Alain Connes

• Institute of Advanced Scientific Studies

In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number theory, naturally extend the classical theory. This generalization transitions from the idele class group, which a...

We integrate in the framework of the semilocal trace formula two recent discoveries on the spectral realization of the zeros of the Riemann zeta function by introducing a semilocal analogue of the prolate wave operator. The latter plays a key role both in the spectral realization of the low lying zeros of zeta—using the positive part of its spectru...

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of \mathbb{S}[\mu_{n,+}] -polynomials and relate this concept to the known notion of number systems. The Riemann–Roch theorem for the ring \mathbb{Z} of the integers that we obtained recently uses the understanding of \mathbb{Z} as a ring of polynomials \mathb...

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of $\mathbb S[\mu_{n,+}]$--polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring $\mathbb Z$ of the integers that we obtained recently uses the understanding of $\mathbb Z$ as a ring of polynomials $\m...

The meeting displayed the cyclic theory as a fundamental mathematical tool with applications in diverse domains such as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory.

We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S . We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wa...

We show that by working over the absolute base $\mathbb S$ (the categorical version of the sphere spectrum) instead of $\mathbb S[\pm 1]$ improves our previous Riemann-Roch formula for $\overline{{\rm Spec\,}\mathbb Z}$. The formula equates the (integer-valued) Euler characteristic of an Arakelov divisor with the sum of the degree of the divisor (u...

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the co...

We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the exampl...

We give a historical perspective on the role of the cyclic category in the development of cyclic theory. This involves a continuous interplay between the extension in characteristic one and in S-algebras, of the traditional development of cyclic theory, and the geometry of the toposes associated with several small categories involved. We clarify th...

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the co...

In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator. The second spectral realization affects only the critical zeros and it is cast in terms of sheaf cohomology....

We describe a remarkable property of the self-adjoint extension of the prolate spheroidal operator introduced in 1998 by A.C. The restriction of this operator to the interval whose characteristic function commutes with it is well known, has a discrete positive spectrum, and is well understood. What we have discovered is that the restriction of the...

We prove a Riemann-Roch formula for Arakelov divisors on $\overline{\text{Spec}\mathbb Z}$ equating the integer valued Euler characteristic with a simple modification of the traditional expression (i. e. the degree of the divisor plus log 2). The integer valued topological side involves besides the ceiling function, the division by log 3. The notio...

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the "zeta sector" of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e. K-theory of endomorphisms) of the "algebraic closure...

In this paper we describe a remarkable new property of the self-adjoint extension W of the prolate spheroidal operator introduced in \cite{college98},\cite{CMbook}. The restriction of this operator to the interval J whose characteristic function commutes with it is well known, has discrete positive spectrum and is well understood. What we have disc...

We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the exampl...

We develop algebraic geometry for general Segal's Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under SpecZ). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in [3] to define F...

We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of Connes (Sel Math (NS) 5(1):29–106, 1999). We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action...

The workshop on “Non-Commutative Geometry and Cyclic Homology” was attended by 16 participants on site. 30 participants could not travel to Oberwolfach because of the pandemia and took advantage of the videoconference tool. This report contains the extended abstracts of the lectures both given on site and externally.

We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wav...

In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional ro...

We introduce the notion of quasi-inner function and show that the product u=ρ∞∏ρv of m+1 ratios of local L-factors ρv(z)=γv(z)/γv(1−z) over a finite set F of places of Q inclusive of the archimedean place is quasi-inner on the left of the critical line ℜ(z)=12 in the following sense. The off diagonal part u21 of the matrix of the multiplication by...

Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use G. Almkvist’s global Witt construction to explore the relation with J. Borger ${\mathbb F}_1$-geometry and compute the Witt functor-ring ${\mathbb W}_0({\mathbb S})$ of the simplest Γ-ring ${\mathbb S}$. We prove that it is isomorphic to the Galois invariant part of...

We reconcile, at the semi-classical level, the original spectral realization of zeros of the Riemann zeta function as an ``absorption'' picture using the ad\`ele class space, with the ``emission'' semi-classical computations of Berry and Keating. We then use the quantized calculus to analyse the recent attempt of X.-J.~Li at proving Weil's positivi...

In this article on mathematics and music, we explain how one can “listen to motives” as rhythmic interpreters. In the simplest instance which is the one we shall consider, the motive is simply the H 1 of the reduction modulo a prime p of an hyperelliptic curve (defined over Q ). The corresponding time onsets are given by the arguments of the comple...

We introduce the notion of {\it quasi-inner} function and show that the product $u=\rho_\infty\prod \rho_v$ of $m+1$ ratios of local {$L$-}factors {$\rho_v(z)=\gamma_v(z)/\gamma_v(1-z)$} over a finite set $F$ of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line $\Re(z)= \f...

We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of the paper "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function". (Selecta Math. 5 (1999), no. 1, 29--106). We explore in great details the simplest case of the single...

In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional ro...

Segal's Gamma-rings provide a natural framework for absolute algebraic geometry. We use Almkvist's global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt functor-ring of Almkvist for the simplest Gamma-ring S. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the clos...

We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann...

We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the semi-classical level between the "absorption" picture of A. Connes and the semiclassical "emission" computations of M. Berry and J. Keating, while the minus sign...

We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi us...

Sir Michael Atiyah was considered as one of the world's foremost mathematicians, He is best known for his work in algebraic topology and the co-development of a branch of mathematics called topological K-theory together with the Atiyah-Singer index theorem for which he received Fields Medal (1966). He received also the Abel Prize (2004) along with...

We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in our previo...

Noncommutative geometry today is a new but mature branch of mathematics shedding light on many other areas from number theory to operator algebras. In the 2018 meeting two of these connections were high-lighted. For once, the applications to mathematical physics, in particular quantum field theory. Indeed, it was quantum theory which told us first...

We define the homology of a simplicial set with coefficients in a Segal's $\Gamma$-set ($\mathbf S$-module). We show the relevance of this new homology with values in $\mathbf S$-modules by proving that taking as coefficients the $\mathbf S$-modules at the archimedean place over the structure sheaf on $\overline{Spec\mathbb Z}$ introduced in our pr...

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit–Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated rep...

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit-Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated rep...

We describe the Riemann–Roch strategy which consists of adapting in characteristic zero Weil’s proof, of RH in positive characteristic, following the ideas of Mattuck–Tate and Grothendieck. As a new step in this strategy we implement the technique of tropical descent that allows one to deduce existence results in characteristic one from the Riemann...

In this note, of recreational nature, on mathematics and music, we explain how one can "listen to motives" as rhythmic interpreters. In the simplest instance which is the one we shall consider, the motive is simply the H1 of the reduction modulo a prime p of an hyperelliptic curve (defined over the rationals). The corresponding times are given by t...

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple (A,H,D) where D is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild c...

We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann...

We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in...

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive...

We study the series s(n, x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n, x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x...

The ideas of noncommutative geometry are deeply rooted in both physics, with the predominant influence of the discovery of Quantum Mechanics, and in mathematics where it emerged from the great variety of examples of “noncommutative spaces” i.e. of geometric spaces which are best encoded algebraically by a noncommutative algebra.

We describe the Riemann-Roch strategy which consists of adapting in characteristic zero Weil's proof, of RH in positive characteristic, following the ideas of Mattuck, Tate and Grothendieck. As a new step in this strategy we implement the technique of tropical descent that allows one to deduce existence results in characteristic one from the Rieman...

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$ . We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

Завершено доказательство теоремы о следах в квантованном исчислении для квазифуксовых групп. Эта теорема была сформулирована на с. 322-325 в книге "Noncommutative geometry" первого автора, где имеется только набросок доказательства. Предложено полное доказательство. Библиография: 35 названий.

We construct the scaling site S by implementing the extension of scalars on the arithmetic site, from the smallest Boolean semifield to the tropical semifield of positive real numbers. The obtained semiringed topos is the Grothendieck topos semi-direct product of the Euclidean half-line and the monoid of positive integers acting by multiplication,...

We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry" of the first author.

We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry" of the first author.

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for ab...

We give a survey of our joint ongoing work with Ali Chamseddine, Slava Mukhanov and Walter van Suijlekom. We show how a problem purely motivated by "how geometry emerges from the quantum formalism" gives rise to a slightly noncommutative structure and a spectral model of gravity coupled with matter which fits with experimental knowledge. This text...

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for ab...

We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of...

The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic” and is based...

