Duality functors for quantum groupoids

TL;DR

This paper develops a formal algebraic framework for quantum groupoids, extending duality principles from quantum groups to these more general structures, with detailed categorical and duality formalism.

Contribution

It introduces suitable notions of quantum groupoids, explores their linear duality, and extends the quantum duality principle from quantum groups to quantum groupoids.

Findings

Establishment of fundamental antiequivalences between categories of quantum groupoids.

Development of a quantum duality principle for quantum groupoids.

Formal algebraic language for quantum deformations of Lie-Rinehart algebras.

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 [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].