# The quantum duality principle

Fabio Gavarini^{[1]}

- [1] Università degli Studi di Roma "Tor Vergata", Dipartimento di Matematica, Via della Ricerca Scientifica 1, 00133 Roma (Italie)

Annales de l’institut Fourier (2002)

- Volume: 52, Issue: 3, page 809-834
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topGavarini, Fabio. "The quantum duality principle." Annales de l’institut Fourier 52.3 (2002): 809-834. <http://eudml.org/doc/115995>.

@article{Gavarini2002,

abstract = {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 $\{\mathcal \{Q\}\}\{\mathcal \{U\}\}\{\mathcal \{E\}\}\{\mathcal \{A\}\}\longrightarrow \{\mathcal \{Q\}\}\{\mathcal \{F\}\}\{\mathcal \{S\}\}\{\mathcal \{H\}\} \{\mathcal \{A\}\}$ and $\{\mathcal \{Q\}\}\{\mathcal \{F\}\}\{\mathcal \{S\}\}\{\mathcal \{H\}\} \{\mathcal \{A\}\}\longrightarrow \{\mathcal \{Q\}\}\{\mathcal \{U\}\}\{\mathcal \{E\}\}\{\mathcal \{A\}\}$, 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.},

affiliation = {Università degli Studi di Roma "Tor Vergata", Dipartimento di Matematica, Via della Ricerca Scientifica 1, 00133 Roma (Italie)},

author = {Gavarini, Fabio},

journal = {Annales de l’institut Fourier},

keywords = {quantum groups; topological Hopf algebras},

language = {eng},

number = {3},

pages = {809-834},

publisher = {Association des Annales de l'Institut Fourier},

title = {The quantum duality principle},

url = {http://eudml.org/doc/115995},

volume = {52},

year = {2002},

}

TY - JOUR

AU - Gavarini, Fabio

TI - The quantum duality principle

JO - Annales de l’institut Fourier

PY - 2002

PB - Association des Annales de l'Institut Fourier

VL - 52

IS - 3

SP - 809

EP - 834

AB - 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 ${\mathcal {Q}}{\mathcal {U}}{\mathcal {E}}{\mathcal {A}}\longrightarrow {\mathcal {Q}}{\mathcal {F}}{\mathcal {S}}{\mathcal {H}} {\mathcal {A}}$ and ${\mathcal {Q}}{\mathcal {F}}{\mathcal {S}}{\mathcal {H}} {\mathcal {A}}\longrightarrow {\mathcal {Q}}{\mathcal {U}}{\mathcal {E}}{\mathcal {A}}$, 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.

LA - eng

KW - quantum groups; topological Hopf algebras

UR - http://eudml.org/doc/115995

ER -

## References

top- N. Abe, Hopf algebras, 74 (1980), Cambridge University Press, Cambridge Zbl0476.16008MR594432
- N. Ciccoli, F. Gavarini, A quantum duality principle for Poisson homogeneous spaces Zbl1296.17006
- V. G. Drinfeld, Quantum groups, Proc. Intern. Congress of Math. (Berkeley, 1986) (1987), 798-820 Zbl0667.16003
- B. Enriquez, Quantization of Lie bialgebras and shuffle algebras of Lie algebras, (2000) Zbl1009.17010MR1868300
- P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, I, Selecta Math. (New Series) 2 (1996), 1-41 Zbl0863.17008MR1403351
- P. Etingof, D. Kazhdan, Symétries quantiques, (Les Houches, 1995) (1995), 935-946, North-Holland, Amsterdam Zbl0962.17008
- L. D. Faddeev, N. Yu. Reshetikhin, L. A. Takhtajan, Quantum groups, Algebraic Analysis (1989), 129-139, Academic Press, Boston
- F. Gavarini, Dual affine quantum groups, Math. Zeitschrift 234 (2000), 9-52 Zbl1015.17014MR1759490
- F. Gavarini, The global quantum duality principle: theory, examples, applications, (2001) Zbl1224.17021MR1877138
- C. Kassel, V. Turaev, Biquantization of Lie bialgebras, Pac. Jour. Math 195 (2000), 297-369 Zbl1040.17008MR1782170
- S. Montgomery, Hopf Algebras and Their Actions on Rings, 82 (1993), American Mathematical Society, Providence, RI Zbl0793.16029MR1243637

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.