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Algebras and Representation Theory

, Volume 18, Issue 5, pp 1267–1297 | Cite as

Automorphisms of the Doubles of Purely Non-Abelian Finite Groups

  • Marc KeilbergEmail author
Article

Abstract

Using a recent classification of End\((\mathcal {D}(G))\), we determine a number of properties for Aut\((\mathcal {D}(G))\), where \(\mathcal {D}(G)\) is the Drinfel’d double of a finite group G. Furthermore, we completely describe Aut\((\mathcal {D}(G))\) for all purely non-abelian finite groups G. A description of the action of Aut\((\mathcal {D}(G))\) on Rep\((\mathcal {D}(G))\) is also given. We are also able to produce a simple proof that \(\mathcal {D}(G)\cong \mathcal {D}(H)\) if and only if \(\mathcal {G}\cong H\), for G and H finite groups.

Keywords

Finite groups Automorphisms Endomorphisms Quantum double Tensor autoequivalences 

Mathematics Subject Classification (2010)

16W20 16T05 20D99 

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References

  1. 1.
    Adney, J.E., Yen, T.: Automorphisms of a p-group. Ill. J. Math. 9, 137–143 (1965). ISSN 0019-2082 http://projecteuclid.org/euclid.ijm/1256067587 zbMATHMathSciNetGoogle Scholar
  2. 2.
    Agore, A.L., Bontea, C.G., Militaru, G.: Classifying bicrossed products of Hopf algebras. Algebra Represent. Theor. 17(1), 227–264 (2014). doi: 10.1007/s10468-012-9396-5. ISSN 1386-923X.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Alev, J., Chamarie, M.: Dérivations et automorphismes de quelques algèbres quantiques. Comm. Algebra 20(6), 1787–1802 (1992). doi: 10.1080/00927879208824431. ISSN 0092-7872zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Alev, J., Dumas, F.: Rigidité des plongements des quotients primitifs minimaux de U q(sl(2)) dans l’algèbre quantique de Weyl-Hayashi. Nagoya Math. J. 143, 119–146 (1996). ISSN 0027-7630 http://projecteuclid.org/euclid.nmj/1118771972 zbMATHMathSciNetGoogle Scholar
  5. 5.
    Andruskiewitsch, N., Dumas, F.: On the automorphisms of \(U_q^{+}(\mathfrak {g})\), Zürich (2008)Google Scholar
  6. 6.
    Bidwell, J.N.S.: Automorphisms of direct products of finite groups. II. Arch. Math. (Basel) 91(2), 111–121 (2008). doi: 10.1007/s00013-008-2653-5. ISSN 0003-889XzbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bidwell, J.N.S., Curran, M.J., McCaughan, D.J.: Automorphisms of direct products of finite groups. Arch. Math. (Basel) 86(6), 481–489 (2006). doi: 10.1007/s00013-005-1547-z. ISSN 0003-889XzbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
  9. 9.
    Courter, R.: Computing Higher Indicators for the Double of a Symmetric Group. PhD thesis, University of Southern California (2012). arXiv:1206.6908
  10. 10.
    Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi Hopf algebras, group cohomology and orbifold models. Nucl. Phys. B, Proc. Suppl. 18B, 60–72 (1991) (1990). doi: 10.1016/0920-5632(91)90123-V. Recent advances in field theory (Annecy-le-Vieux, 1990)zbMATHMathSciNetGoogle Scholar
  11. 11.
    GAP. GAP – Groups, Algorithms, and Programming, Version 4.6.5. The GAP Group, 2013. http://www.gap-system.org
  12. 12.
    Christopher, G., Geoffrey M., Siu-Hung, Ng.: On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2), 849–875 (2007). doi: 10.1016/j.jalgebra.2006.10.022. ISSN 0021-8693zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Robert G., Montgomery, S.: Frobenius-Schur indicators for subgroups and the Drinfel′d double of Weyl groups. Trans. Amer. Math. Soc. 361(7), 3611–3632 (2009). doi: 10.1090/S0002-9947-09-04659-5. ISSN 0002-9947zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hall, P.: The classification of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940). ISSN 0075-4102MathSciNetGoogle Scholar
  15. 15.
    Iovanov, M., Mason, G., Montgomery, S.: FSZ-groups and Frobenius-Schur indicators of quantum doubles. Math. Res. Lett. 4, 1–23 (2014)MathSciNetGoogle Scholar
  16. 16.
    Noboru, I.: Remarks on factorizable groups. Acta Sci. Math. Szeged 14, 83–84 (1951). ISSN 0001-6969zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kashina, Y., Sommerhäuser, Y., Zhu, Y.: On higher Frobenius-Schur indicators. Mem. Amer. Math. Soc. 181(855), viii+65 (2006). doi: 10.1090/memo/0855. ISSN 0065-9266Google Scholar
  18. 18.
    Marc, K.: Higher indicators for some groups and their doubles. J. Algebra Appl. 11(2), 1250030. doi: 10.1142/S0219498811005543. 38, 2012. ISSN 0219-4988.
  19. 19.
    Keilberg, M.: Higher indicators for the doubles of some totally orthogonal groups. Comm. Algebra 42(7), 2969–2998 (2014). doi: 10.1080/00927872.2013.775651. ISSN 0092-7872zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Linchenko, V., Montgomery, S.: A Frobenius-Schur theorem for Hopf algebras. Algebr. Represent. Theory 3(4), 347–355 (2000). doi: 10.1023/A:1009949909889. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthdayzbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Montgomery, S.: Hopf algebras and their actions on rings, volume 82 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. ISBN 0-8218-0738-2.Google Scholar
  22. 22.
    Nikshych, D., Riepel, B.: Categorical Lagrangian Grassmannians and Brauer–Picard groups of pointed fusion categories. J. Algebra 411, 191–214 (2014). doi: 10.1016/j.jalgebra.2014.04.013. ISSN 0021-8693zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Radford, D.E.: The group of automorphisms of a semisimple Hopf algebra over a field of characteristic 0 is finite. Amer. J. Math. 112, 331–357 (1990). doi: 10.2307/2374718. ISSN 0002-9327zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Rotman, J.J.: An introduction to the theory of groups, volume 148 of Graduate Texts in Mathematics, 4th. Springer-Verlag, New York (1995). ISBN 0-387-94285-8CrossRefGoogle Scholar
  25. 25.
    Daniel S.S., Vega, M.D.: Twisted Frobenius-Schur indicators for Hopf algebras. J. Algebra 354, 136–147 (2012). doi: 10.1016/j.jalgebra.2011.12.026. ISSN 0021-8693zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Shoda, K.: Über die automorphismen einer endlichen abelschen gruppe. Math. Ann. 100(1), 674–686 (1928)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras. Comm. Algebra 9(8), 841–882 (1981). doi: 10.1080/00927878108822621. ISSN 0092-7872zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Wiegold, J., Williamson, A.G.: The factorisation of the alternating and symmetric groups. Math. Z. 175(2), 171–179 (1980). doi: 10.1007/BF01674447. ISSN 0025-5874zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Faculté des Sciences Mirande, Institut de Mathématiques de BourgogneUniversité de BourgogneDijon CedexFrance

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