 CNN, trained from
scratch with learning rate set to 0.0001 and SGD optimizer. Even in this case,
the accuracy of our proposed solution was higher, by about 9% on average on
the test set.

[width=0.9]figures/CRC-crop.pdf

Figure : Removal rate of spurious samples with _W2WNet_ (left) and
classification accuracy comparison of _W2WNet_ and DenseNet (right) on the CRC
dataset. Error bars represent standard deviation of values among different
classes.


# On the quantization of $C^{\infty}(\mathbb{R}^{d})$

Murray Gerstenhaber  Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104-6395, USA<EMAIL_ADDRESS>To the memory of my
beloved wife, Ruth P. Zager M.D., z”l

###### Abstract.

The Basic Universal Deformation Formula is proven and applied to show that
Weyl algebras, which encode Heisenberg’s uncertainty principle, are effective
deformations of polynomial rings, and that uncertainty is necessary for
stability. Deformation problems may have associated modular groups an
algebraic example of which is given. Poisson structures on
$C^{\infty}(\mathbf{R}^{d})$, shown by Kontsevich to be infinitesimal
deformations integrable to full deformations, here are shown to be the skew
forms of those infinitesimal deformations of $C^{\infty}(\mathbf{R}^{d})$ with
vanishing primary obstructions. In dimension 3 any smooth multiple of such an
infinitesimal again has vanishing primary obstruction. This exceptional
property suggests that in our universe a large disturbance like the Big Bang
can be confined to an arbitrarily small interval in time and almost completely
confined to an arbitrarily small region in space.

###### Key words and phrases:

deformation, obstruction, universal deformation formulas

###### 2020 Mathematics Subject Classification:

16S80, 16E40

The author thanks Jim Stasheff for his help in the preparation of this paper.

##  1\. History

Algebra deformation theory has a long background in the deformation theory of
complex analytic structures but in its present form originated in the author’s
papers [11], [13] and was further developed in [14], [15], [16]. Riemann
studied the ‘moduli’ on which compact Riemann surfaces depend which may be
considered as deforming when the moduli are varied. When different points in
the parameter space yield structures isomorphic to the original there is a
modular group operating on that space. An algebraic example is given here,
analogous to the classical modular group associated with deformations of
compact Riemann surfaces of genus one. Infinitesimal deformations of Riemann
surfaces were first formalized by Teichmüller, [37] as quadratic differential
but the break-through definition of infinitesimal deformations for a complex
manifold of arbitrary dimension as elements of the cohomology in dimension one
of the manifold with coefficients in its sheaf of germs of holomorphic tangent
vectors is due to Frölicher and Nijenhuis, [9]. This led to the comprehensive
works of Kodaira and Spencer; for an overview see [25]. However, obstructions
as well as jump deformations, [4], [16],[20, §7, p. 20ff], were not clearly
understood until the development of algebraic deformation theory. These are
discussed in §6. Deformation Quantization, an extraordinary advance in
algebraic deformation theory introduced by Bayen, Flato, Frønsdal,
Lichnerowitz, and Sternheimer, [1], allows quantum theory to be understood and
the spectrum of the hydrogen atom to be computed without the use of wave
mechanics or Schrödinger’s equation. For a summary and some later
developments, see [35] and [6]. Subsequent to [1], in the context of
algebras,“quantization” has become become almost synonymous with ”deformation”
and is so used here.

##  2\. Overview

Algebraic deformation theory asks the following question: Given an algebra
$\mathcal{A}$ over some commutative unital ring $\mathbf{k}$, in what ways can
one create a “deformation” of that algebra with multiplication of the form

(1) $a\star b=ab+\hbar F_{1}(a,b)+\hbar^{2}F_{2}(a,b)+\cdots\quad$

while remaining in the same “equationally defined category”, so called in
[14], or in present terminology, category of algebras defined over a
particular operad. The notation and term “star product” were introduced in
[1]. The symbol $\hbar$ originally denoted the reduced Planck’s constant,
$h/2\pi$, where $h\approx 6.626176\times 10^{-34}$ joule-seconds is Planck’s
original constant. Its use to denote a deformation parameter is now common and
derives from the fact that Planck’s constant may, in fact, be viewed as such,
cf [1], [17].111Correction to Theorem 8 of [17]: In view of the remark at the
end of §2 (_not_ §1) it should read, “…the coherent twist of the path algebra
induced by $\omega$ is trivial _only if_ the class of $\omega$ is trivial as
an element of $H^{2}(\mathcal{M},\mathbb{R}/\tau\mathbb{R})$.”

For associative algebras, the $F_{i}$ in (1) are Hochschild 2-cochains of
$\mathcal{A}$ with coefficients in $\mathcal{A}$ itself, tacitly extended to
be defined over $\mathcal{A}[[\hbar]]$; we may write $F_{0}$ for the original
multiplication in $\mathcal{A}$. Here $F_{1}$ must be a 2-cocycle, usually
called the infinitesimal of the deformation and a main problem is to determine
which infinitesimals give rise to full deformations. Note that if
$D_{1},D_{2}$ are derivations then $D_{1}\smile D_{2}$ is a 2-cocycle. Given a
2-cocycle $F_{1}$, there generally is a sequence of cohomological obstructions
to constructing the necessary $F_{i},i\geq 2$. The analogue of the foregoing
in the Kodaira-Spencer approach to the deformation of complex analytic
manifolds is already present in an insufficiently recognized paper by Douady
[8]. A full deformation as in (1), if it exists, is often called an integral
of $F_{1}$, something which depends only on the cohomology class of $F_{1}$.
While it is really the class which is the infinitesimal, of which $F_{1}$ is
only a representative, it will be convenient also to call $F_{1}$ ‘the’
infinitesimal of the deformation.

A Universal Deformation Formula (UDF) exhibits in closed form an explicit
integral for some class of 2-cocycles, considered as infinitesimal
deformations. The first such, called here the Basic UDF, was introduced in
[15, Ch. II, Lemma 1, p. 13] and stated there for “composition complexes”,
amongst which are the cohomology groups of algebras, coalgebras, and
simplicial complexes, but the first proof appears in [21]. While needed here
only for deformations of associative algebras the proof in §3 holds for all
composition complexes.

The Basic UDF asserts that if $D_{1},D_{2}$ are commuting derivations of an
algebra $\mathcal{A}$ then a suitably defined exponential of
$D_{1}\\!\\!\smile\\!\\!D_{2}$ is an integral. The star product of (1) is,
however, only a formal power series, while what is generally wanted is that
the deformed algebra be defined over the original coefficient ring
$\mathbf{k}$. It may not be meaningful, a priori, to specialize the
deformation parameter $\hbar$ to an element of $\mathbf{k}$. When this is so
for a particular UDF operating on some algebra, the operation will be called
effective. In §4 it is shown, using the Basic UDF, that the first Weyl
algebra, which encapsulates Heisenberg’s uncertainty relation between position
and momentum, is an effective deformation of the polynomial ring in two
variables; although (1) is formally a power series, when applied to any pair
of terms in the original algebra it terminates in a finite number of steps.
Clifford algebras are, likewise, effective deformations of graded polynomial
rings, when degree is taken into account. The Basic UDF, §9, also gives
another approach to the first UDF with non-commuting derivations, [5].

Kontsevich’s UDF, [26], provides an explicit, but again purely formal,
integral for any Poisson infinitesimal deformation in the form of a power
series in partial derivatives. It uses the fundamental works of Stasheff,
[34], and Schlessinger and Stasheff, [32], which introduced the concepts of
homotopy associative algebras and homotopy Lie algebras, respectively. The
concept of algebras up to homotopy is now also understood for algebras over
operads. Tamarkin, [36], almost simultaneously gave another proof of the
integrability of Poisson structures, but by quite different methods. Cattaneo,
Felder, and Tomassini, [3], extended these results to the quantization of
Poisson structures on a manifold. The coefficients in Kontevich’s UDF are not
rational and are difficult to compute. Dolgushev, [7], later showed that if a
solution exists, then there is a rational one. Determining the Poisson
structures on $C^{\infty}(\mathbb{R}^{d})$ remains an open problem. A power
series integral to an infinitesimal deformation generally does not converge
when the infinitesimal is smooth but not analytic. However, effective
quantization is sometimes possible by using Fourier series, §5.

An infinitesimal deformation, viewed as a cohomology class, has a unique
representative 2-cocycle which is antisymmetric or skew. The primary
obstruction to integrating an infinitesimal deformation of a commutative
algebra $\mathcal{A}$ in which 6 is a unit is shown to vanish if and only if
its unique skew representative is Poisson. Any infinitesimal deformation of
$C^{\infty}(\mathbb{R}^{d})$ with vanishing primary obstruction is therefore,
by the work of Kontsevich, integrable to a formal deformation. It is shown
also that the primary obstruction to an infinitesimal deformation of
$C^{\infty}(\mathbb{R}^{d})$ vanishes if and only if the coefficients of a
representative cocycle satisfy a system of $\binom{d}{3}$ non-linear partial
differential obstruction equations. When $d=2$ there are no obstructions; all
infinitesimal deformations give rise to quantizations. For $d=3$, the single
obstruction equation has a property not present in higher dimensions; it
allows an arbitrarily large deformation to be localized to an arbitrarily
small region in space and an arbitrarily small interval of time, §10. This may
help us to understand the Big Bang.

##  3\. The Basic Universal Deformation Formula

Throughout, $\mathbf{k}$ will denote a commutative unital ground ring and
$\mathcal{A}$ will be a $\mathbf{k}$-algebra. When we have a deformation of
$\mathcal{A}$ with star product as in (1), note that if $a,b\in\mathcal{A}$
then $a\star b$ lies not in $\mathcal{A}$ but in $\mathcal{A}[[\hbar]]$, the
ring of power series in $\hbar$ with coefficients in $\mathcal{A}$. While in
general it is not possible to specialize $\hbar$ to an element of the ground
ring $\mathbf{k}$, it will be shown that this is possible in some important
cases.

The condition that star products must remain associative can be expressed
using special cases of the composition products $\circ_{i}$ and $\circ$
introduced in [11]. If $F,G$ are 2-cochains of $\mathcal{A}$ then
$F\circ_{1}G,\,F\circ_{2}G,\,F\circ G$ are the 3-cochains defined by setting,
respectively,

(2) $(F\circ_{1}G)(a,b,c)=F(G(a,b),c),\quad(F\circ_{2}G)(a,b,c)=F(a,G(b,c)),$

and

(3) $(F\circ
G)(a,b,c)=(F\circ_{1}G-F\circ_{2}G)(a,b,c)=F(G(a,b),c)-F(a,G(b,c)).$

With these, the requirement that $a\star(b\star c)-(a\star b)\star c$ = 0 can
be expressed by the equations

(4) $\sum_{i=0}^{n}F_{i}\circ F_{n-i}\quad=0\quad\text{for all}\quad n.$

Transposing to the right those terms where either $i=0$ or $i=n$, (4) can be
rewritten as

(5) $\sum_{i+j=n,\,i,j>0}F_{i}\circ F_{j}=-\delta F_{n},$

where $\delta$ is the Hochschild coboundary operator. When $n=1$ the left side
vanishes, showing that $F_{1}$ is, as mentioned, a 2-cocycle.

