×

Representations of matched pairs of groupoids and applications to weak Hopf algebras. (English) Zbl 1100.16032

de la Peña, José A. (ed.) et al., Algebraic structures and their representations. Proceedings of ‘XV coloquio Latinoamericano de álgebra’, Cocoyoc, Morelos, México, July 20–26, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3630-7/pbk). Contemporary Mathematics 376, 127-173 (2005).
A matched pair of groupoids is a pair \((V,H)\) of groupoids (\(V\) for vertical, \(H\) for horizontal) with a common base \(P\) together with actions of each groupoid on the other one satisfying some axioms. A representation of \((V,H)\) is a quiver over \(P\), i.e., a set \(E\) and two maps of \(E\) to \(P\) together with an action of \(H\) and a grading over \(V\) which satisfy some compatibility conditions. The category \(\text{Rep}(V,H)\) is a monoidal category, and the forgetful functor \(F\) from \(\text{Rep}(V,H)\) to \(\text{Quiv}(P)\), the category of quivers over \(P\), is monoidal. Morphisms between matched pairs are defined. Each such morphism induces a monoidal functor between the representation categories, called restriction, which preserves \(F\).
The authors’ first main result is that any monoidal functor between representation categories is a restriction functor. Next, they study the centralizer of such monoidal functors, including the center of \(\text{Rep}(V,H)\), which is the centralizer of the identity functor. Their main result here is that any such centralizer is the category of representations of a (different) matched pair, which they call a generalized double. In particular, the center of \(\text{Rep}(V,H)\) is the category of representations of the double \(D(V,H)\), and the centralizer of \(F\) is the category of representations of the dual of \((V,H)\). They show that the constructions of generalized doubles and of duals commute.
They then classify braidings in \(\text{Rep}(V,H)\). The result involves the notion of a rotation for \((V,H)\), which is a morphism from \(V\) to \(H\) satisfying compatibility conditions with the actions of \(V\) and \(H\) on each other. This leads to the notion of a matched pair of rotations, and the result is that gradings are in bijective correspondence with matched pairs of rotations.
These ideas are used to continue the study of the construction of the second author and S. Natale of a weak Hopf algebra (or quantum groupoid) \(k(V,H)\), \(k\) a field of characteristic zero, from a matched pair \((V,H)\) of finite groupoids [Publ. Mat. Urug. 10, 11-51 (2005; Zbl 1092.16021)]. There is a monoidal functor from \(\text{Rep}(V,H)\) to the category of modules over \(k(V,H)\), called the linearization functor. It is shown that a matched pair of rotations gives rise to a quasitriangular structure on \(k(V,H)\), with the \(R\)-matrix given explicitly. It is shown that the construction of \(k(V,H)\) commutes with doubles (it was known to commute with duals). An explicit description is given of the Drinfeld double of \(k(V,H)\). Constructions of these quasitriangular structures were given by various authors in the case of matched pairs of groups. One of these used certain pairs which turn out to be matched pairs of rotations. The authors state that their approach is different from these other approaches, since it is based on calculation of the center of the category of representations.
For the entire collection see [Zbl 1067.16001].

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)

Citations:

Zbl 1092.16021
PDFBibTeX XMLCite
Full Text: arXiv