0000000000398192
AUTHOR
Philippe Bonneau
showing 9 related works from this author
TOPOLOGICAL QUANTUM DOUBLE
1994
Following a preceding paper showing how the introduction of a t.v.s. topology on quantum groups led 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. A la suite d'un précédent article montrant comment l'introduction d'une topologie d'e.v.t. sur les groupes quantiques permet une unification et une rigidification remarquables des différentes définitions,…
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.
Cohomology and associated deformations for not necessarily co-associative bialgebras
1992
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.
On the geometry of the characteristic class of a star product on a symplectic manifold
2001
The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. 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 natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter $\hbar$. We then s…
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.
Topological Hopf Algebras, Quantum Groups and Deformation Quantization
2019
After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologi es on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities a nd provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.
Compact Topological Quantum Groups
1995
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.
Topological Hopf algebras, quantum groups and deformation quantization
2003
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
The hidden group structure of Quantum Groups:strong duality,rigidity and preferred deformations
1994
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