Star products $a\star b=\sum\hbar^{i}F_{i}$ and
$a\star^{\prime}b=\sum\hbar^{i}F_{i}^{\prime}$ are called gauge equivalent if
there is a one-parameter family $\gamma$ of $\mathbf{k}$-linear automorphisms
of $\mathcal{A}$ of the form
$\gamma(a)=a+\hbar\gamma_{1}(a)+\hbar^{2}\gamma_{2}(a_{)}+\cdots$, where the
$\gamma_{i}$ are linear maps from $\mathcal{A}$ to itself, again tacitly
extended to be defined over $\mathcal{A}[[\hbar]]$, such that
$a\star^{\prime}b=\gamma^{-1}(\gamma(a)\star\gamma(b))$. The deformed algebras
that $\star$ and $\star^{\prime}$ define on the underlying $\mathbf{k}$-space
of $\mathcal{A}$ are then isomorphic, and one has
$F_{1}^{\prime}=F_{1}+\delta\gamma_{1}$. As $\gamma_{1}$ can be arbitrary,
$F_{1}$ can be replaced in a gauge equivalent deformation by any cohomologous
2-cocycle, showing as remarked, that the integrability of $F_{1}$ depends only
on its cohomology class.

When $F_{1}$ is a cocycle then so is $F_{1}\circ F_{1}$. Its cohomology class
in $H^{3}(\mathcal{A},\mathcal{A})$, which depends only on the class of
$F_{1}$, is its primary obstruction, a term also commonly applied to
$F_{1}\circ F_{1}$ itself. If $F_{1}\circ F_{1}$ is a coboundary then one can
choose an $F_{2}$ with $-\delta F_{2}=F_{1}\circ F_{1}$ and one can ask if an
$F_{3}$ exists so that one can continue building the series (1). However, (5)
with $n=3$ shows that one may encounter another obstruction in
$H^{3}(\mathcal{A},\mathcal{A})$, and so on indefinitely. If, with a given
2-cocycle $F_{1}$, we are able to construct a series such as that in (1), then
that integral is also said to quantize $\mathcal{A}$. When $\mathcal{A}$ is
the algebra $C^{\infty}(\mathbb{R}^{d})$ of smooth functions on
$\mathbb{R}^{d}$ the integral is also said to quantize $\mathbb{R}^{d}$. While
in principle one may encounter an infinite sequence of obstructions,
Kontsevich’s work, [26], together with what will be shown here, implies that
an infinitesimal deformation $F_{1}$ of $C^{\infty}(\mathbb{R}^{d})$ has only
the primary obstruction; if that one obstruction vanishes, then there is a
star product with the given $F_{1}$.

An algebra $\mathcal{A}$ for which all deformations are gauge equivalent to
the trivial deformation, i.e., the one where the star product is just the
original multiplication, is called rigid. This will be the case if every
infinitesimal deformation is ultimately obstructed and certainly if
$H^{2}(A)=0$, in which case the algebra is called absolutely rigid or,
preferably, stable in the terminology of [9]. A tensor product of stable
algebras need not be stable, as will be seen.

When $H^{3}(\mathcal{A},\mathcal{A})=0$, every infinitesimal deformation is
integrable. In particular, this is the case for $C^{\infty}(\mathbb{R}^{2})$
by the Hochschild-Kostant-Rosenberg (HKR) Theorem, [24]. That theorem asserts
that $H^{*}(C^{\infty}(\mathbb{R}^{d}))$ is isomorphic, as a module over
$C^{\infty}(\mathbb{R}^{d})$, to the exterior algebra generated by the partial
derivatives, $\partial_{1},\dots,\partial_{d}$ with respect to the coordinates
$x_{1},\dots,x_{d}$. In particular, when $d=2$, one has
$H^{3}(C^{\infty}(\mathbb{R}^{2}))=0$, so there are no obstructions to any
infinitesimal deformation of $C^{\infty}(\mathbb{R}^{2})$. (The original
formulation of the HKR Theorem was more restrictive, for a proof of the
present form, see Roger, [30].) Note, however, that the module isomorphism
does not carry the exterior product on
$\wedge^{*}\operatorname{Der}{\mathcal{A}}$ to the cup product on
$H^{*}(\mathcal{A},\mathcal{A})$, cf [2].

If $D_{1},\dots,D_{n}$ is a sequence of derivations of $\mathcal{A}$ (which
need not commute or be distinct), and $I=(i_{1},\dots,i_{r})$ is a subsequence
of $(1,\dots,d)$, set $D_{I}=D_{i_{1}}D_{i_{2}}\cdots D_{i_{r}}$ and let
$I^{c}$ denote the ordered complement of $I$. Then $\delta(D_{1}D_{2}\cdots
D_{n})=-\sum_{I}D_{I}\\!\\!\smile\\!\\!D_{I^{c}}$, where the sum is over all
ordered proper subsequences of $(i_{1},\dots,i_{r})$, i.e., neither empty nor
the whole (a sum over “unshuffles”). In particular, for any pair of
derivations $D_{1},D_{2}$ of $\mathcal{A}$, we have
$\delta(D_{1}D_{2})=-(D_{1}\\!\\!\smile\\!\\!D_{2}+D_{2}\\!\\!\smile\\!\\!D_{1})$.
Therefore, $D_{1}\\!\\!\smile\\!\\!D_{2}$ is cohomologous to
$-D_{2}\\!\\!\smile\\!\\!D_{1}$ and also to
$(1/2)(D_{1}\\!\\!\smile\\!\\!D_{2}-D_{2}\\!\\!\smile\\!\\!D_{1})$, provided
that 2 is a unit. It follows from the HKR Theorem that every 2-cocycle of
$C^{\infty}(\mathbb{R}^{d})$ is cohomologous both to one in normal form
$\sum_{1\leq i<j\leq d}a_{ij}\partial_{i}\\!\\!\smile\\!\\!\partial_{j}$ and
to one in skew form $\sum_{1\leq i<j\leq
d}(a_{ij}/2)(\partial_{i}\\!\\!\smile\\!\\!\partial_{j}-\partial_{j}\\!\\!\smile\\!\\!\partial_{i})$.

All Hochschild cohomology groups of an algebra $\mathcal{A}$ considered here
will, unless stated otherwise, have coefficients in $\mathcal{A}$ itself as a
bimodule. That cohomology is frequently called the regular (Hochschild)
cohomology and will be denoted simply $H^{*}(\mathcal{A})$, and similarly for
cochains. Note in what follows that if $a$ is a central element of
$\mathcal{A}$ and $F$ a cocycle of any dimension, then $aF$ is again a cocycle
and if $a$ is central and $D$ is a derivation, then $Da$ is again central.

The product of $1$-cochains of an associative algebra $\mathcal{A}$ is always
well defined as their composition. For $2$-cochains of the form
$f\\!\\!\smile\\!\\!g$, where $f,g$ are $1$-cochains, one can not generally
define a product by setting
$(f_{1}\\!\\!\smile\\!\\!g_{1})(f_{2}\\!\\!\smile\\!\\!g_{2})=f_{1}f_{2}\\!\\!\smile\\!\\!g_{1}g_{2}$,
for if $a$ is a central element of $\mathcal{A}$, then as $2$-cochains one has
$a(f\\!\\!\smile\\!\\!g)=af\\!\\!\smile\\!\\!g=f\\!\\!\smile\\!\\!ag$ but such
changes in representation will usually change the product. Suppose, however,
that we have a set of commuting derivations $\\{D_{i}\\}$ of $\mathcal{A}$.
When $I=\\{i_{1},\dots,i_{r}\\}$ is a set of indices (repetitions allowed) of
the $D_{i}$, set, as before, $D_{I}=D_{i_{1}}\cdots D_{i_{r}}$, and similarly
for some $D_{J}$. As the derivations now are assumed to commute, the ordering
does not matter. (When the set $I$ is empty, interpret $D_{I}$ as the identity
map $\mathcal{A}$ and similarly for $D_{J}$; when $I$ and $J$ are both empty
then $D_{I}\\!\\!\smile\\!\\!D_{J}$ will denote the multiplication map.) The
foregoing problem then does not arise when multiplication is restricted to
$2$-cochains of the form $D_{I}\\!\\!\smile\\!\\!D_{J}$. In particular, if
$D_{1},D_{2}$ are commuting derivations of $\mathcal{A}$ then
$(D_{1}\\!\\!\smile\\!\\!D_{2})^{n}=D_{1}^{n}\\!\\!\smile\\!\\!D_{2}^{n}$ is
well defined, as is
$\exp(D_{1}\\!\\!\smile\\!\\!D_{2})=\sum_{n=0}^{\infty}\frac{1}{n!}(D_{1}\\!\\!\smile\\!\\!D_{2})^{n}$
when $\mathcal{A}$ is defined over $\mathbb{Q}$.

In the following, ‘formal’ indicates that no assertion is made that the
deformation parameter $\hbar$ in the series defining $a\star b$ for
$a,b\in\mathcal{A}$ can be specialized to some set of values in the
coefficient ring. When this is possible for a particular UDF and algebra we
will say that the UDF is effective on that algebra. An important example is
that of the deformation of $\mathbb{C}[x,y]$ to the first Weyl algebra,
discussed in §4.

######  Theorem 1 (The Basic UDF).

If $D_{1},\,D_{2}$ are commuting derivations of an associative algebra
$\mathcal{A}$ over a ring $\mathbf{k}$ containing the rationals, $\mathbb{Q}$,
then $\exp\hbar(D_{1}\\!\\!\smile\\!\\!D_{2})$ integrates
$(D_{1}\\!\\!\smile\\!\\!D_{2})$, i.e., the multiplication defined on
$\mathcal{A}[[\hbar]]$ by

(6) $a\star
b\,=\,\exp\hbar(D_{1}\\!\\!\smile\\!\\!D_{2})(\,a,b)\,=\,\sum_{n=0}^{\infty}\frac{\hbar^{n}}{n!}D_{1}^{n}a\cdot
D_{2}^{n}b$

is associative and defines a formal deformation of $\mathcal{A}$.

