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Tensor categories attached to double groupoids. (English) Zbl 1099.16016
A (finite) double groupoid is a groupoid object in the category of (finite) groupoids. They have been around since the 1960’s, arising originally in homotopy theory. Roughly speaking, they consist of a finite number of boxes with vertical and horizontal compositions and with groupoid compositions on the sides, satisfying some compatibility conditions. Let $kT$ ($k$ a field of characteristic zero) be the vector space with the boxes of $T$ as basis. The authors previously considered an algebra structure on $kT$ coming from the vertical composition and a coalgebra structure from the horizontal composition, and studied when this produced a weak Hopf algebra (also called a quantum groupoid) [Publ. Mat. Urug. 10, 11-51 (2005; Zbl 1092.16021)]. A necessary and sufficient condition was given (called vacancy of $T$), and the resulting quantum groupoid was described as an Abelian bicrossed product. In the paper under review, the coalgebra structure is modified. The resulting construction is shown to be a quantum groupoid if $T$ satisfies a condition called the filling condition. The resulting quantum groupoid does not seem to be describable as a bicrossed product, but it is semisimple and its finite-dimensional representation category is a semisimple rigid monoidal category. The authors show that several tensor categories fit their construction. One is the category of $R$-bimodules over a separable algebra $R$.

16W30Hopf algebras (associative rings and algebras) (MSC2000)
18D05Double categories, $2$-categories, bicategories and generalizations
18B40Groupoids, semigroupoids, semigroups, groups (viewed as categories)
Full Text: DOI arXiv
[1] N. Andruskiewitsch, S. Natale, Double categories and quantum groupoids, preprint math.QA/0308228 (2003), Publ. Mat. Urug., to appear. · Zbl 1092.16021
[2] Böhm, G.; Nill, F.; Szlachányi, K.: Weak Hopf algebras I. Integral theory and C*-structure. J. algebra 221, 385-438 (1999) · Zbl 0949.16037
[3] Böhm, G.; Szlachányi, K.: A coassociative C*-quantum group with nonintegral dimensions. Lett. math. Phys. 35, 437-456 (1996) · Zbl 0872.16022
[4] Böhm, G.; Szlachányi, K.: Weak Hopf algebras. II: representation theory, dimensions, and the Markov trace. J. algebra 233, 156-212 (2000) · Zbl 0980.16028
[5] R. Brown, Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems, Fields Inst. Commun. 43 101-130; American Mathematical Society, Providence, RI, 2004.
[6] Brown, R.; Mackenzie, K.: Determination of a double Lie groupoid by its core diagram. J. pure appl. Algebra 80, 237-272 (1992) · Zbl 0766.22001
[7] R. Brown, C. Spencer, Double groupoids and crossed modules, Cahiers de Topologie et Geo. Differentielle Categoriques, vol. XVII, 1976, pp. 343-364. · Zbl 0344.18004
[8] Ehresmann, C.: Catégories doubles et catégories structurées. CR acad. Sci. Paris 256, 1198-1201 (1963) · Zbl 0132.25702
[9] P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, preprint math.QA/0203060 (2002). · Zbl 1125.16025
[10] F. Hausser, F. Nill, Integral theory for quasi-Hopf algebras, preprint math.QA/9904164 (1999). · Zbl 0936.17011
[11] T. Hayashi, A brief introduction to face algebras, in: New trends in Hopf Algebra Theory, Contemp. Math. 267 (2000) 161-176. · Zbl 0974.16032
[12] Mackenzie, K.: Double Lie algebroids and second-order geometry, I. Adv. math. 94, 180-239 (1992) · Zbl 0765.57025
[13] Mackenzie, K.: Double Lie algebroids and second-order geometry, II. Adv. math. 154, 46-75 (2000) · Zbl 0971.58015
[14] Majid, S.: Foundations of quantum group theory. (1995) · Zbl 0857.17009
[15] S. Natale, Frobenius-Schur indicators for a class of fusion categories, preprint math.QA/0312466 (2003), Pacific J. Math., to appear.
[16] Nikshych, D.: Semisimple weak Hopf algebras. J. algebra 275, 639-667 (2004) · Zbl 1066.16042
[17] Nikshych, D.; Vainerman, L.: Finite quantum groupoids and their applications. Math. sci. Res. inst. Publ. 43, 211-262 (2002) · Zbl 1026.17017
[18] V. Ostrik, Module categories over the Drinfeld double of a finite group, Internat. Math. Res. Not. 2003 (2003) 1507-1520, preprint math.QA/0202130. · Zbl 1044.18005
[19] M. Takeuchi, Survey on matched pairs of groups, An Elementary Approach to the ESS-LYZ Theory, Banach Center Publications, vol. 61, 2003, pp. 305-331. · Zbl 1066.16044