We review the developments in the past twenty years (which are based on our deformation philosophy of physical theories) dealing with elementary particles composed of singletons in anti De Sitter space-time. The study starts with the kinematical aspects (especially for massless particles) and extends to the beginning of a field theory of composite elementary particles and its relations with conformal field theory (including very recent developments).

We write the integral formula of Tarasov-Varchenko type for the solutions to the quantum Knizhnik-Zamolodchikov equation associated with a tensor product of the vector representations of sln. We consider the case where the deformation parameter q satisfies |q| = 1. We use the bosonization of the type II vertex operators in order to find the hypergeometric pairing in this setting.

The study starts with the kinematical aspects of singletons and massless particles. It extends to the beginning of a field theory of composite elementary particles and its relations with conformal field theory, including very recent developments and speculations about a possible interpretation of neutrino oscillations and CP violation in this context. This framework was developed during the past twenty years. Based on our deformation philosophy of physical theories, it deals with elementary particles composed of singletons in anti-De Sitter spacetime.

We present briefly the deformation philosophy and indicate, with references, how it was applied to the quantization of Nambu mechanics and to particle physics in anti De Sitter space.

We indicate similarities in the structure of two types of infinite-dimensional algebras, one introduced 28 years ago in connection with the mass problem of elementary particles and the other seven years ago in connection with spin systems (XY models). We show that these algebras can be considered as representations of a single ‘Grand Algebra’, the enveloping algebra of an affine Kac–Moody algebra built on the Poincaré Lie algebra. As an associative and coassociative bialgebra of operators, the latter representation of the grand algebra is a preferred nontrivial deformation of the Ising case bialgebra.

There is a chance that singleton fields, that in the context of strings and membranes have been regarded as topological gauge fields that can interact only at the boundary of anti-De Sitter space, at spatial infinity, may have a more physical manifestation as constituents of massless fields in spacetime. The composite character of massless fields is expressed by field-current identities that relate ordinary massless field operators to singleton currents and stress-energy tensors. Naive versions of such identities do not make sense, but when the singletons are described in terms of dipole structures, then such constructions are at least formally possible. The new proposal includes and generalizes an early composite version of QED, and includes quantum gravity, super gravity and models of QCD. Unitarity of such theories is conjectural.

We start with an overview of the "generalized Hamiltonian dynamics" introduced in 1973 by Y. Nambu, its motivations, mathematical background and subsequent developments -- all of it on the classical level. This includes the notion (not present in Nambu's work) of a generalization of the Jacobi identity called Fundamental Identity. We then briefly describe the difficulties encountered in the quantization of such $n$-ary structures, explain their reason and present the recently obtained solution combining deformation quantization with a "second quantization" type of approach on ${\Bbb R}^n$. The solution is called "Zariski quantization" because it is based on the factorization of (real) polynomials into irreducibles. Since we want to quantize composition laws of the determinant (Jacobian) type and need a Leibniz rule, we need to take care also of derivatives and this requires going one step further (Taylor developments of polynomials over polynomials). We also discuss a (closer to the root, "first quantized") approach in various circumstances, especially in the case of covariant star products (exemplified by the case of su(2)). Finally we address the question of equivalence and triviality of such deformation quantizations of a new type (the deformations of algebras are more general than those considered by Gerstenhaber).

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

We study Abelian generalized deformations of the usual product of polynomials introduced in an earlier work. We construct an explicit example for the case of \(\mathfrak{s}\mathfrak{u} \)(2) which provides a tentative quantum-mechanical description of Nambu mechanics on \(\mathbb{R}^3 \). By introducing the notions of strong and weak triviality of generalized deformations, we show that the Zariski product is never trivial in either sense, while the example constructed here in a quantum-mechanical context is strongly nontrivial.

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra $A$ together with a Lie algebra $L$ mapped into the derivations of $A$. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.

In this article we present an announcement of results concerning: a) A solution to the Cauchy problem for the M-D equations, namely global existence, for small initial data at t = 0, of solutions for the M-D equations. b) Arguments from which asymptotic completeness for the M-D equations follows. c) Cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron in M-D classical field theory. The full detailed results will be published elsewhere.

A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients andC ∞-functions. Strong rigidity (H bi 2 ={0}) under deformations in the category of bialgebras is proved and consequences are deduced.

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called " closed star products " and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called "closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.

This short summary of recent developments in quantum compact groups and star products is divided into 2 parts. In the first one we recast star products in a more abstract form as deformations and review its recent developments. The second part starts with a rapid presentation of standard quantum group theory and its problems, then moves to their completion by introduction of suitable Montel topologies well adapted to duality. Preferred deformations (by star products and unchanged coproducts) of Hopf algebras of functions on compact groups and their duals, are of special interest. Connection with the usual models of quantum groups and the quantum double is then presented.

Illustrates developments in the many areas of contemporary physics which have benefitted from the work and influence of Yvonne Choquet-Bruhat. The work considers the role of geometry of manifolds in space-time physics and that of functional analysis in quantum mechanics and quantum field theory.

We consider quantum deformations of the real symplectic (or anti-De Sitter) algebra sp(4), ℝ ≅ spin(3, 2) and of its singleton and (4-dimensional) zero-mass representations. For q a root of −1, these representations admit finite-dimensional unitary subrepresentations. It is pointed out that Uq(sp(4, ℝ)), unlike Uq(su(2, 2)), contains Uq(sl 2 ) as a quantum subalgebra.