Proof. Setting $F_{i}=D_{1}^{i}\\!\\!\smile\\!\\!D_{2}^{i}$ in (4), what must
be shown is that, for all $n$, one has the following relation among 3-cochains
of $\mathcal{A}$:

(7)
$\sum_{i=0}^{n}\frac{1}{i!(n-i)!}D_{1}^{i}\\!\\!\smile\\!\\!D_{2}^{i}(D_{1}^{n-i}\\!\\!\smile\\!\\!D_{2}^{n-i})=\sum_{i=0}^{n}\frac{1}{i!(n-i)!}D_{1}^{i}(D_{1}^{n-i}\\!\\!\smile\\!\\!D_{2}^{n-i})\\!\\!\smile\\!\\!D_{2}^{i}.$

By Leibniz’ rule, the left side of (7) can be written as

(8)
$\sum_{i=0}^{n}\frac{1}{i!(n-i)!}\sum_{j=0}^{i}\binom{i}{j}D_{1}^{i}\\!\\!\smile\\!\\!D_{2}^{j}D_{1}^{n-i}\\!\\!\smile\\!\\!D_{2}^{i-j}D_{2}^{n-i}\\\
=\sum_{i,j=0,\dots,n,\,i\geq
j}\frac{1}{(n-i)!j!(i-j)!}D_{1}^{i}\\!\\!\smile\\!\\!D_{2}^{j}D_{1}^{n-i}\\!\\!\smile\\!\\!D_{2}^{n-j},$

and the right side of (7) can be written as

(9)
$\sum_{i=0}^{n}\frac{1}{i!(n-i)!}\sum_{j=0}^{i}\binom{i}{j}D_{1}^{i-j}D_{1}^{n-i}\\!\\!\smile\\!\\!D_{1}^{j}D_{2}^{n-i}\\!\\!\smile\\!\\!D_{2}^{i}\\\
=\sum_{i,j=0,\dots,n,\,i\geq
j}\frac{1}{(n-i)!j!(i-j)!}D_{1}^{n-j}\\!\\!\smile\\!\\!D_{1}^{j}D_{2}^{n-i}\\!\\!\smile\\!\\!D_{2}^{i}.$

On the right side of (9) one can now replace the dummy variable $i$ by $n-j$
and $j$ by $n-i$. The sum continues to be over the same set of indices since
$i\geq j$ if and only if $n-j\geq n-i.$ As $D_{1}$ and $D_{2}$ commute, the
right side of (9) then becomes identical to the right side of (8), proving the
assertion. $\Box$

The Basic UDF has the following immediate extension.

######  Corollary 1.

If $D_{i},\,i=1,\dots,n$ are commuting derivations (not necessarily all
distinct) of an algebra $\mathcal{A}$ over $\mathbb{Q}$ and
$c_{ij},i,j=1,\dots,n$ are central elements which are constants for all the
$D_{i}$, i.e., central elements of $\mathcal{A}$ with $D_{k}c_{ij}=0$ for all
$i,j,k$ then $\exp\sum_{i,j=1}^{n}c_{ij}(D_{i}\smile D_{j})$ quantizes
$\mathcal{A}$. $\Box$

Proof. The case $n=2$ is immediate from the foregoing. Suppose it is true for
a given $n$. A derivation which commutes with all $D_{1},\dots,D_{n}$ remains
a derivation in the deformed multiplication given by (6). Therefore, if we now
have commuting derivations $D^{\prime},D^{\prime\prime}$ which commute with
all $D_{,}\dots,D_{n}$ and a central element $c$ which is a constant for
$D_{1},\dots,D_{n},D^{\prime},D^{\prime\prime}$, then $a\star^{\prime}b:=\exp
c(D^{\prime}\\!\\!\smile\\!\\!_{\star}D^{\prime\prime})(a,b)$ is a deformation
of the previously defined star product, where $\,\\!\\!\smile\\!\\!_{\star}$
indicates that the cup product is to be taken with the star multiplication. As
the derivations commute, the product of the exponentials involved is the
exponential of the sum of the exponents. $\Box$

##  4\. Weyl and Clifford algebras

Heisenberg, [23], stated his uncertainty principle for a particle moving in
one dimension in the form

$pq-qp=\frac{h}{2\pi i},$

where $p$ is the position of the particle and $q$ its momentum, considered as
functions in phase space. Its algebraic meaning is codified in the first Weyl
algebra, $\mathcal{W}$, which can be defined as the tensor algebra $T(V)$ on a
2-dimensional vector space $V$ with basis $x,y$, modulo the ideal generated by
$xy-yx-1$. This can be done more generally with a commutative unital ground
ring $\mathbf{k}$ over the rationals. The resulting algebra will be denoted in
the following by $\mathcal{W}(\mathbf{k})$.

######  Theorem 2.

The Weyl algebra $\mathcal{W}(\mathbf{k})$ can be obtained as an effective
deformation of the polynomial ring $\mathbf{k}[x,y]$, by taking, in (6),
$D_{1}=\partial_{x},\,D_{2}=\partial_{y}$. The infinitesimal deformation of
$\mathbf{k}[x,y]$ is then $\partial_{x}\\!\\!\smile\\!\\!\partial_{y}$, and
the deformed star multiplication is given, for $a,b$ in $\mathbf{k}[x,y]$, by

(10) $a\star
b=ab+\hbar\partial_{x}a\partial_{y}b+\frac{\hbar^{2}}{2!}\partial_{x}^{2}a\,\partial_{y}^{2}b+\frac{\hbar^{3}}{3!}\partial_{x}^{3}a\,\partial_{y}^{3}b+\cdots.$

Proof. While the right side of (10) is formally an infinite series, it must
terminate for any fixed $a$ and $b$. It follows that the deformation
parameter, $\hbar$, can be specialized to any value in $\mathbf{k}$, so the
deformation is effective. From (10), one sees that $x\star y=xy+\hbar$ while
$y\star x=yx=xy$, so the star commutator, $[x,y]_{\star}$, is $x\star y-y\star
x=\hbar$. Setting $\hbar=1$, the Weyl algebra $\mathcal{W}(\mathbf{k})$ is
thus a deformation of $\mathbf{k}[x,y]$. $\Box$

The $n$th Weyl algebra, $\mathcal{W}_{n}=\mathcal{W}^{\otimes n}$, is
similarly an effective deformation of
$\mathbb{C}[x_{1},y_{1},\dots,x_{n},y_{n}]$ with infinitesimal
$\sum_{i=1}^{n}\partial_{x_{i}}\\!\\!\smile\\!\\!\partial_{y_{i}}$. It encodes
the uncertainty principle for a particle moving in $n$ dimensions with
position and momentum coordinates $p_{i},q_{i},\,i=1,\dots,n$ and commutation
relations $[p_{i},p_{j}]=[q_{i},q_{j}]=0,[p_{i},q_{j}]=\delta_{ij}$.

Clifford algebras can similarly be viewed as effective deformations of
exterior algebras. The latter are graded algebras, so one must follow Koszul’s
rule of signs. In the smallest case, let $\mathcal{A}$ be the four dimensional
exterior algebra over a field $\mathbf{k}$ generated by a two dimensional
vector space spanned by $x$ and $y$, each of which has degree 1. For
simplicity, assume that the characteristic of $\mathbf{k}$ is not 2. Here
$\partial_{x},\partial_{y}$ are both derivations of $\mathcal{A}$ of degree -1
which commute in the sense that
$\partial_{x}\partial_{y}=-\partial_{y}\partial_{x}$, and each has square
equal to zero. (The square of a derivation of odd degree always must be zero
when the characteristic is not 2, since it commutes with itself.) The
exponential $\exp(\partial_{x}\\!\\!\smile\\!\\!\partial_{y})$ reduces to
$\operatorname{id}\\!\\!\smile\\!\\!\operatorname{id}+\,\partial_{x}\\!\\!\smile\\!\\!\partial_{y}$.
With the star product it induces one still has $x\star x=y\star y=0$ but now
$x\star y=xy+1$, while $y\star x=yx$, so $x\star y+y\star x=1$. Therefore,
$(x+y)^{\star 2}=1$ and $(x-y)^{\star 2}=-1$, so the deformed algebra is the
Clifford algebra ${\text{C}l}_{1,1}(\mathbf{k})$.

##  5\. Fourier series

Consider again the algebra $C^{\infty}(\mathbb{C}^{2})$ of smooth functions of
two complex variables, $x,y$, with commuting derivations
$D_{1}=\partial_{x},D_{2}=\partial_{y}$ and infinitesimal deformation
$\partial_{x}\\!\\!\smile\\!\\!\partial_{y}$. It contains a subalgebra
$\mathcal{F}$ of for which the associativity of the resulting star product as
well as its effectiveness can be seen without the use of Theorem 1, namely
that generated by functions of the form $e^{\lambda x+\mu y}$. For these,
applying (10) with $a=e^{\lambda_{1}x+\mu_{1}y},\,b=e^{\lambda_{2}x+\mu_{2}y}$
one has first

$e^{\lambda_{1}x+\mu_{1}y}\star
e^{\lambda_{2}x+\mu_{2}y}=e^{\mu_{1}y+\lambda_{2}x}(e^{\lambda_{1}x}\star
e^{\mu_{2}y})$

because $y$ is a constant for $\partial_{x}$ and $x$ for $\partial_{y}$, after
which computation gives

(11) $e^{\lambda_{1}x}\star
e^{\mu_{2}y}=e^{\hbar\lambda_{1}\mu_{2}}e^{\lambda_{1}x+\mu_{2}y}.$

Together one has

(12) $e^{\lambda_{1}x+\mu_{1}y}\star
e^{\lambda_{2}x+\mu_{2}y}=e^{\hbar(\lambda_{1}\mu_{2})}e^{(\lambda_{1}+\lambda_{2})x+(\mu_{1}+\mu_{2})y},$

where the right side is so written to exhibit its second factor as the
undeformed multiplication and its first as the effect of the deformation. From
(12) one may see first that it independently shows the associativity of this
star multiplication for functions of the special form $e^{\lambda x+\mu y}$
and second that the deformation is effective since one can specialize $\hbar$
to any complex value. The star product may still be viewed as local, even
though its value at a point depends on the global behavior of the factors,
since those factors are analytic and determined globally by their germs at any
point.

Functions of the form $e^{\lambda x+\mu y}$ are doubly periodic but those in
$\mathcal{F}$, which are linear combinations of such, are generally only
almost periodic because the periods of the summands may not be commensurable.
Doubly periodic functions in $x$ and $y$ with fixed periods in each, say
$2\pi$, can, with suitable restrictions, be represented as double Fourier
series using complex exponentials. Those with absolutely convergent double
series form a subring, $\mathcal{F}_{\text{ac}}(2\pi,2\pi)$, on which the
foregoing star product is still well-defined and effective: If in (11),
$\lambda_{1}$ and $\mu_{2}$ are both purely imaginary and and one replaces
$\hbar$ by $i\hbar$ then the deformation factor will have absolute value 1, so
the star product will still be absolutely convergent for all now real
(previously purely imaginary) $\hbar$. The functions in
$\mathcal{F}_{\text{ac}}(2\pi,2\pi)$ are generally not analytic and the
deformed product is no longer local but is still effective. This should be
extendable to Fourier transforms.

The foregoing allows a wide class of effective deformations stemming from the
Basic UDF. While Kontsevich’s UDF, [26], applies to any Poisson structure it
remains purely formal; it can not define a star product on functions which are
not analytic such as those above.

##  6\. Jump deformations, rigidity, and stability

The Weyl algebra is also an example of a jump deformation, [20, §7, p. 20ff],
[18], i.e., one in which the algebras defined for all values of $\hbar$ other
than zero are isomorphic. In a jump deformation, the infinitesimal of the
deformation becomes a coboundary in the star multiplication, see e.g., [18].

The cohomology of a $\mathbf{k}$ algebra $\mathcal{A}$ and that of a
deformation, $\mathcal{A}_{\hbar}$, of $\mathcal{A}$ are closely related,
[16], as follows. The definition of the star product as an infinite series in
$\hbar$ requires that the underlying coefficient ring of $\mathcal{A}_{\hbar}$
be extended at least to the power series ring $\mathbf{k}[[\hbar]]$, but to
remain over a field, let it be extended to Laurent series,
$\mathbf{k}((\hbar))$, in which a finite number of negative powers of $\hbar$
are allowed. An $n$-cocycle $z$ of $\mathcal{A}$ is liftable to
$\mathcal{A}_{\hbar}$ if there are $n$-cochains $z_{2},z_{3},\dots$ such that
$z+\hbar z_{1}+\hbar^{2}z_{2}+\cdots$ is an $n$-cocycle of
$\mathcal{A}_{\hbar}$. The cohomology of $A_{\hbar}$ then consists of the
liftable cocycles modulo those which lift to coboundaries.