We construct the scaling site S by implementing the extension of scalars on the arithmetic site, from the smallest Boolean semifield to the tropical semifield of positive real numbers. The obtained semiringed topos is the Grothendieck topos semi-direct product of the Euclidean half-line and the monoid of positive integers acting by multiplication,...

Text We show that the basic categorical concept of an S-algebra as derived from the theory of Segal's Γ-sets provides a unifying description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoïds, semirings and hyperrings as well as the development by means...

The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is "analytic" and is based...

We analyze the running at one-loop of the gauge couplings in the spectral Pati-Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario's for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different...

We investigate the semi-ringed topos obtained by extension of scalars from the arithmetic site of our previous work, by replacing the smallest Boolean semifield by the tropical semifield of real numbers with the max-plus operations. The obtained site is the semi-direct product of the Euclidean half-line by the action of the monoid of positive integ...

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold...

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of max-plus integers ℤ max . Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the...

We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adele class space of the field of rational numbers by the maximal compact subgroup of the idele class group, which we had previously shown to yield...

We show that the basic categorical concept of an s-algebra as derived from the theory of Segal's Gamma-sets provides a unified description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoids, semirings and hyperrings as well as the development by means of...

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case...

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold...

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring, provides the right theory to obtain, using λ-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of max-plus integers. An object of thi...

We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the "Arithmetic Site". This site involves the tropical semiring viewed as a sheaf on the topos which is the dual of the multiplicative semigroup of positive integers. We prove that the set of points of the arithmetic site over th...

This book gets to the heart of science by asking a fundamental question about its essence: what is the true nature of space and time? Both defy modern physics and scientists find themselves continually searching for answers. This unique volume brings together world leaders in cosmology, particle physics, quantum gravity, mathematics, philosophy and...

Given a point p of the topos of simplicial sets and the corresponding flat covariant functor F from the small category Delta to the category of sets, we determine the extensions of F to the cyclic category. We show that to each such cyclic structure on a point p of the topos of simplicial sets corresponds a group G(p), that such groups can be nonco...

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield F of "max-plus integers". Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one...

Given a point p of the topos of simplicial sets and the corresponding flat covariant functor F from the small category Delta to the category of sets, we determine the extensions of F to the cyclic category. We show that to each such cyclic structure on a point p of the topos of simplicial sets corresponds a group G(p), that such groups can be nonco...

The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition...

We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra A and which extends the unitary group of A. This has a key application in noncommutative sp...

Noncommutative Geometry applies ideas from geometry to mathematical structures determined by noncommuting variables. This meeting emphasized the connections of Noncommutative Geometry to number theory and ergodic theory.

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring of infinite adeles of K, provides the right theory to obtain, using the lambda-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

We show that the inconsistency between the spectral Standard Model and the experimental value of the Higgs mass is resolved by the presence of a real scalar field strongly coupled to the Higgs field. This scalar field was already present in the spectral model and we wrongly neglected it in our previous computations. It was shown recently by several...

General Relativity describes spacetime as far as large scales are concerned and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincaré, Einstein and Minkowski by a curved spacetime metric whose components form the gravitational potential. The basic equations are Einstein equations (Figure 4.1 in t...

We define the universal 1-adic thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the "dequantization" process and yields...

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator...

In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding σ of the multiplicative group of an algebraic closure of \mathbb Fp{\mathbb {F}_p} as complex roots of unity, we construct a p-adic indecomposable representation πσ of th...

We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general case of Robertson-Walker metrics. We check the terms of the expansion up to a_6 against the known universal for...

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of an algebraic closure of F_p. The obtained representations are the p-adic analogues of the complex, e...

TextWe show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K× of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R× is a subgroup of its multiplicative group) is a hyperring...

We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In pa...

These reports contain an account of 2015’s meeting on noncommutative geometry. Noncommutative geometry has developed itself over the years to a completely new branch of mathematics shedding light on many other areas as number theory, differential geometry and operator algebras. A connection that was highlighted in particular in this meeting was the...

I present here some recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) about the “characteristic $1$” limit case. The main goal is to prove that the adèle class space of a global field, which, up to now, has only been considered as a non-commutative space, has in fact a natural algebraic structure. We will also see that t...

I present here some recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) about the "characteristic 1" limit case. The main goal is to prove that the adèle class space of a global field, which, up to now, has only been considered as a noncommutative space, has in fact a natural algebraic structure. We will also see that the...