Duality functors for quantum groupoids
Sophie Chemla
UPMC Université Paris 06, FranceFabio Gavarini
Università di Roma Tor Vergata, Italy
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Abstract
We present a formal algebraic language to deal with quantum deformations of Lie–Rinehart algebras — or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [34], 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. [8], §7) and later detailed in [12].
Cite this article
Sophie Chemla, Fabio Gavarini, Duality functors for quantum groupoids. J. Noncommut. Geom. 9 (2015), no. 2, pp. 287–358
DOI 10.4171/JNCG/194