By the HKR theorem, the cohomology of $\mathbb{C}[x,y]$ is isomorphic, as a
vector space, to the exterior algebra on $\partial_{x},\partial_{y}$. Both
$\partial_{x}$ and $\partial_{y}$ lift, in fact to themselves, as derivations
of $\mathcal{W}$, but become inner since, e.g., in $\mathcal{W}$ one has
$\partial_{y}y=\hbar^{-1}[x,y]_{\star}=1$, whence
$\partial_{y}a=\hbar^{-1}[x,a]_{\hbar}$ for all $a\in\mathcal{W}$. It follows
that the cohomology of $\mathcal{W}$ vanishes in all positive dimensions, in
particular, it is absolutely rigid in the terminology of [13], or in the
better terminology of Froelicher and Nijenhuis, [9], stable. This was first
proven in Ramaiyengar Sridharan’s thesis under S. Eilenberg, [33].

As the Weyl algebra is a stable deformation of the polynomial ring in two
variables, which itself is not stable, this might suggest that classical
physical laws tend to deform to stable ones but the laws themselves do not
change, it is our understanding of them that evolves. It is also remarkable
that, at least in this context, uncertainty seems to be necessary for
stability.

If $\mathcal{A}_{1},\mathcal{A}_{2}$ are algebras over the same field
$\mathbf{k}$ with vanishing regular cohomology (i.e., with coefficients in
themselves, cf §1) in all positive dimensions, then the same is true of their
tensor product, a special case of [27, Theorem 4, p. 207]. It follows that if
in $\mathbb{C}^{\infty}[q_{1},\dots,q_{d},\,p_{1},\dots,p_{d}]$ one takes as
infinitesimal deformation $i\hbar\sum_{i=1}^{d}(\partial
q_{i}\\!\\!\smile\\!\\!\partial p_{i})$, then the resulting deformed algebra,
in which $q_{i}p_{i}-p_{i}q_{i}=i\hbar$ for all $i$ but $p_{i}$ and $q_{j}$
still commute for $i\neq j$, has no regular cohomology in positive dimensions.
In particular, it is stable.

By contrast, a tensor product of rigid algebras need not be rigid. As an
example, consider the twisting of a tensor product: Suppose that we have
algebras $\mathcal{A}_{1},\mathcal{A}_{2}$ defined over a ring $\mathbf{k}$
containing $\mathbb{Q}$ with respective derivations $D_{1},D_{2}$. Extend
$D_{1},D_{2}$ to derivations of
$\mathcal{A}_{1}\otimes_{\mathbf{k}}\mathcal{A}_{2}$ by setting
$D_{1}(a_{1}\otimes a_{2})=D_{1}a_{1}\otimes a_{2}$ and $D_{2}(a_{1}\otimes
a_{2})=a_{1}\otimes D_{2}a_{2}$. These extensions commute, so
$D_{1}\\!\\!\smile\\!\\!D_{2}$ can be exponentiated to a full deformation of
$\mathcal{A}_{1}\otimes\mathcal{A}_{2}$. If either $D_{1}$ or $D_{2}$ is inner
then the infinitesimal deformation $D_{1}\\!\\!\smile\\!\\!D_{2}$ will be a
coboundary and the resulting deformation will be trivial. However, if neither
is inner, then this is an example where
$\mathcal{A}_{1}\otimes_{\mathbf{k}}\mathcal{A}_{2}$ can be deformed even when
both $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are rigid.

Theorem 1 was first explicitly stated not only for associative algebras but
for all composition complexes as defined in [15, Lemma 1, p.14] but is already
implicit in two important papers on quantum mechanics, Groenewold, [22], 1946,
and Moyal, [29], 1949. The algebra being deformed in both is the algebra of
observables $\mathbb{C}[p,\,q]$ in phase space, where, as remarked, before
deformation the position variable $q$ and momentum variable $p$ commute, but
after deformation satisfy Heisenberg’s uncertainty relation $qp-pq=i\hbar$.

Infinitesimal deformations of $C^{\infty}(\mathbf{R}^{d})$ of the form
$\sum_{i<j}c_{ij}f_{i}\partial_{i}\\!\\!\smile\\!\\!f_{j}\partial_{j}$, with
all $c_{ij}\in\mathbb{R}$ and each $f_{i}$ a smooth function only of the one
variable $x_{i}$, will be called basic. From the corollary to Theorem 1, the
following is immediate.

######  Theorem 3.

Basic infinitesimal deformations of $C^{\infty}(\mathbb{R}^{d})$ are
integrable by exponentiation. $\Box$

To shorten some lengthy expressions, in what follows we will write $(i|j)$ for
$\partial_{i}\\!\\!\smile\\!\\!\partial_{j}$, $(ij|k)$ for
$\partial_{i}\partial_{j}\\!\\!\smile\\!\\!\partial_{k}$, $(i|j|k)$ for
$\partial_{i}\\!\\!\smile\\!\\!\partial_{j}\\!\\!\smile\\!\\!\partial_{k}$,
and likewise for similar expressions. With this notation, simple computation
yields the following.

######  Lemma 1.

For all $1\leq i<j\leq d,1\leq\,k<l\leq d$, one has

$\displaystyle a_{ij}(i|j)a_{kl}(k|l)$
$\displaystyle=a_{ij}a_{kl}(ik|jl)\quad\text{if }i\neq k,j\neq l$
$\displaystyle a_{ij}(i|j)a_{il}(i|l)$
$\displaystyle=a_{ij}\partial_{i}a_{il}(i|jl)+a_{ij}a_{il}(ii|jl)\quad\text{
if $j\neq l$;}$ $\displaystyle a_{ij}(i|j)a_{kj}(k|j)$
$\displaystyle=a_{ij}\partial_{j}a_{kj}(ik|j)+a_{ij}a_{kj}(ik|jj)\quad\text{if
$i\neq k$;}$ $\displaystyle a_{ii}(i|i)a_{ii}(i|i)$
$\displaystyle=\partial_{i}a_{ii}\partial_{i}a_{ii}(i|i)+a_{ii}\partial_{i}a_{ii}(ii|i)+a_{ii}\partial_{i}a_{ii}(i|ii)+a_{ii}a_{ii}(ii|ii).\Box$

This lemma allows one to write all the powers, and hence the exponential, of
an infinitesimal $F_{1}=\sum_{ij}a_{ij}(i|j)$ in terms of the $a_{ij}$ and
their various derivatives. In particular, for a basic infinitesimal
$F_{1}=\sum_{i<j}c_{ij}f_{i}\partial_{i}\\!\\!\smile\\!\\!f_{j}\partial_{j}$,
writing $a_{ij}$ for $c_{ij}f_{i}f_{j}$, these relations allow one to write
$F_{1}^{2}$, and hence all powers $F_{1}^{n}$ as well as $\exp{F_{1}}$, in
terms of the $a_{ij}$ and their derivatives without reference to the $f_{i}$.
However, if $F_{1}=\sum_{1\leq i<j\leq d}a_{ij}(i|j)$, is not basic then
$\exp{F_{1}}$ will generally not be an integral of $F_{1}$. For it to be an
integral, by (5) we must have $F_{1}\circ
F_{2}=\delta(-\frac{1}{2}F_{1}^{2})$, but it is the difference between the
left and right that gives rise to the obstruction equations discussed later.

##  7\. Differential subcomplex

There is a differential subcomplex,
$C_{\text{diff}}^{\bullet}(C^{\infty}(\mathbb{R}^{d}))$, of the Hochschild
complex $C^{\bullet}(C^{\infty}(\mathbb{R}^{d}))$ of
$C^{\infty}(\mathbb{R}^{d})$ generated by the derivations
$\partial_{1},\dots\partial_{d}$ with respect to its coordinates
$x_{1},\dots,x_{d}$; . Its 0-cochains are just the elements of
$C^{\infty}(\mathbb{R}^{d})$. The module of n-multiderivations is spanned by
the $n$-fold cup products of these. Simple 1-cochains are of the form
$a\partial_{i_{1}}\partial_{i_{2}}\cdots\partial_{i_{r}}$, where $a\in
C^{\infty}(\mathbb{R}^{d})$, $r$ is arbitrary, and there may be duplications
amongst the indices. The module of 1-cochains is composed of sums of such
cochains of various orders. Simple $n$-cochains are cup products of simple
1-cochains. The differential order of such a cup product is the sum of the
orders of its cup factors. A general $n$-cochain is a sum of simple ones,
possibly of different differential orders; it is homogeneous if the orders are
the same.

The Hochschild coboundary operator preserves differential order. While the
differential subcomplex is closed under the composition product, the
composition product of homogeneous cochains is generally no longer
homogeneous. The HKR theorem implies that the inclusion of the differential
subcomplex into the full Hochschild complex induces an isomorphism of
cohomology.

Using the notation preceding Theorem 3, the HKR theorem also implies that
$H^{r}(\mathcal{A})$ can be identified with the $\mathcal{A}$ module spanned
(using the notation of §6) by all forms $(i_{1}|i_{2}|\cdots|i_{r})$ with
$i_{1}<i_{2}<\cdots<i_{r}$, since the class of any skew form also has a unique
representative in that module.

##  8\. Poisson structure

A Poisson structure on a commutative algebra $\mathcal{A}$ is a second,
’bracket’, multiplication $[a,b],\,a,b\in\mathcal{A}$, which is skew,
$[a,b]=-[b,a]$, a biderivation, $[ab,c]=a[b,c]+b[a,c]$, the same holding on
the right by skewness, and which satisfies the Jacobi identity,

(13) $J(a,b,c):=[a,\,[b,\,c]]+[b,\,[c,\,a]]+[c,\,[a,\,b]]=0,$

making $\mathcal{A}$ with this second multiplication into a Lie algebra. The
Poisson structure is an infinitesimal deformation of the algebra $\mathcal{A}$
since a biderivation of a commutative algebra is clearly 2-cocycle. The
‘Jacobiator’, $J$, is skew in all three variables. Let $\mathfrak{g}$ be a
finite dimensional Lie algebra over a field $\bf k$ of arbitrary
characteristic with basis $x_{1},\dots,x_{d}$. The symmetric algebra on the
underlying vector space of $\mathfrak{g}$, which is just the polynomial ring
$\bf k[x_{1},\dots,x_{d}]$, carries a Poisson structure defined by extending
the Lie bracket on $\mathfrak{g}$ to be a biderivation of $\bf
k[x_{1},\dots,x_{d}]$.

