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Contemporary Mathematics Representations of matched pairs of groupoids and
 

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 set-theoretic 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 groupoid-theoretic 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

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics