Abstract
A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with the natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations of the normalized 3-cocycles and the quasi-bicharacters of finite abelian groups which are direct product of two cyclic groups.
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References
Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. (2) 171(1), 375–417 (2010)
Angiono, I.E.: Basic quasi-Hopf algebras over cyclic groups. Adv. Math. 225(6), 3545–3575 (2010)
Bulacu, D., Caenepeel, S., Torrecillas, B.: The braided monoidal structures on the category of vector spaces graded by the Klein group. Proc. Edinb. Math. Soc. (2) 54(3), 613–641 (2011)
Etingof, P., Ostrik, V.: Finite tensor categories. Mosc. Math. J. 4(3), 627–654 (2004)
Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. (2) 162(2), 581–642 (2005)
Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. In: Lecture Note for the MIT Course 18.769 (2009). Available at www-math.mit.edu/etingof/tenscat.pdf
Hoàng X.S.: Gr-catégories. Universite Paris VII, Thèse de doctorat (1975). Available at http://www.math.rwth-aachen.de/~kuenzer/sinh.html
Huang, H.-L.: Quiver approaches to quasi-Hopf algebras. J. Math. Phys. 50(4), 043501, 9 (2009)
Huang, H.-L.: From projective representations to quasi-quantum groups. Sci. China Math. 55(10), 2067–2080 (2012)
Huang, H.-L., Liu, G., Ye, Y.: Quivers, quasi-quantum groups and finite tensor categories. Commun. Math. Phys. 303(3), 595–612 (2011)
Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)
Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)
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Supported by the NSFC grants 10971206 and 11071111, the SDNSF grant 2009ZRA01128 and the IIFSDU grant 2010TS021.
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Huang, HL., Liu, G. & Ye, Y. The Braided Monoidal Structures on a Class of Linear Gr-Categories. Algebr Represent Theor 17, 1249–1265 (2014). https://doi.org/10.1007/s10468-013-9445-8
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DOI: https://doi.org/10.1007/s10468-013-9445-8