A Poisson manifold $\mathcal{M}$ is one with a Poisson structure on its
algebra of smooth functions. The dual $\mathfrak{g}^{*}$ of a finite-
dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ or $\mathbb{C}$ is an
example. It carries a Poisson bracket called variously a Kirillov-Poisson,
Lie-Poisson, or Kirillov-Kostant-Souriau (KKS) structure described as follows.
If $f$ is a differentiable function on a finite dimensional vector space $V$
and $p$ a point of $V$, then the differential of $f$ evaluated at $p$,
$d_{p}f$, is an element of the dual space $V^{*}$. Suppose now that $f,g$ are
functions on the dual, $\mathfrak{g}^{*}$, of a finite dimensional Lie algebra
$\mathfrak{g}$ and $\xi$ a point of $\mathfrak{g}^{*}$. One can then take the
Lie product $[d_{\xi}f,\,d_{\xi}g]_{\mathfrak{g}}$ in $\mathfrak{g}$ of their
differentials at $\xi$ and define the value at $\xi$ of the KKS Lie product
$[f,g]_{\mathfrak{g}^{*}}$ of $f$ and $g$ to be
$\xi([d_{\xi}f,\,d_{\xi}g]_{\mathfrak{g}})$. To calculate the infinitesimal
deformation which the Kirillov Poisson structure defines on
$\mathfrak{g}^{*}$, choose a basis $x_{1},\dots,x_{d}$ of $\mathfrak{g}$, let
$\xi_{1},\dots,\xi_{d}$ be its dual basis in $\mathfrak{g}^{*}$, let $\xi=\sum
a_{i}\xi_{i}$ be point in $\mathfrak{g}^{*}$ and suppose that
$[x_{i},x_{j}]=\sum c_{ij}^{k}x_{k}$. Then

$df=\sum_{1}^{d}x_{i}\frac{\partial_{f}}{\partial_{\xi_{i}}},\quad[df,dg]=\sum
c_{ij}^{k}x_{k}\frac{\partial_{f}}{\partial_{\xi_{i}}}\frac{\partial_{g}}{\partial_{\xi_{j}}},\,\text{and}\,\,\xi([df,dg])=\sum
c_{ij}^{k}a_{k}\frac{\partial_{f}}{\partial_{\xi_{i}}}|_{\xi}\frac{\partial_{g}}{\partial_{\xi_{j}}}|_{\xi}$

where all indices run from 1 to d and the partial derivatives in the last term
on the right are evaluated at $\xi=\sum a_{i}\xi_{i}$. To represent this in
the form

$(\sum\lambda_{rs}(x_{r}\\!\\!\smile\\!\\!x_{s})(f,g))|_{\xi}=\sum\lambda_{rs}x_{r}(f)|_{\xi}x_{s}(g)|_{\xi}$

take $f=\xi_{i},\,g=\xi_{j}$; comparing with the preceding gives

(14) $\lambda_{ij}=\sum c_{ij}^{k}a_{k}.$

The KKS product of linear functions on $\mathfrak{g}^{*}$ is linear.
Conversely, any Poisson manifold in which the product of linear functions is
linear arises in this way. If $\xi_{i}$ is a central element of $\mathfrak{g}$
then all $c_{ij}^{k}$ are zero so if $\mathfrak{g}$ is Abelian, then the KKS
structure on $\mathbb{C}^{\infty}(\mathfrak{g}*)$ vanishes and induces no
deformation. Nevertheless, as the elements $x$ of the Lie algebra
$\mathfrak{g}$ act as derivations on $\mathbb{C}^{\infty}(\mathfrak{g}^{*})$
by $\partial_{x}f|_{\xi}=x(f)_{\xi}=\xi[x,d_{\xi}f]_{g}$, if $\mathfrak{g}$
has two elements which commute then one can apply the Basic UDF to deform
$\mathbb{C}^{\infty}(\mathfrak{g}^{*})$.

For any $\xi\in\mathfrak{g}^{*}$ and differentiable $f$ on $\mathfrak{g}^{*}$
there is a linear function with the same differential at $\xi$ as $f$, so the
Poisson bracket on $\mathbb{C}^{\infty}(\mathfrak{g}^{*})$ is already
determined by that on its linear functions.

##  9\. Coll-Gerstenhaber-Giaquinto UDF

Theorem 3 provides another approach to the Coll-Gerstenhaber-Giaquinto (CGG)
UDF, [5], the first with non-commuting derivations. In the following,
$[D_{2}]_{n}$ will denote the “descending factorial”,
$[D_{2}]_{n}=D_{2}(D_{2}-1)\cdots(D_{2}-n+1)$ and $n$ will mean
$n\operatorname{id}_{\mathcal{A}}$, so, e.g., $[D_{2}]_{2}$ means
$D_{2}(D_{2}-\operatorname{id}_{\mathcal{A}})$.

######  Theorem 4.

Suppose that $D_{1},D_{2}$ are derivations of an algebra $\mathcal{A}$ over
$\mathbb{Q}$ such that $[D_{1},D_{2}]=D_{1}$. Then
$D_{1}\\!\\!\smile\\!\\!D_{2}$ is an integrable infinitesimal deformation of
$\mathcal{A}$ with integral

$e(\hbar,D_{1},D_{2}):=\\\
id_{\mathcal{A}}\\!\\!\smile\\!\\!id_{\mathcal{A}}+\hbar
D_{1}\\!\\!\smile\\!\\!D_{2}+(\hbar^{2}/2!)D_{1}^{2}\\!\\!\smile\\!\\![D_{2}]_{2}+\cdots+(\hbar^{n}/n!)D_{1}^{n}\\!\\!\smile\\!\\![D_{2}]_{n}+\cdots\quad.$

Proof. The UDF asserted here is essentially a sequence of formal identities in
$D_{1},D_{2}$ within the universal enveloping algebra of the two-dimensional
Lie algebra generated by $D_{1},D_{2}$. If we can exhibit an explicit
associative algebra $\mathcal{A}$ with derivations $D_{1},D_{2}$ having an
isomorphic universal enveloping algebra then this will serve as a model in the
sense that any proposition about the universal enveloping algebra which holds
in this model must be true in general. We can model $D_{1}$ and $D_{2}$ as
derivations of the algebra of smooth functions of $x$ and $y$ by mapping
$D_{1}$ to $e^{-y}\partial_{x}$ and $D_{2}$ to $\partial_{y}$, but must show
that the natural map of the universal enveloping algebra of the Lie algebra
generated by $D_{1}$ and $D_{2}$ onto the algebra of operators generated by
$e^{-y}\partial_{x}$ and $\partial_{y}$ is an isomorphism. The commutation
relation between $D_{1}$ and $D_{2}$ implies that the universal enveloping
algebra is, as a module over the ground ring, free with generators all
monomials of the form $D_{1}^{m}D_{2}^{n}$, with $m,n\geq 0$. The analogue
clearly also holds for the algebra of operators generated by
$e^{-y}\partial_{x},\partial_{y}$, so they are isomorphic modules. Now
$D_{1}\\!\\!\smile\\!\\!D_{2}=e^{-y}\partial_{x}\\!\\!\smile\\!\\!\partial_{y}$,
but this can also be written as
$\partial_{x}\\!\\!\smile\\!\\!e^{-y}\partial_{y}$, which is a basic
infinitesimal with integral

(15)
$\operatorname{exp}(\hbar\partial_{x}\\!\\!\smile\\!\\!e^{-y}\partial_{y})=\sum_{n=0}^{\infty}(\hbar^{n}/n!)(\partial_{x})^{n}\\!\\!\smile\\!\\!(e^{-y}\partial_{y})^{n}.$

It is a simple induction to show that for every $\lambda\in\mathbb{C}$ and
positive integer $n$ one has

$(e^{-\lambda
y}\partial_{y})^{n}=e^{-ny}\partial_{y}(\partial_{y}-\lambda)(\partial_{y}-2\lambda)\cdots(\partial_{y}-(n-1)\lambda).$

In particular, for $\lambda=1$, one has
$(e^{-y}\partial_{y})^{n}=e^{-ny}[\partial_{y}]_{n}.$ The right side of (15)
can therefore be written as
$\sum_{n=0}^{\infty}(\hbar^{n}/n!)(e^{-y}\partial_{x})^{n}\\!\\!\smile\\!\\![\partial_{y}]_{n}$,
proving the theorem. $\Box$

An important property of this UDF is that it gives an explicit integral for
the KKS structure (§8) on the dual of the unique non-abelian solvable two-
dimensional Lie algebra $\mathfrak{s}$. One can take a basis,
$\\{x_{1},x_{2}\\}$, for $\mathfrak{s}$ with $[x_{1},x_{2}]=x_{1}$. With this,
in (14), one has $c_{12}^{1}=1,c_{12}^{2}=0$, which completely determines
$\Pi$, since it is skew;
$\Pi=\partial_{x_{1}}\\!\\!\smile\\!\\!\partial_{x_{2}}-\partial_{x_{2}}\\!\\!\smile\\!\\!\partial_{x_{1}}$.
Theorem 4 integrates this since it is cohomologous to
$2\,\partial_{x_{1}}\\!\\!\smile\\!\\!\partial_{x_{2}}$. The integral is
rational and computable term by term.

##  10\. The primary obstruction

As remarked in §8, a biderivation of a commutative algebra is a 2-cocycle,
hence an infinitesimal deformation. As a partial converse, one has the
following.

######  Lemma 2.

Let $f$ be a skew 2-cocycle of a commutative algebra $\mathcal{A}$ over a ring
$\mathbf{k}$ in which 2 is a unit. Then $f$ is a biderivation.

Proof. For a skew 2-cochain $f$ of a commutative algebra one has the identity

$(\delta f)(a,b,c)+(\delta f)(c,a,b)-(\delta
f)(a,c,b)=2[af(b,c)-f(ab,c)+bf(a,c)].\,\Box$

Lemma 2 suggests that an alternating $n$-cocycle (one which changes sign under
an odd permutation of its arguments) of a commutative algebra should be a
multiderivation when $n!$ is a unit. Recall from the HKR theorem that every
element of $H^{2}(C^{\infty}(\mathbb{R}^{d}))$ has a unique skew
representative 2-cocycle, which by Lemma 2 is a biderivation, and from §3 that
the primary obstruction to an infinitesimal deformation $F_{1}$ of an algebra
is $F_{1}\circ F_{1}$.

In what follows, for a function $f$ of three variables, set

$\sum_{\circlearrowleft}f(a,b,c):=f(a,b,,c)+f(b,c,a)+f(c,a,b).$

Note that if $f$ is skew in any pair of variables then
$\sum_{\circlearrowleft}f$ is skew in every pair.

######  Theorem 5.

1\. Let $\Pi$ be a skew 2-cocycle of a commutative algebra $\mathcal{A}$
defined over a ring $\mathbf{k}$ in which $3!$ is a unit. Viewing $\Pi$ as an
infinitesimal deformation of $\mathcal{A}$, if its primary obstruction,
$\Pi\circ\Pi$, vanishes then $\Pi$ is Poisson. 2. An infinitesimal deformation
of a commutative algebra $\mathcal{A}$ defined over $\mathbb{Q}$ is integrable
to a full deformation if and only if its primary obstruction vanishes.

