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

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Abstract

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.

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References

  1. Abella, A.: Cosemisimple coalgebras. Ann. Sci. Math. Québec 30(2), 119–133 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  3. Abella, A., Ferrer, W., Haim, M.: Some constructions of compact quantum groups. São Paulo J. Math. Sci. 6(1), 1–40 (2012)

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/s40863-016-0053-5

    Article  MathSciNet  MATH  Google Scholar 

  5. Aldjaeff, E., Etingoff, P., Gelaki, S., Nyshych, D.: On twisting of finite-dimensional Hopf algebras. J. Algebra 256(2), 484–501 (2002)

    Article  MathSciNet  Google Scholar 

  6. Andruskiewitsch, N.: Compact involutions of semisimple quantum groups. Czech. J. Phys. 44, 963–972 (1994)

    Article  MathSciNet  Google Scholar 

  7. Andruskiewitsch, N., García, A.: arXiv:1512.01696v2 (to appear in Annali dell’ Università di Ferrara)

  8. Andruskiewitsch, N., Natale, S.: Examples of self-dual Hopf algebras. J. Math. Sci. Univ. Tokio 6(1), 181–215 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Beattie, M.: A survey of Hopf algebras of low dimension. Acta Appl. Math. 108(1), 19–31 (2009)

    Article  MathSciNet  Google Scholar 

  10. Beattie, M., Garcia, G.A.: Classifying Hopf algebras of a given dimension. Contemp. Math. 585, 125–152 (2013)

    Article  MathSciNet  Google Scholar 

  11. Dascalescu, S., Nastasescu, C., Raianu, S.: Hopf Algebras: An Introduction. Monographs and Textbooks in Pure and Applied Matehmatics. Marcel Dekker, New York (2001)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. Drinfeld, V.: Quantum groups. In: Proceedings of the International Congress of Mathematics, vol. 1, no. 2 (Berkeley 1987), pp. 798–820 (1987)

  14. Drinfeld, V.: On almost cocommutative Hopf algebras. Leningr. Math. J. 1, 321–342 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Dijkhuizen, M., Koornwinder, T.: CQG algebras: a direct algebraic approach to compact quantum groups. Lett. Math. Phys. 32(4), 315–330 (1994)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Book  Google Scholar 

  18. Fukuda, D.: Classification of Hopf algebras of dimension 18. Israel J. Math. 168, 119–123 (2008)

    Article  MathSciNet  Google Scholar 

  19. Gelaki, S.: Quantum groups of dimension \(pq^2\). Israel J. Math. 102, 227–267 (1997)

    Google Scholar 

  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)

  21. Guillot, P., Kassel, C.: Cohomology of invariant Drinfeld twists on group algebras. Int. Math. Res. Not. 2010(10), 1894–1939 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Kobayashi, T., Masuoka, A.: A result extended from groups to Hopf algebras. Tsukuba J. Math. 21(1), 55–58 (1997)

    Article  MathSciNet  Google Scholar 

  23. Kac, G.I.: Extension of groups to ring groups. Math. USSR Sbornik 5(3), 451–474 (1968)

    Article  Google Scholar 

  24. Kac, G., Paljutkin, V.: Finite ring groups. Trans. Moscow Math. Soc. 5, 251–294 (1966)

    Google Scholar 

  25. Kashina, Y.: Classification of semisimple Hopf algebras of dimension 16. J. Algebra 232, 617–663 (2000)

    Article  MathSciNet  Google Scholar 

  26. Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  27. Masuoka, A.: Semisimple Hopf algebras of dimension 6, 8. Israel J. Math. 92(1), 361–373 (1995)

    Article  MathSciNet  Google Scholar 

  28. Masuoka, A.: Some further classifications results on semisimple Hopf algebras. Commun. Algebra 24(1), 307–329 (1996)

    Article  MathSciNet  Google Scholar 

  29. Masuoka, A.: Faithfully flat forms and cohomology of Hopf algebra extensions. Commun. Algebra 25, 1169–1197 (1997)

    Article  MathSciNet  Google Scholar 

  30. Masuoka, A.: Extensions of Hopf algebras, Trabajos de Matemtica 41/99 (Lecture Notes taken by Matias Graña) FaMAF. Universidad Nacional de Cordoba (1999)

  31. Masuoka, A.: Hopf algebra extensions and cohomology. In: New Directions in Hopf Algebras. vol. 43, pp. 167–210. MSRI Publications (2002)

  32. Masuoka, A.: Classification of semisimple Hopf algebras. Handb. Algebra 5, 429–455 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Montgomery, S.: Hopf algebras and their actions on rings. CBMS, vol. 28. American Mathematical Society, Providence (1993)

    Book  Google Scholar 

  34. Movshev, M.: Twisting in group algebras of finite groups. Funct. Anal. Appl. 27, 240–244 (1994)

    Article  MathSciNet  Google Scholar 

  35. Natale, S.: On semisimple Hopf algebras of dimension \(pq^2\). J. Algebra 221, 242–278 (1999)

    Google Scholar 

  36. Natale, S.: On semisimple Hopf algebras of dimension \(pq^r\). Algebras Represent. Theory 7(2), 173–188 (2004)

    Google Scholar 

  37. Sweedler, M.: Cocommutative Hopf algebras with antipode. Bull. Am. Math. Soc. 73(1), 126–128 (1967)

    Article  MathSciNet  Google Scholar 

  38. Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (1979)

    Book  Google Scholar 

  39. Takeuchi, M.: Matched pair of groups and bismash products of Hopf algebras. Commun. Algebra 9(8), 841–882 (1981)

    Article  MathSciNet  Google Scholar 

  40. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys 111, 613–665 (1987)

    Article  MathSciNet  Google Scholar 

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  1. Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400, Montevideo, Uruguay

    Andrés Abella

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Abella, A. Some advances about the existence of compact involutions in semisimple Hopf algebras. São Paulo J. Math. Sci. 13, 628–651 (2019). https://doi.org/10.1007/s40863-018-0109-9

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