The quantum duality principle

Gavarini, 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 -