Proof. Assertion 2. follows immediately from 1. since if $\Pi$ is Poisson then
by [26] it is integrable, so in particular its primary obstruction vanishes.
Suppose now that the primary obstruction to $\Pi\circ\Pi$ vanishes, i.e., that
it is a coboundary. In view of Lemma 2 it is sufficient now to show that
$\sum_{\circlearrowleft}\Pi\circ\Pi=0$. Since $\Pi\circ\Pi$ is a coboundary so
is $\sum_{\circlearrowleft}\Pi\circ\Pi$. Since $\Pi$ is skew one has that
$\Pi\circ\Pi(a,b,c)=\Pi(\Pi(a,b),c)-\Pi(a,\Pi(b,c))$ is skew in it first and
third variables, $a$ and $c$. It follows that
$\sum_{\circlearrowleft}\Pi\circ\Pi$, which is again a coboundary, is skew in
all three variables. From the decomposition of commutative algebra cohomology
introduced in [19] it must be identically zero: There it was shown that if
$n!$ is a unit then the $n$th cohomology $H^{n}$ of a commutative algebra with
coefficients in a commutative module decomposes into a direct sum

$H^{n}=H^{1,n-1}+H^{2,n-2}+\cdots+H^{n,0}$

where the ‘top’ component, $H^{n,0}$ consists of skew multiderivations each
the unique representative of its cohomology class; a skew mutiderivation
therefore can not be a coboundary unless it is identically zero. $\Box$

The UDF given by Kontsevitch in [26] has the property that the higher order
terms come from various partial derivatives of the coefficients in the
infinitesimal deformation, so the full deformation it constructs will be
trivial in any open set where all coefficients of its infinitesimal vanish.

##  11\. Moduli spaces and modular groups

A deformation of an algebra or other structure is usually presented initially
as dependent on one or more parameters. The moduli space of the deformation is
its parameter space with points representing isomorphic structures identified.
It carries the quotient topology, in which a set is open if and only if its
preimage is open. The parameter space can sometimes be discrete, as in [17],
and the moduli space can even be non-Hausdorff. This is the case with a jump
deformation, where the moduli space consists of two points, one closed
representing the original algebra, and one open representing the deformed one.
A deformation problem will have an associated modular group when
specializations of the deformation parameter yielding structures isomorphic to
the original are discrete. The modular group then operates on the parameter
space to identify points corresponding to isomorphic objects.

An example of a modular group associated with the deformation of the
polynomial ring in two variables to the quantum plane is given below. It is
analogous to the classical modular group associated to the deformation of
compact Riemann surfaces of genus one or complex tori. For these and compact
Riemann surfaces of higher genera there is a Teichmüller space whose points
are essentially the conformal equivalence classes of the surfaces each with a
choice of integral generators for the homology group of its underlying
manifold. In the case of genus one, choosing one surface with marked
generators of its integral homology groups, the others are then marked by the
elements of the special linear group $\mathbf{SL}(2,\mathbb{Z})$ which is
isomorphic to the free product $\mathbb{Z}/4\star\mathbb{Z}/6$. In detail,
every complex torus can be represented as the quotient $\mathbb{C}/L$ of the
complex plane $\mathbb{C}$ by a lattice $L$, which one may view as generated
by an unordered pair of real non-collinear real vectors
$\begin{bmatrix}x_{1}\\\ y_{1}\end{bmatrix}$, $\begin{bmatrix}x_{2}\\\
y_{2}\end{bmatrix};$ they can be given a natural order by requiring that the
determinant of $\begin{bmatrix}x_{1}&x_{2}\\\ y_{1}&y_{2}\\\ \end{bmatrix}$ be
positive. The angle from the first vector to the second in the positive
direction is then less than $\pi$. The parallelogram they generate is a
fundamental domain for $L$ considered as a group operating on $\mathbb{C}$ and
the torus may be viewed as the fundamental domain with opposite sides
identified. The sides become a pair of circles intersecting in a single point
and are generators for the first homology group of the torus. The special
linear group $\mathbf{SL}_{2}(\mathbb{Z})$ operates on these pairs of vectors,
identifying pairs which generate isomorphic latices. Each of the lattices has
an automorphism of order two obtained by rotating the parallelogram through a
half circle around its center. Two have larger groups of automorphisms, namely
that in which the parallelogram is a square where the automorphism group is
$\mathbb{Z}/4$ and that in which it has equal sides separated by an angle of
$\pi/3$ whose automorphism group is $\mathbb{Z}/6$. Accordingly,
$\mathbf{SL}_{2}(\mathbb{Z})$ is an amalgam; it is the free product of
$\mathbb{Z}/4$ and $\mathbb{Z}/6$ with the subgroups of order 2 identified,
$\mathbb{Z}/2\star_{\mathbb{Z}/2}\mathbb{Z}/6$, cf [10]. Its center is the
consolidated group $\mathbb{Z}/2$. As matrices in
$\mathbf{SL}_{2}(\mathbb{Z})$ it consists of the identity and its negative.

Viewing vectors in the plane as complex numbers, any complex torus
$\mathbb{C}/L$ is biholomorphic to one where the first generating vector of
$L$ is 1 and the second therefore a complex number $\tau$ with positive
imaginary part. The parameter space for complex tori is thus the upper half
plane. The projective special linear group $\mathbf{PSL}_{2}(\mathbb{Z})$
consisting of all maps from $\mathbb{C}$ to itself of the form
$\tau\mapsto(a\tau+b)/(c\tau+d),\,a,b,c,d\in Z,ad-bc=1$ acts on it to identify
values of $\tau$ which represent the same complex torus. One can visualize the
deformation of a torus as a stretching of the fundamental domain onto a new
one by the motion of $\tau$ but the underlying structure of a manifold is
preserved. This classical modular group, associated to the deformation of
complex tori, is the quotient of $\mathbf{SL}_{2}(\mathbb{Z})$ by its center
and is isomorphic to $\mathbb{Z}/2\star\mathbb{Z}/3$. The classical moduli
space is smooth and biholomorphic to the punctured sphere, i.e., Riemann
sphere with one point removed.

For an analogous purely algebraic modular group and moduli space consider now
the deformation of $\mathbb{C}[x,y]$ with infinitesimal
$x\partial_{x}\\!\\!\smile\\!\\!y\partial_{y}$ and star product given by
$\exp(\hbar\,x\partial_{x}\\!\\!\smile\\!\\!y\partial_{y})$. This is again an
example of the twisting of a tensor product of $\mathbf{k}[x]$ with itself.
Previously the derivation chosen in each tensor factor was $\partial_{x}$
while it is now $x\partial_{x}$. With the resulting 2-cocycle
$x\partial_{x}\\!\\!\smile\\!\\!y\partial_{y}$ one has

$x\star
y=xy+\hbar(x\partial_{x})x\,(y\partial_{y})y+\frac{\hbar^{2}}{2}(x\partial_{x})^{2}x\,(y\partial_{y})^{2}y+\frac{\hbar^{3}}{3!}(x\partial_{x})^{3}x\,(y\partial_{y})^{3}y+\cdots.$

As $(x\partial_{x})^{n}x=x$ for all $n\geq 0$, and similarly with $y$, one has
$x\star y=e^{\hbar}xy$, while $y\star x=yx=xy$. Therefore, with this
deformation,

(16) $x\star y=e^{\hbar}y\star x.$

Writing $q$ for $e^{\hbar}$ one has $x\star y=qy\star x$, the defining
equation of the quantum plane. The parameter space is all of $\mathbb{C}$. The
modular group, $\mathbb{G}$, is generated by transformations
$\sigma:\hbar\mapsto\hbar+2n\pi i,n\in\mathbb{Z}$ and
$\tau:\hbar\mapsto-\hbar$, the latter corresponding to the interchange of $x$
and $y$. There is an exact sequence

(17)
$0\longrightarrow\mathbb{Z}\longrightarrow\mathbb{G}\longrightarrow\mathbb{Z}/2\longrightarrow
0$

which is split by sending the non-identity element of $\mathbb{Z}/2$ to
$\tau$, so $\mathbb{G}$ is the semidirect product
$\mathbb{Z}/2\rtimes\mathbb{Z}$, the infinite dihedral group frequently
denoted $\mathbf{D}_{\infty}$. There is an underlying structure preserved by
the deformations, namely that of the tensor product of two vector spaces. Two
algebras amongst the foregoing deformations have non-trivial automorphisms
which preserve this structure, namely the original polynomial algebra and the
anticommutative one with $xy=-yx$. In each case the automorphism is the
interchange of tensor factors, so by [10] the modular group for this
deformation problem is $\mathbb{Z}/2\star\mathbb{Z}/2$ which is identical with
$\mathbf{D}_{\infty}$.

The quotient of $\mathbb{C}$ by $\mathbb{Z}$ (appearing in (17) as multiples
of $\sigma:\hbar\to\hbar+2\pi i$) is an infinite cylinder, which is conformal
by the exponential map to $\mathbb{C}\backslash 0$, i.e., $\mathbb{C}$ with
zero removed. This is evident in (16). The group $\mathbb{Z}/2$ operates on
$\mathbb{C}\backslash 0$ by sending $z$ to $1/z$, in (16), $e^{\hbar}\to
e^{-\hbar}$, so the quotient, the moduli space, is again the punctured sphere.
This moduli space, like the classical one, has no singularities. However, the
moduli space for Riemann surfaces of genus 2, which has complex dimension
three, has a singularity at the point representing the surface given by
$y^{2}=x^{6}-1$; there is a discussion of the modular group in genus 2 in
[31].

The deformations of $\mathbb{C}[x,y]$ to the Weyl algebra, to the quantum
plane, $W_{qp}=\mathbb{C}\\{x,y\\}/(xy-qyx)$ and to the $q$-Weyl algebra
$W_{q}=\mathbb{C}\\{x,y\\}/(xy-qyx-1)$ are discussed in [18] where the
cohomology of each of the latter two algebras are computed. The generic
$q$-Weyl algebra is not rigid while the Weyl algebra, its specialization at
$q=1$ is absolutely rigid or stable. Note, as observed in [18], that the
algebra $W_{q}(\hbar)=\mathbb{C}{x,y}/(xy-qyx-\hbar)$ is isomorphic to
$W_{q}$: replace $\hbar$ by $x^{n}\hbar$ for any $n>0$ and $y$ by
$y+x^{n-1}\hbar/(1-q)$. The results of [18] suggest that the moduli space of
the quantum Weyl algebras consists of two copies of that of the quantum plane,
the first with its usual topology and the second with the discrete topology,
but the points of the first are not closed in the whole space, the closure of
a point in the first copy containing the corresponding point in the second.

