  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.
