We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum duality principle" for quantum groupoids, which extends the one for quantum groups - dealing with Hopf algebras - originally introduced by Drinfeld (cf. [V. G. Drinfeld, "Quantum groups", Proc. ICM (Berkeley, 1986), 1987, pp. 798-820], sec. 7) and later detailed in [F. Gavarini, "The quantum duality principle", Annales de l'Institut Fourier 53 (2002), 809-834].

Chemla, S., Gavarini, F. (2015). Duality functors for quantum groupoids. JOURNAL OF NONCOMMUTATIVE GEOMETRY, 9(2), 287-358 [10.4171/JNCG/194].

Duality functors for quantum groupoids

GAVARINI, FABIO
2015-01-01

Abstract

We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum duality principle" for quantum groupoids, which extends the one for quantum groups - dealing with Hopf algebras - originally introduced by Drinfeld (cf. [V. G. Drinfeld, "Quantum groups", Proc. ICM (Berkeley, 1986), 1987, pp. 798-820], sec. 7) and later detailed in [F. Gavarini, "The quantum duality principle", Annales de l'Institut Fourier 53 (2002), 809-834].
2015
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/02 - ALGEBRA
Settore MAT/03 - GEOMETRIA
English
Con Impact Factor ISI
Bialgebroids; Lie-Rinehart (bi)algebras; Quantum Groupoids; Quantum Duality
http://www.mat.uniroma2.it/~gavarini/page-web_files/page-web_ENG_files/page-web_ENG-2_data/publ-ENG.html
Chemla, S., Gavarini, F. (2015). Duality functors for quantum groupoids. JOURNAL OF NONCOMMUTATIVE GEOMETRY, 9(2), 287-358 [10.4171/JNCG/194].
Chemla, S; Gavarini, F
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/83048
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