##  12\. Obstruction equations

There are $\binom{d}{3}$ non-linear partial differential ‘obstruction
equations’ in the coefficients $a_{ij}$ of an infinitesimal deformation
$F_{1}=\sum_{1\leq i<j\leq d}a_{ij}(i|j)$. These arise from the difference
between $F_{1}\circ F_{1}$ and $\delta(-\frac{1}{2}F_{1}^{2})$; the primary
obstruction to $F_{1}$ vanishes if and only they are satisfied. The terms in
$F_{1}\circ F_{1}$ are in natural one-to-one correspondence with those in
$F_{1}^{2}$ with $a_{ij}(i|j)\circ a_{kl}(k|l)$ in the former corresponding,
in the notation of Section 3, to $a_{ij}(i|j)a_{kl}(k|l)$ in the latter. The
terms of differential order 4 in $F_{1}\circ F_{1}$ and
$\delta(-\frac{1}{2}F_{1}^{2})$ must coincide since that would be the case if
all the coefficients were constants. While $F_{1}^{2}$ contains terms of
differential order 2, these are cocycles and will not contribute to its
coboundary. The obstruction equations arise, therefore, from the difference
between the terms of order 3 of $F_{1}\circ F_{1}$ and
$\delta(-\frac{1}{2}F_{1}^{2})$. Analogous to the products in Theorem 1, there
are the following four kinds of terms in $F_{1}\circ F_{1}:$

$\displaystyle(1)$ $\displaystyle\sum_{1\leq i<j\leq d,1\leq k<l\leq
d}a_{ij}(i|j)\circ a_{kl}(k|l),\,\text{where}\,i,j,k,l\,\text{are all
distinct}$ $\displaystyle(2)$ $\displaystyle\sum_{1\leq i<j\leq d,1\leq
i<l\leq d}a_{i|j}(i|j)\circ a_{il}(i|l),\,\text{where}\,j,l\,\text{are
distinct}$ $\displaystyle(3)$ $\displaystyle\sum_{1\leq i<j\leq d,1\leq
k<j\leq d}a_{ij}(i|j)\circ a_{kj}(k|j),\,\text{where}\,i,k\,\text{are
distinct}$ $\displaystyle(4)$ $\displaystyle\,\,\,\,\,\,\,\,\,\,\,\sum_{1\leq
i<j\leq d}\,\,\,\,\,\,\,\,\,\,\,a_{ij}(i|j)\circ a_{ij}(i|j).$

Each composition product above gives rise to two terms of the form
$b_{ijk}(i|j|k)$, where there may be a single repetition among $i,j,k$. A
brief examination will show the following.

######  Lemma 3.

The sum of all the terms $b_{ijk}(i|j|k)$ in $F_{1}\circ F_{1}$ in which there
is a repetition amongst $i,j,k$ is precisely $\delta(-\frac{1}{2}F_{1}^{2})$.
$\Box$

It follows that $F_{1}\circ F_{1}-\delta(-\frac{1}{2}F_{1}^{2})$ is just the
sum of those terms without repetitions. The vanishing of the primary
obstruction to a general infinitesimal $F_{1}$ is therefore equivalent to
having the sum of those terms without repetitions be a coboundary. When the
dimension $d=2$, the HKR theorem implies that there can be no obstructions;
every infinitesimal deformation is then integrable. For every obstruction must
lie in $H^{3}(C^{\infty}(\mathbb{R}^{2}))$, whose elements are uniquely
represented by cocycles of the form $\sum_{1\leq i<j<k\leq d}a_{ijk}(i|j|k)$,
but there can be no such cocycles when $d=2$. When $d\geq 3$, for every triple
$(i,j,k)$ with $1\leq i<j<k\leq d$ a term of the form $b_{ijk}(i|j|k)$ appears
amongst the composition products above, e.g., in $a_{ii}(i|i)\circ
a_{jk}(j|k)$. If $(i^{\prime},j^{\prime},k^{\prime})$ is a permutation of
$(i,j,k$) then $(i^{\prime}|j^{\prime}|k^{\prime})$ is cohomologous to
$(i|j|k)$ if the permutation is even, and to $-(i|j|k)$ if the permutation is
odd. Therefore, the sum of the terms without repetitions is cohomologous to a
unique sum of the form $\sum_{1\leq i<j<k\leq d}a_{ijk}(i|j|k)$. Here the
$a_{ijk}$ are expressions in the coefficients $a_{ij}$ of $F_{1}=\sum_{1\leq
i<j\leq d}a_{ij}(i|j)$ and their partial derivatives with respect to the
variables $x_{1},\dots,x_{d}$. For the primary obstruction to vanish, each of
the $\binom{d}{3}$ coefficients $a_{ijk}$ must vanish, giving rise to
$\binom{d}{3}$ non-linear partial differential equations in the coefficients
of $F_{1}$.

##  13\. The obstruction equation in dimension $d=3$

In dimension 3, an infinitesimal deformation has the form
$F_{1}=a_{12}(1|2)+a_{13}(1|3)+a_{23}(2|3)$ and there is just one obstruction
equation. It has a remarkable property, expressed in Theorem 6, below. One has

$\displaystyle F_{1}\circ F_{1}=$ $\displaystyle a_{12}(1|2)\circ
a_{12}(1|2)+a_{13}(1|3)\circ a_{13}(1|3)+a_{23}(2|3)\circ a_{23}(2|3)+$
$\displaystyle a_{12}(1|2)\circ a_{13}(1|3)+a_{13}(1|3)\circ a_{12}(1|2)+$
$\displaystyle a_{12}(1|2)\circ a_{23}(2|3)+a_{23}(2|3)\circ a_{12}(1|2)+$
$\displaystyle a_{13}(1|3)\circ a_{23}(2|3)+a_{23}(2|3)\circ(a_{13}(1|3).$

After expanding the composition products above gives

$\displaystyle F_{1}\circ F_{1}-\delta(-\frac{1}{2}F_{1}^{2})=$
$\displaystyle\,\,\,\,\,\,\,\,(a_{12}\partial_{1}a_{13})(1|3|2)+(a_{13}\partial_{1}a_{12})(1|2|3)+$
$\displaystyle-(a_{12}\partial_{2}a_{23})(1|2|3)+(a_{23}\partial_{2}a_{12})(1|2|3)+$
$\displaystyle-(a_{13}\partial_{3}a_{23})(1|2|3)-(a_{23}\partial_{3}a_{13})(2|1|3).$

Since $(1|3|2)=-(1|2|3)+\delta(1|23)$ and $(2|1|3)=-(1|2|3)-\delta(12|3),$ we
can write the sum of the foregoing terms as

$[-a_{12}\partial_{1}a_{13}+a_{13}\partial_{1}a_{12}-a_{12}\partial_{2}a_{23}+a_{23}\partial_{2}a_{12}-a_{13}\partial_{3}a_{23}+a_{23}\partial_{3}a_{13}](1|2|3)\\\
+\delta(a_{12}\partial_{1}a_{13}+a_{23}\partial_{3}a_{13})(1|23).$

The single obstruction equation in dimension 3 is therefore

(18)
$-a_{12}\partial_{1}a_{13}+a_{13}\partial_{1}a_{12}-a_{12}\partial_{2}a_{23}+a_{23}\partial_{2}a_{12}-a_{13}\partial_{3}a_{23}+a_{23}\partial_{3}a_{13}=0.$

In any region where none of $a_{12},a_{13}.a_{23}$ vanish, (18) can be
rewritten as

(19)
$a_{12}^{2}\partial_{1}(\frac{a_{13}}{a_{12}})+a_{23}^{2}\partial_{2}(\frac{a_{12}}{a_{23}})+a_{13}^{2}\partial_{3}(\frac{a_{13}}{a_{23}})=0.$

######  Theorem 6.

If $F_{1}$ is an infinitesimal deformation of $C^{\infty}(\mathbb{R}^{3})$
whose primary obstruction vanishes then the same is true of $\varphi F_{1}$
for any $\varphi$ in $C^{\infty}(\mathbb{R}^{3})$. In particular, if a skew
biderivation $F_{1}$ of $C^{\infty}(\mathbb{R}^{3})$ is Poisson, then so is
$\varphi F_{1}$.

Proof. In (18), if $F_{1}$ is replaced by $\varphi F_{1}$, where $\varphi$ is
a smooth function of $x_{1},x_{2},x_{3}$, then the terms in which $\varphi$ is
differentiated cancel, so the summand on the left side is just multiplied by
$\varphi^{2}$, which may be more easily seen in (19). Therefore, if it is
satisfied by a triple of smooth functions $(a_{12},a_{13},a_{23})$ of
$x_{1},x_{2},x_{3}$, then $(\varphi a_{12},\varphi a_{13},\varphi a_{23})$ is
also a solution. $\Box$

The extraordinary property of dimension 3 presented in Theorem 6 does not
appear in dimension 4 and therefore neither in any higher dimension. In
dimension $d=4$, to see whether an infinitesimal deformation
$F_{1}=\sum_{1\leq i<j\leq 4}a_{ij}(i|j)$ has vanishing primary obstruction,
we need only examine those terms in $F_{1}\circ F_{1}$ which have differential
order three. The sum of these can be written as
$b_{123}(1|2|3)+b_{124}(1|2|4)+b_{134}(1|3|4)+b_{234}(2|3|4)+$(coboundaries).
The four obstruction equations assert the vanishing of the four $b_{ijk}$. It
will be sufficient to show for one of them, say $b_{123}$, that vanishing for
some $F_{1}$ does not imply that it also does so for all $\varphi F_{1}$,
where $\varphi$ is a smooth function of $x_{1},\dots,x_{4}$. The contributions
to $b_{123}$ come first from those terms in $F_{1}\circ F_{1}$ of the form
$a_{ij}(i|j)\circ a_{kl}(k|l)$ in which none of the indices $i,j,k,l$ is 4,
and second from from $a_{14}(1|4)\circ a_{23}(2|3),a_{24}(2|4)\circ
a_{13}(1|3)$, and $a_{34}(3|4)\circ a_{12}(1|2)$. In the first terms, if
$F_{1}$ is replaced by $\varphi F_{1}$ then, as we have seen with the single
obstruction equation in dimension three, the contribution from those terms is
just multiplied by $\varphi^{2}$. This is not the case with the second terms,
so the statement fails in dimension 4 and hence in all higher dimensions.

##  14\. Poisson structures in dimension three

It is an open problem to determine all Poisson structures on
$C^{\infty}(\mathbb{R}^{d})$ or equivalently to determine those infinitesimal
deformations whose primary obstructions vanish, since by Theorem 5 their skew
forms are precisely the Poisson structures. One can, however, construct a
large set of such in the special case of dimension $d=3$. As before, letting
$F_{1}=a_{12}(1|2)+a_{13}(1|3)+a_{23}(23)$ it will be seen that any one of the
functions $a_{12},a_{13},a_{23}$ can be chosen to be an arbitrary smooth
function of $x_{1},x_{2},x_{2}$. Because of the symmetry in the variables
suppose it is $a_{12}$. By Theorem 6 it will be sufficient then to give
examples where $a_{12}=1$ but note that those so obtained need not be all.
Writing again $\partial_{i}$ for $\partial_{x_{i}}$, (18) then becomes

(20)
$-\partial_{1}a_{13}-\partial_{2}a_{23}-a_{13}\partial_{3}a_{23}+a_{23}\partial_{3}a_{13}=0,$

which will be called the reduced obstruction equation in dimension 3. It can
be rewritten as

(21)
$(\partial_{2}+a_{13}\partial_{3})a_{23}-(\partial_{3}a_{13})a_{23}=-\partial_{1}a_{13}.$

To find solutions to this equation, observe that if we first choose for
$a_{13}$ an arbitrary smooth function of $x_{1},x_{2},x_{3}$ then (21) would
resemble a first order ordinary differential equation in a single variable $x$
of the form

