0000000000013941
AUTHOR
Moshé Flato
showing 15 related works from this author
Closedness of Star Products and Cohomologies
1994
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.
Cohomology and Deformation of Leibniz Pairs
1995
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.
A natural and rigid model of quantum groups
1992
We introduce a natural (Frechet-Hopf) algebra A containing all generic Jimbo algebras U t (sl(2)) (as dense subalgebras). The Hopf structures on A extend (in a continuous way) the Hopf structures of generic U t (sl(2)). The Universal R-matrices converge in A\(\hat \otimes \)A. Using the (topological) dual of A, we recover the formalism of functions of noncommutative arguments. In addition, we show that all these Hopf structures on A are isomorphic (as bialgebras), and rigid in the category of bialgebras.
Non linear representations of Lie Groups
1977
International audience
On global solutions of the Maxwell-Dirac equations
1987
We prove, for the Maxwell-Dirac equations in 1+3 dimensions, that modified wave operators exist on a domain of small entire test functions of exponential type and that the Cauchy problem, inR+×R3, has a unique solution for each initial condition (att=0) which is in the image of the wave operator. The modification of the wave operator, which eliminates infrared divergences, is given by approximate solutions of the Hamilton-Jacobi equation, for a relativistic electron in an electromagnetic potential. The modified wave operator linearizes the Maxwell-Dirac equations to their linear part.
Quantum Groups, Star Products and Cyclic Cohomology
1993
After some historical remarks, we start with a rapid overview of the star-product theory (deformation of algebras of functions on phase space) and its applications to deformation-quantization. We then concentrate on Poisson-Lie groups and their “quantization”, give a star-product realization of quantum groups and discuss uniqueness and the rigidity as bialgebra of a universal model for the quantum SL(2) groups. In the last part we develop the notion of closed star-product (for which a trace can be defined on the algebra), show that it is classified by cyclic cohomology, permits to define a character and that there always exists one; finally we show that the pseudodifferential calculus on a …
THE MAXWELL–DIRAC EQUATIONS: ASYMPTOTIC COMPLETENESS AND THE INFRARED PROBLEM
1994
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.
On a possible origin of quantum groups
1991
A Poisson bracket structure having the commutation relations of the quantum group SLq(2) is quantized by means of the Moyal star-product on C∞(ℝ2), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra Uq(sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of Uq(sl(2)).
From where do quantum groups come?
1993
The phase space realizations of quantum groups are discussed using *-products. We show that on phase space, quantum groups appear necessarily as two-parameter deformation structures, one parameter (v) being concerned with the quantization in phase space, the other (η) expressing the quantum groups as “deformation” of their Lie counterparts. Introducing a strong invariance condition, we show the uniqueness of the η-deformation. This suggests that the strong invariance condition is a possible origin of the quantum groups.
Closed star products and cyclic cohomology
1992
We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy t…
Three-dimensional singletons
1990
The three-dimensional analog of singleton gauge theory turns out to be related to the topological gauge theory of Schwartz and Witten. It is a fully-fledged gauge theory, though it involves only a single scalar field. Real, physical degrees of freedom propagate in 3-space, but they are ‘confined’ in the sense that they cannot be detected locally. The physical Hamiltonian density is not zero, but it is concentrated on the boundary at spatial infinity. This boundary surface, a torus, supports a two-dimensional conformal field theory.
THREE-D SINGLETONS AND 2-D C.F.T.
1992
Two-dimensional Wess-Zumino-Novikov-Witten theory is extended to three dimensions, where it becomes a scalar gauge theory of the singleton type. The three-dimensional formulation involves a scalar field valued in a compact group G, a Nakanishi-Lautrup field valued in Lie (G) and Faddeev-Popov ghosts. The physical sector, characterized by the vanishing of the Nakanishi-Lautrup field, coincides with the WZNW theory of the group G. Three-dimensional space-time structure involves a generalized metric, but only its boundary values are of consequence. An alternative formulation in terms of left and right movers (in three dimensions!) is also possible.
The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations
1994
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.
Remarks on quantum groups
1991
We give a Poisson-bracket realization of SL q (2) in the phase space ℝ2. We then discuss the physical meaning of such a realization in terms of a modified (regularized) toy model, the nonregularized version of which is due to Klauder. Some general remarks and suggestions are also presented in this Letter.
Quantum deformations of singletons and of free zero-mass fields
1993
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(sl2) as a quantum subalgebra.