Quasi-coassociative C⁎-quantum groupoids of type A and modular C⁎-categories

For this, we give a general construction of quantum groupoids for complex simple Lie algebras g≠E8 and certain roots of unity. Our main tools here are Drinfeld's coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfeld's associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case g=slN we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.