no. 3
The quantum duality principle
[Le principe de dualité quantique]
Gavarini, Fabio1
1 Università degli Studi di Roma "Tor Vergata", Dipartimento di Matematica, Via della Ricerca Scientifica 1, 00133 Roma (Italie)
Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 809-834.

Le “principe de dualité quantique” affirme que la quantification d’une bigèbre de Lie - moyennant une algèbre enveloppante universelle quantifiée (en abrégé, une QUEA) - donne aussi une quantification de la bigèbre de Lie duale (via son groupe de Poisson formel associé) - moyennant une algèbre de Hopf de séries formelles (QFSHA) - et, vice versa, une QFSHA associée à une bigèbre de Lie (via son groupe de Poisson formel associé) donne également une QUEA pour la bigèbre de Lie duale. Plus précisément, il existe deux foncteurs 𝒬𝒰ℰ𝒜⟶𝒬ℱ𝒮ℋ𝒜 et 𝒬ℱ𝒮ℋ𝒜⟶𝒬𝒰ℰ𝒜, inverses l’un de l’autre, tels que dans les deux cas la bigèbre de Lie associée à l’objet cible est la duale de celle de l’objet source. Une version plus faible de ce type de résultats avait été annoncée par Drinfeld, mais aucune preuve ne figure dans la littérature : j’en donne ici une preuve complète et détaillée.

The “quantum duality principle” states that the quantization of a Lie bialgebra – via a quantum universal enveloping algebra (in short, QUEA) – also provides a quantization of the dual Lie bialgebra (through its associated formal Poisson group) – via a quantum formal series Hopf algebra (QFSHA) — and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more in detail, there exist functors 𝒬𝒰ℰ𝒜⟶𝒬ℱ𝒮ℋ𝒜 and 𝒬ℱ𝒮ℋ𝒜⟶𝒬𝒰ℰ𝒜, inverse to each other, such that in both cases the Lie bialgebra associated to the target object is the dual of that of the source object. Such a result was claimed true by Drinfeld, but seems to be unproved in the literature: I give here a thorough detailed proof of it.

DOI : 10.5802/aif.1902
Classification : 17B37, 20G42, 81R50, 16W30
Keywords: quantum groups, topological Hopf algebras
Mots-clés : groupes quantiques, algèbres de Hopf topologiques

Gavarini, Fabio 1

1 Università degli Studi di Roma "Tor Vergata", Dipartimento di Matematica, Via della Ricerca Scientifica 1, 00133 Roma (Italie) 
@article{AIF_2002__52_3_809_0,
     author = {Gavarini, Fabio},
     title = {The quantum duality principle},
     journal = {Annales de l'Institut Fourier},
     pages = {809--834},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {3},
     year = {2002},
     doi = {10.5802/aif.1902},
     zbl = {1054.17011},
     mrnumber = {1907388},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1902/}
}
TY  - JOUR
AU  - Gavarini, Fabio
TI  - The quantum duality principle
JO  - Annales de l'Institut Fourier
PY  - 2002
SP  - 809
EP  - 834
VL  - 52
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1902/
DO  - 10.5802/aif.1902
LA  - en
ID  - AIF_2002__52_3_809_0
ER  - 
%0 Journal Article
%A Gavarini, Fabio
%T The quantum duality principle
%J Annales de l'Institut Fourier
%D 2002
%P 809-834
%V 52
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1902/
%R 10.5802/aif.1902
%G en
%F AIF_2002__52_3_809_0
Gavarini, Fabio. The quantum duality principle. Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 809-834. doi : 10.5802/aif.1902. https://aif.centre-mersenne.org/articles/10.5802/aif.1902/

[A] N. Abe Hopf algebras, Cambridge Tracts in Mathematics, 74, Cambridge University Press, Cambridge, 1980 | MR | Zbl

[CG] N. Ciccoli; F. Gavarini A quantum duality principle for Poisson homogeneous spaces (Preprint)

[Dr] V. G. Drinfeld Quantum groups, Proc. Intern. Congress of Math. (Berkeley, 1986) (1987), pp. 798-820 | Zbl

[E] B. Enriquez Quantization of Lie bialgebras and shuffle algebras of Lie algebras (2000) (e-print, math.QA/0008128) | MR

[EK1] P. Etingof; D. Kazhdan Quantization of Lie bialgebras, I, Selecta Math. (New Series), Volume 2 (1996), pp. 1-41 | DOI | MR | Zbl

[EK2] P. Etingof; D. Kazhdan Symétries quantiques, (Les Houches, 1995) (1995), pp. 935-946 | Zbl

[FRT] L. D. Faddeev; N. Yu. Reshetikhin; L. A. Takhtajan; M. Kashiwara, T. Kawai (eds.) Quantum groups, Algebraic Analysis (1989), pp. 129-139 | Zbl

[Ga1] F. Gavarini Dual affine quantum groups, Math. Zeitschrift, Volume 234 (2000), pp. 9-52 | DOI | MR | Zbl

[Ga2] F. Gavarini The global quantum duality principle: theory, examples, applications (2001) (e-print, math.QA/0108015) | MR | Zbl

[KT] C. Kassel; V. Turaev Biquantization of Lie bialgebras, Pac. Jour. Math, Volume 195 (2000), pp. 297-369 | DOI | MR | Zbl

[M] S. Montgomery Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, 82, American Mathematical Society, Providence, RI, 1993 | MR | Zbl

Cité par Sources :