(22) $\frac{dy}{dx}+p(x)y=q(x),$

with the differential operator $(\partial_{2}+a_{13}\partial_{3})$ in place of
$\frac{d}{dx}$, $-(\partial_{3}a_{13})$ in place of $p$, $-\partial_{1}a_{13}$
in place of $q$, and where $a_{23}$ is in place of the unknown $y$. Letting
$P$ be any function with $P^{\prime}=p$, equation (22) has, as a particular
solution,

(23) $y=\exp(-P(x))\int\exp(P(x))\,q(x)dx$

from which all others can be obtained by adding a solution to the homogeneous
equation $\frac{dy}{dx}+p(x)y=0$ These all have the form $c\,\exp(-P(x))$
where $c$ is a constant, so the general solution is

(24) $y=\exp(-P(x))\int\exp(P(x))\,q(x)dx+c\,\exp(-P(x)).$

The only properties of the exponential function needed for (24) are that for
any differentiable function $f$ of $x$ we have $(\exp f)^{\prime}=(\exp
f)f^{\prime}$ and $exp(-f)=(expf)^{-1}$. This determines the exponential
function up to a sign; note that (24) would remain correct if the exponential
were replaced by its negative. More generally, suppose that we have a
commutative unital ring $\mathcal{R}$ of arbitrary characteristic with a
derivation $\mathcal{D}$ and a function $\exp:\mathcal{R}\to\mathcal{R}$ such
that (i) for any $r\in\mathcal{R}$ we have $\exp(-r)=(\exp r)^{-1}$ and (ii)
$\mathcal{D}(\exp(r))=\exp r\,\mathcal{D}r$. If $p,q\in\mathcal{R}$ then the
solutions to the equation $\mathcal{D}y+py=q$ have exactly the same form as in
(24) but with the usual exponential function replaced by this $\exp$ and with
the integral replaced by any fixed preimage under $\mathcal{D}$. One then has

######  Theorem 7.

Let $\mathcal{R}$ be a commutative unital ring with a derivation $\mathcal{D}$
relative to which there is an ‘exponential function’ $\exp$ such that for any
$r\in\mathcal{R}$ we have (i) $\exp(-r)=(\exp r)^{-1}$ and (ii)
$\mathcal{D}(\exp r)=\exp(r)\,\mathcal{D}r$. If $P,q\in\mathcal{R}$ and
$\mathcal{D}P=p$ then a particular solution to

(25) $\mathcal{D}y+py=q$

is given by

(26) $y=\exp(-P)\int(\exp P)\,q,$

from which all other solutions may be obtained by adding a function of the
form $c\,\exp(-P)$, where $c$ is a constant for, i.e. annihilated by,
$\mathcal{D}$. $\Box$

Theorem 7 may apply when instead of functions of one real variable we have
functions of several. For ‘$\exp$’ we may then take the usual exponential
function. In prime characteristics one can use the Artin-Hasse exponential,
for a discussion of which cf e.g. [28]. In the classical case of a single
variable the derivation $\frac{d}{dx}$ is surjective on
$\mathbb{C}^{\infty}(\mathbb{R})$ so there is no restriction on $p$. This is
not the case in (21) since it is required that
$\partial_{3}a_{13}=(\partial_{2}+a_{13}\partial_{3})P$ for some
$P\in\mathbb{C}^{\infty}(\mathbb{R}^{3})$. However, if we take for $a_{13}$ a
function only of $x_{1}$ and $x_{2}$ then then $\partial_{3}$ will commute
with $\partial_{2}+a_{13}\partial_{3}$. We can then take any
$P_{1}\in\mathbb{C}^{\infty}(\mathbb{R}^{3})$ and set $P=\partial_{3}P_{1}$.
Applying Theorem 7 then gives a solution.

Applying Theorem 6 to the Poisson structure on the dual of the simple three
dimensional Lie algebra, cf §8, also gives a large family of integrable
infinitesimal deformations on $\mathbb{C}^{3}$. Theorem 6 gives a partial
order on integrable infinitesimals of $C^{\infty}(\mathbb{R}^{3})$, with
$F_{1}\prec F_{1}^{\prime}$ whenever $F_{1}^{\prime}=\varphi F_{1}$ for some
$\varphi$, and an equivalence relation when also $F_{1}\prec
F_{1}^{\prime}\prec F_{1}$. It also implies, for dimension 3, that if $F_{1}$
is a basic infinitesimal of $C^{\infty}(\mathbb{R}^{3})$ then $\varphi F_{1}$
is integrable for any $\varphi$ in $C^{\infty}(\mathbb{R}^{3})$; such
infinitesimals will be called quasibasic. For a basic infinitesimal
$F_{1}=f_{1}f_{2}\partial_{1}\\!\\!\smile\\!\\!\partial_{2}+f_{1}f_{3}\partial_{1}\\!\\!\smile\\!\\!\partial_{3}+f_{2}f_{3}\partial_{2}\\!\\!\smile\\!\\!\partial_{3}$
the stronger equations

(27)
$\partial_{1}(a_{12}/a_{13})=\partial_{3}(a_{13}/a_{23})=\partial_{2}(a_{23}/a_{12})=0$

hold. The analog is true in all dimensions.

######  Theorem 8.

Let $F_{1}=\sum_{i<j}a_{ij}\partial_{i}\\!\\!\smile\\!\\!\partial_{j}$ be an
infinitesimal deformation of $C^{\infty}(\mathbb{R}^{d}),$ where $d\geq 3$.
Suppose that the $a_{ij}$ are all invertible, and that for all $\\{i,j,k\\}$
with $i<j<k$ we have
$\partial_{i}(a_{ij}/a_{ik})=\partial_{j}(a_{ij}/a_{jk})=\partial_{k}(a_{ik}/a_{jk})=0$.
Denote $a_{ij}(x_{1},\dots,x_{i-1},0,x_{i+1},\dots,x_{d})$ by
$a_{ij}(\hat{x_{i}})$. Then $a_{ij}/a_{ij}(\hat{x_{i}})$ can be written in the
form $1+x_{i}u_{i}$, where $u_{i}$ is not a function of $x_{i}$. Further, if
the dimension is 3 then $F_{1}$ is quasibasic; there are single variable
functions $f_{1},f_{2},f_{3}$ of $x_{1},x_{2},x_{3}$, respectively, and a
function $\varphi$ such that

$a_{12}=\varphi f_{1}f_{2},\quad a_{13}=\varphi f_{1}f_{3}\quad a_{23}=\varphi
f_{2}f_{3}.$

Proof. For the first part, we can write
$a_{ij}/a_{ij}(\hat{x_{i}})=1+x_{i}u_{ij}$ for certain functions $u_{ij}$.
Suppose that $i<j<k$; the other cases are similar. By the hypotheses,
$[a_{ij}/a_{ij}(\hat{x_{i}})]/[a_{ik}/a_{ik}(\hat{x_{i}})]$ is not a function
of $x_{i}$ but equals $(1+x_{i}u_{ij})/(1+x_{i}u_{ik})$, so the latter is not
a function of $x_{i}$. However, when $x_{i}=0$ it is equal to 1, so in fact
$(1+x_{i}u_{ij})/(1+x_{i}u_{ik})$ is identically equal to 1, implying that
$u_{ij}=u_{ik}$. We may therefore write simply $u_{i}$ for $u_{ij}$.

Now let $d=3$, so
$F_{1}=a_{12}\partial_{1}\\!\\!\smile\\!\\!\partial_{2}+a_{13}\partial_{1}\\!\\!\smile\\!\\!\partial_{3}+a_{23}\partial_{2}\\!\\!\smile\\!\\!\partial_{3}.$
We may assume, without loss of generality, that $a_{12}=1$. From the
hypotheses it then follows that $a_{13}$ is not a function of $x_{1}$ and
$a_{23}$ is not a function of $x_{2}$. It follows that $a_{13}(\hat{x_{3}})$
is a function only of $x_{2}$, which we write as $1/f_{2}$, and that
$a_{23}(\hat{x_{3}})$ is a function only of $x_{1}$, which we write as
$1/f_{1}$. Then $a_{13}=(1+x_{3}u_{3})/f_{2}$, where $u_{3}$ can be a function
only of $x_{2}$ and $x_{3}$, and similarly $a_{23}=(1+x_{3}u_{3})/f_{2}$,
where $u_{3}$ now can be a function only of $x_{1}$ and $x_{3}$. Therefore, it
is a function of $x_{3}$ alone. Denoting $(1+x_{3}u_{3})$ by $f_{3}$, the
original $F_{1}$ is therefore equivalent to an infinitesimal with
$a_{12}=1,a_{13}=f_{3}/f_{2},a_{23}=f_{3}/f_{1}$. Multiplying by $f_{1}f_{2}$
gives the desired result. $\Box$

We do not know if the analogous result holds in higher dimensions.

##  15\. The Big Bang

Theorem 6 indicates an important property of the exceptional dimension 3,
which is the smallest in which obstructions can appear. It allows the
construction of infinitesimal deformations of
$\mathbb{C}^{\infty}(\mathbb{R}^{3})$ which are arbitrarily large somewhere
inside a bounded open set $U$ but which vanish outside $U$. For example, let
$\Pi$ be a non-zero Poisson structure on
$\mathbb{C}^{\infty}(\mathbb{R}^{3})$, e.g., the Kirillov-Kostant-Souriau
Poisson structure on the dual of the Lie algebra ${sl}_{2}$ of $2\times 2$
matrices of trace zero, and $\varphi$ be a smooth function which is
arbitrarily large in a smaller open set and which vanishes outside $U$; take
the infinitesimal to be $\varphi\Pi$.

The question now arises of whether there is an integral of the foregoing
$\varphi\Pi$ to a full deformation of $\mathbb{C}^{\infty}(\mathbb{R}^{3})$
which shares the property of vanishing outside $U$ and being arbitrarily large
somewhere inside. The integral provided by Kontsevich’s UDF has this property
but is purely formal; to be physically meaningful, the integral must in
addition be effective. If such exists, then by taking $U$ arbitrarily small
one can have a deformation of $\mathbb{C}^{\infty}(\mathbb{R}^{3})$ where
outside $U$ the products of smooth functions are unchanged but inside this
arbitrarily small $U$ the deformation is large. The problem is that the chosen
$\varphi$ can not be analytic, hence neither is $\varphi\Pi$, and Kontsevich’s
UDF is not effective for such infinitesimal deformations. However, as an
approximation, take $\varphi$ to be a highly peaked Gaussian distribution
which becomes very small outside $U$. The effect of the infinitesimal
deformation $\varphi\Pi$ may be said to be largely confined to $U$. It is
analytic and one can apply Kontsevich’s UDF.

Now let $\psi(t)$ be a smooth function of time. Then
$\psi(t)\varphi(x_{1},x_{2},x_{3})\Pi$ will again be integrable as an
infinitesimal deformation of $C^{\infty}(x_{1},x_{2},x_{3},t)$ since $\Pi$
involved only the space variables. We may take $\psi(t)$ to be arbitrarily
large at some point $t_{0}$ but vanishing outside an arbitrarily small
neighborhood of $t_{0}$. Taking increasingly peaked Gaussian distributions and
smaller intervals of time, in the limit, if that were meaningful, one would
have an extreme disturbance localized at a point in space and instant of time.
This might yield some insight into the nature of three-dimensional space and
the Big Bang.

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