Abstract. We introduce a natural (Fréchet - Hopf) algebra
A containing all generic Jimbo algebras Ut (sl(2)) (as dense
subalgebras). The Hopf structures on A extend (in a continuous way) the
Hopf structures of generic Ut(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.
"Cohomology and associated deformations for non necessarily coassociative
bialgebras",
Lett. Math. Phys. 26 (1992), 277-283.
Abstract. In this Letter, a cohomology and an associated theory
of deformations for (not necessarily co-associative) bialgebras are studied.
The cohomology was introduced in a previous paper (Lett. Math. Phys.
25, 75-84(1992)). This theory has several advantages, especially
in calculating cohomology spaces and in its adaptability to deformations
of quasi-co-associative (qca) bialgebras and even quasi-triangular qca
bialgebras.
"Hidden group structure of quantum groups : strong duality, rigidity
and preferred deformations", with M. FLATO, M. GERSTENHABER and G.
PINCZON,
Comm. Math. Phys. 16 (1994), 125-156.
Abstract. 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 and C^\infty-functions.
Strong rigidity (Hbi2= {0}) under deformations in
the category of bialgebras is proved and consequences are deduced.
"Topological quantum double",
Rev. Math. Phys. vol 16, n°2 (1994), 305-318.
Abstract. Following a preceding paper showing how the introduction
of a t.v.s. topology on quantum groups leads to a remarkable unification
and rigidification of the different definitions, we adapt here, in
the same way, the definition of quantum double. This topological double
is dualizable and reflexive (even for infinite dimensional
algebras).
In a simple case we show, considering the double as the "zero class"
of an extension theory, the uniqueness of the double structure as a quasi-Hopf
algebra.
"Compact topological quantum groups",
in Modern Group Theoretical Methods in Physics, (1995),
87-96. J. Bertrand et al.(eds.), Kluwer.
Abstract. Using vector spaces topologies we unify the different
models of quantum groups. Duality and reflexivity are built in. The Drinfeld
deformation can be extended to the distributions on a simple compact Lie
group and dually to the infinitely differentiable functions. The topological
quantum double is similarly defined and a uniqueness result is obtained.
"Star-Products and Quantum Groups", ( dvi
or ps)
in Proceedings of the 1995 Bialowieza Workshop, Polish Scientific
Publishers.
Abstract. We show that the theory of quantum groups is a part
of the theory of star-products and make a review of the results obtained
with this approach.
"Fedosov star-products and 1-differentiable deformations", (ps)
Lett. Math. Phys. 45 (1998), 363-376.
Abstract. We show that every star product on a symplectic manifold
defines uniquely a 1-differentiable deformation of the Poisson bracket. Explicit formulas
are given. As a corollary we can identify the characteristic class
of any star product as a part of its explicit (Fedosov) expression.
"Classifications of star products
and deformations of Poisson brackets",
Banach Center Publ. 51 (2000), 25-29.
Abstract.
On the algebra of functions on a symplectic manifold we consider
the pointwise product and the Poisson bracket: after a brief review of the
classifications of the deformations of these structures,
we give explicit formulas relating a star product with its classifying formal
Poisson bivector.
"On the geometry of the characteristic class of
a star product on a symplectic manifold",with P. BIELIAVSKY,
Reviews in Mathematical Physics.
Abstract.
The characteristic class of a star product on a symplectic manifold
obtained by Fedosov's method
appears as the class of a deformation of a given symplectic connection.
In contrast, one usually thinks of the characteristic class of a star
product as the class of a deformation of the Poisson
structure (as in Kontsevich's work).
In this paper, we present, in the symplectic framework,
a universal procedure for constructing a star product
by directly quantizing a perturbation of the symplectic structure.
We then show that the class of the resulting star product coincides
with the de Rham class of the symplectic perturbation we started with.
Moreover, within a given class, equivalences of star products are realized
as infinite jets of one-parameter families of diffeomorphisms obtained from
a Moser argument at the classical level.
"Topological Hopf algebras, quantum groups and deformation
quantization", with D. STERNHEIMER,
Proceedings de la conférence
"Hopf Algebras in Noncommutative Geometry and Physics", Bruxelles, 2002.
Abstract. After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologies on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities and provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.