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Examples of Weak Hopf Algebras Arising from Vacant Double Groupoids

Published online by Cambridge University Press:  11 January 2016

Nicolás Andruskiewitsch  and
Juan Martín Mombelli
Nicolás Andruskiewitsch
Affiliation:
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM - CONICET, Medina Allende s/n, (5000) Ciudad Universitaria, Córdoba, Argentina andrus@mate.uncor.edu
Juan Martín Mombelli
Affiliation:
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM - CONICET, Medina Allende s/n, (5000) Ciudad Universitaria, Córdoba, Argentina mombelli@mate.uncor.edu
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Abstract

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We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi [H]) via vacant double groupoids as explained in [AN]. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.

Keywords

17B3781R50

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 181 , March 2006 , pp. 1 - 27
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[AN] Andruskiewitsch, N. and Natale, S., Double categories and quantum groupoids, Publ. Mat. Urug., 10 (2005), 11–51.Google Scholar
[AM] Adem, A. and Milgram, R. J., Cohomology of Finite Groups, Springer-Verlag, 1994.Google Scholar
[BaSV] Baaj, S., Skandalis, G. and Vaes, S., Measurable Kac cohomology for bicrossed products, Trans. Amer. Math. Soc., 357 (2005), 1497–1524.Google Scholar
[BNS] Böohm, G., Nill, F. and Szlacháanyi, K., Weak Hopf algebras I. Integral theory and C* -structure, J. Algebra, 221 (1999), 385–438.Google Scholar
[BS] Böohm, G. and Szlacháanyi, K., A coassociative C* -quantum group with nonintegral dimensions, Lett. Math. Phys., 35 (1996), 437–456.Google Scholar
[BW] Brzezinski, T. and Wisbauer, R., Corings and Comodules, Cambridge University Press, 2003.CrossRefGoogle Scholar
[CE] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, 1956.Google Scholar
[ENO] Etingof, P., Nikshych, D. and Ostrik, V., On fusion categories, Annals of Math., 162 (2005), 581–642.Google Scholar
[H] Hayashi, T., A brief introduction to face algebras, New trends in Hopf Algebra Theory, Contemp. Math. 267 (2000), pp. 161–176.Google Scholar
[K] Kassel, C., Quantum groups, Springer-Verlag, 1995.Google Scholar
[Ma] Mackenzie, K., Double Lie algebroides and Second-order Geometry I, Adv. Math., 94 (1992), 180–239.Google Scholar
[M1] Masuoka, A., Hopf algebra extensions and cohomology, Math. Sci. Res. Inst. Publ., 43 (2002), 167–209.Google Scholar
[M2] Masuoka, A., Calculations of Some Groups of Hopf Algebra Extensions, J. Algebra, 191 (1997), 568–588.Google Scholar
[NV] Nikshych, D. and Vainerman, L., Finite quantum groupoids and their applications, Recent developments in Hopf algebra theory, Math. Sci. Res. Inst. Publ. 43 (2002), pp. 211–262.Google Scholar
[R] Renault, J., A groupoid approach to C* -algebras, Lect. Notes Math. 793, Springer-Verlag, Berlin, 1980.Google Scholar
[W] Weibel, C., An introduction to homological algebra, Cambridge University Press, 1994.Google Scholar
[WW] Wiegold, J. and Williamson, A. G., The Factorizations of the Alternating and Symmetric Groups, Math. Z., 175 (1980), 171–179.Google Scholar