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

Abstract: We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{i\pi/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({\mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({\mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C^*-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras {\mathfrak g}\neq E_8 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 {\mathfrak g}={\mathfrak sl}_N 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.