 
Summary: Contemporary Mathematics
Representations of matched pairs of groupoids and
applications to weak Hopf algebras
Marcelo Aguiar and NicolŽas Andruskiewitsch
Abstract. We introduce the category of settheoretic representations of a
matched pair of groupoids. This is a monoidal category endowed with a
monoidal functor f to the category of quivers over the common base of the
groupoids in the matched pair. We study monoidal functors between two such
categories of representations which preserve the functor f . We show that the
centralizer of such a monoidal functor is the category of representations of a
new matched pair, which we construct explicitly. We introduce the notions of
double of a matched pair of groupoids and generalized double of a morphism of
matched pairs. We show that the centralizer of f is the category of representa
tions of the dual matched pair, and the centralizer of the identity functor (the
center) is the category of representations of the double. We use these construc
tions to classify the braidings in the category of representations of a matched
pair. Such braidings are parametrized by certain groupoidtheoretic structures
which we call matched pairs of rotations. Finally, we express our results in
terms of the weak Hopf algebra associated to a matched pair of groupoids.
A matched pair of rotations gives rise to a quasitriangular structure for the
