Dualité tannakienne pour les quasi-groupoïdes quantiques

Abstract

This paper is the sequel of a previous one [2] where we extended the Tannaka-Krein duality results to the non-commutative situation, i.e. to ‘quantum groupoids’. Here we extend those results to the quasi-monoidal situation, corresponding to ‘quasi-quantum groupoids’ as defined in [3] (‘quasi-’ stands for quasi-associativity a la Drinfeld). More precisely, let B be a commutative algebra over a field k. Given a tensor autonomous category τ,. we define the notion of a quasi-fibre functor ω:τ-proj B (here, ‘quasi-’ means without compatibility to associativity constraints). On the other hand, we define the notion of a transitive quasi-quantum groupoid over B. We then show that the category of tensor autonomous categories equipped with a quasi-fibre functor (with suitable morphisms), is equivalent to the category of transitive quasi-quantum groupoids (5.4.2) Moreover, we classify quasi-fibre functors for a semisimple tensor autonomous category (6.1.2), and give a few examples : a family of quantum groups having the same tensor category of representations as Sl2(C), but with non-isornorphic underlying coalgebras, constructed by means of an R-matrix introduced by Gurevich ([9]) in a manner suggested to the author by Lyubashenko (6.2.1 and 6.2.2), and quasi-quantum groups which cannot be obtained from quantum groups by a Drinfeld twist (6.2.1)