São Paulo Journal of Mathematical Sciences

, Volume 13, Issue 2, pp 628–651 | Cite as

Some advances about the existence of compact involutions in semisimple Hopf algebras

  • Andrés AbellaEmail author
Original Article


In this paper we show that all complex semisimple Hopf algebras of dimension less than 24 are compact quantum groups. To do this, we survey all the above algebras and show explicitly that they can be described by bicrossed products of group algebras and its duals. We also study the behaviour under twisting of compact quantum groups. Using this we show that certain families of triangular semisimple Hopf algebras are compact quantum groups.


Compact quantum group Twisted Hopf algebra Semisimple Hopf algebra 



  1. 1.
    Abella, A.: Cosemisimple coalgebras. Ann. Sci. Math. Québec 30(2), 119–133 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abella, A., Ferrer, W., Haim, M.: Compact coalgebras, compact quantum groups and the positive antipode. São Paulo J. Math. Sci. 3(1), 191–227 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Abella, A., Ferrer, W., Haim, M.: Some constructions of compact quantum groups. São Paulo J. Math. Sci. 6(1), 1–40 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abella, A., Freitas, D., Morgado, A.: Almost-involutive Hopf-Ore extensions of low dimension. São Paulo J. Math. Sci. 11(1), 133–147 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aldjaeff, E., Etingoff, P., Gelaki, S., Nyshych, D.: On twisting of finite-dimensional Hopf algebras. J. Algebra 256(2), 484–501 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Andruskiewitsch, N.: Compact involutions of semisimple quantum groups. Czech. J. Phys. 44, 963–972 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Andruskiewitsch, N., García, A.: arXiv:1512.01696v2 (to appear in Annali dell’ Università di Ferrara)
  8. 8.
    Andruskiewitsch, N., Natale, S.: Examples of self-dual Hopf algebras. J. Math. Sci. Univ. Tokio 6(1), 181–215 (1999)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Beattie, M.: A survey of Hopf algebras of low dimension. Acta Appl. Math. 108(1), 19–31 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Beattie, M., Garcia, G.A.: Classifying Hopf algebras of a given dimension. Contemp. Math. 585, 125–152 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dascalescu, S., Nastasescu, C., Raianu, S.: Hopf Algebras: An Introduction. Monographs and Textbooks in Pure and Applied Matehmatics. Marcel Dekker, New York (2001)zbMATHGoogle Scholar
  12. 12.
    David, M., Thiéry, N.M.: Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions. J. Algebra Appl. 10(5), 995–1106 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Drinfeld, V.: Quantum groups. In: Proceedings of the International Congress of Mathematics, vol. 1, no. 2 (Berkeley 1987), pp. 798–820 (1987)Google Scholar
  14. 14.
    Drinfeld, V.: On almost cocommutative Hopf algebras. Leningr. Math. J. 1, 321–342 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dijkhuizen, M., Koornwinder, T.: CQG algebras: a direct algebraic approach to compact quantum groups. Lett. Math. Phys. 32(4), 315–330 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Etingof, P., Gelaki, S.: The classification of triangular semismple and cosemisimple Hopf algebras over an algebraically closed field. Intern. Math. Res. Not. 5, 223–234 (2000)CrossRefGoogle Scholar
  17. 17.
    Enock, M., Schwartz, J.M.: Kac Algebras and Duality of Locally Compact Groups. Springer, Berlin (1992). (With a preface by Alain Connes, With a postface by Adrian Ocneanu)CrossRefGoogle Scholar
  18. 18.
    Fukuda, D.: Classification of Hopf algebras of dimension 18. Israel J. Math. 168, 119–123 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gelaki, S.: Quantum groups of dimension \(pq^2\). Israel J. Math. 102, 227–267 (1997)Google Scholar
  20. 20.
    Gelaki, S.: On the classsification of finite-dimensional triangular Hopf algebras. In: New directions in Hopf algebras. vol. 43, pp. 69–116, MSRI Publications (2002)Google Scholar
  21. 21.
    Guillot, P., Kassel, C.: Cohomology of invariant Drinfeld twists on group algebras. Int. Math. Res. Not. 2010(10), 1894–1939 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kobayashi, T., Masuoka, A.: A result extended from groups to Hopf algebras. Tsukuba J. Math. 21(1), 55–58 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kac, G.I.: Extension of groups to ring groups. Math. USSR Sbornik 5(3), 451–474 (1968)CrossRefGoogle Scholar
  24. 24.
    Kac, G., Paljutkin, V.: Finite ring groups. Trans. Moscow Math. Soc. 5, 251–294 (1966)Google Scholar
  25. 25.
    Kashina, Y.: Classification of semisimple Hopf algebras of dimension 16. J. Algebra 232, 617–663 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  27. 27.
    Masuoka, A.: Semisimple Hopf algebras of dimension 6, 8. Israel J. Math. 92(1), 361–373 (1995)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Masuoka, A.: Some further classifications results on semisimple Hopf algebras. Commun. Algebra 24(1), 307–329 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Masuoka, A.: Faithfully flat forms and cohomology of Hopf algebra extensions. Commun. Algebra 25, 1169–1197 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Masuoka, A.: Extensions of Hopf algebras, Trabajos de Matemtica 41/99 (Lecture Notes taken by Matias Graña) FaMAF. Universidad Nacional de Cordoba (1999)Google Scholar
  31. 31.
    Masuoka, A.: Hopf algebra extensions and cohomology. In: New Directions in Hopf Algebras. vol. 43, pp. 167–210. MSRI Publications (2002)Google Scholar
  32. 32.
    Masuoka, A.: Classification of semisimple Hopf algebras. Handb. Algebra 5, 429–455 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Montgomery, S.: Hopf algebras and their actions on rings. CBMS, vol. 28. American Mathematical Society, Providence (1993)CrossRefGoogle Scholar
  34. 34.
    Movshev, M.: Twisting in group algebras of finite groups. Funct. Anal. Appl. 27, 240–244 (1994)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Natale, S.: On semisimple Hopf algebras of dimension \(pq^2\). J. Algebra 221, 242–278 (1999)Google Scholar
  36. 36.
    Natale, S.: On semisimple Hopf algebras of dimension \(pq^r\). Algebras Represent. Theory 7(2), 173–188 (2004)Google Scholar
  37. 37.
    Sweedler, M.: Cocommutative Hopf algebras with antipode. Bull. Am. Math. Soc. 73(1), 126–128 (1967)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (1979)CrossRefGoogle Scholar
  39. 39.
    Takeuchi, M.: Matched pair of groups and bismash products of Hopf algebras. Commun. Algebra 9(8), 841–882 (1981)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys 111, 613–665 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2018

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad de la RepúblicaMontevideoUruguay

Personalised recommendations