Algebras and Representation Theory

, Volume 17, Issue 1, pp 227–264 | Cite as

Classifying Bicrossed Products of Hopf Algebras



Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object \({\mathcal H}^{2} (A, H)\). Equivalently, we classify up to a left A-linear Hopf algebra isomorphism, the set of all bicrossed products A ⋈ H associated to all possible matched pairs of Hopf algebras \((A, H, \triangleleft, \triangleright)\) that can be defined between A and H. In the construction of \({\mathcal H}^{2} (A, H)\) the key role is played by special elements of \(CoZ^{1} (H, A) \times {\rm Aut}\,_{\rm CoAlg}^1 (H)\), where CoZ 1 (H, A) is the group of unitary cocentral maps and \({\rm Aut}\,_{\rm CoAlg}^1 (H)\) is the group of unitary automorphisms of the coalgebra H. Among several applications and examples, all bicrossed products H 4 ⋈ k[C n ] are described by generators and relations and classified: they are quantum groups at roots of unity H 4n, ω which are classified by pure arithmetic properties of the ring ℤ n . The Dirichlet’s theorem on primes is used to count the number of types of isomorphisms of this family of 4n-dimensional quantum groups. As a consequence of our approach the group Aut Hopf(H 4n, ω ) of Hopf algebra automorphisms is fully described.


Bicrossed product Factorization problem Classification of Hopf algebras 

Mathematics Subject Classifications (2010)

16T05 16S40 


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  1. 1.
    Agore, A.L.: Crossed product of Hopf algebras. Commun. Algebra (2012). Preprint available at arXiv:1203.2454
  2. 2.
    Agore, A.L., Chirvasitu, A., Ion, B., Militaru, G.: Bicrossed products for finite groups. Algebr. Represent. Theory 12, 481–488 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Agore, A.L., Militaru, G.: Deformations and descent type theory for Hopf algebras (2012). Preprint available at arXiv:1205.6564
  4. 4.
    Aguiar, M., Andruskiewitsch, N.: Representations of matched pairs of groupoids and applications to weak Hopf algebras. Algebraic structures and their representations. Contemp. math. - Am. Math. Soc., Providence 376, 127–173 (2005)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Andruskiewitsch, N., Devoto, J.: Extensions of Hopf algebras. Algebra Anal. 7, 22–61 (1995)MATHMathSciNetGoogle Scholar
  6. 6.
    Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 171, 375–417 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Baaj, S., Skandalis, G., Vaes, S.: Measurable Kac cohomology for bicrossed products. Trans. Am. Math. Soc. 357, 1497–1524 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Beattie, M.: A survey of Hopf algebras of low dimension. In: Proceedings of Groups, rings, Lie and Hopf algebras, 2007. Acta Applicandae Mathematicae, vol. 108, pp. 19–31. Bonne Bay, Nfld (2009)Google Scholar
  9. 9.
    Bontea, C.G.: Classifying bicrossed products of two Swedler’s Hopf algebras (2012). Preprint available at arXiv:1205.7010
  10. 10.
    Brzezinski, T., Majid, S.: Coalgebra bundles. Commun. Math. Phys. 191, 467–492 (1998)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Brzezinski, T., Majid, S.: Quantum geometry of algebra factorisations and coalgebra bundles. Commun. Math. Phys. 213, 491–521 (2000)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Brzezinski, T.: Deformation of algebra factorisations. Commun. Algebra 29, 737–748 (2001)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Burciu, S.: On complements and the factorization problem for Hopf algebras. Cent. Eur. J. Math. 9, 905–914 (2011)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Caenepeel, S., Ion, B., Militaru, G., Zhu, S.: The factorization problem and the smash biproduct of algebras and coalgebras. Algebr. Represent. Theory 3(1), 19–42 (2000)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Cap, A., Schichl, H., Vanzura, J.: On twisted tensor product of algebras. Commun. Algebra 23, 4701–4735 (1995)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Chandrasekharan, K.: Introduction to Analytic Number Theory. GTM 148, Springer (1968)Google Scholar
  17. 17.
    Cirio, L.S., Pagani, C.: Deformation of tensor product (co)algebras via non-(co)normal twists (2011). Preprint available at arXiv:1112.2992
  18. 18.
    Cohn, P.M.: A remark on the general product of two infinite Cyclic groups. Arch. Math. (Basel) 7, 94–99 (1956)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Darafsheh, M.R.: Finite groups which factor as product of an alternating group and symmetric group. Commun. Algebra 32, 637–647 (2004)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Douglas, J.: On finite groups with two independent generators. I, II, III, IV. Proc. Nat. Acad. Sci. U.S.A. 37, 604–610, 677–691, 749–760, 808–813 (1951)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Doi, Y., Takeuchi, M.: Multiplication alteration by two-cocycles. Commun. Algebra 22, 5715–5732 (1994)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Guccione, J.A., Guccione, J.J., Valqui, C.: Twisted Planes. Commun. Algebra 38, 1930–1956 (2010)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Hilgemann, M., Ng, S.-H.: Hopf algebras of dimension 2p 2. J. Lond. Math. Soc. 80(2), 295–310 (2009)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Hungerford, T.W.: Algebra. Graduate Text in Mathematics, vol. 73. Springer (1974)Google Scholar
  25. 25.
    Ito, N.: Über das Produkt von zwei abelschen Gruppen. Math. Z. 62, 400–401 (1955)MATHGoogle Scholar
  26. 26.
    Kassel, C.: Quantum groups. Graduate Texts in Mathematics, vol. 155. Springer-Verlag, New York (1995)Google Scholar
  27. 27.
    Krotz, B.: A novel characterization of the Iwasawa decomposition of a simple lie group. Springer Lect. Notes Phys. 723, 195–201 (2007)Google Scholar
  28. 28.
    López Peña, J., Navarro, G.: On the classification and properties of noncommutative duplicates. K-Theory 38(2), 223–234 (2008)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    López Peña, J., Stefan, D.: On the classification of twisting maps between K n and K m. Algebr. Represent. Theory 14, 869–895 (2011)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Maillet, E.: Sur les groupes échangeables et les groupes décomposables. Bull. Soc. Math. Fr. 28, 7–16 (1900)MATHMathSciNetGoogle Scholar
  31. 31.
    Majard,D.: On DoubleGroups and the Poincare group (2011). Preprint available at arXiv:1112.6208
  32. 32.
    Majid, S.: Matched Pairs of Lie Groups and Hopf Algebra Bicrossproducts. Nucl. Phys., B Proc. Suppl. 6, 422–424 (1989)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Majid, S.: Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130, 17–64 (1990)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Majid, S.: Foundations of Quantum Groups Theory. Cambridge University Press (1995)Google Scholar
  35. 35.
    Masuoka, A.: Hopf algebra extensions and cohomology. In: New Directions in Hopf Algebras, vol. 43, pp. 167–209. MSRI Publ. (2002)Google Scholar
  36. 36.
    Masuoka, A.: Classification of semisimple Hopf algebras. In: Handbook of Algebra 5. Amsterdam, Elsevier/Noth-Holland (2008)Google Scholar
  37. 37.
    Michor, P.W.: Knit products of graded Lie algebras and groups. In: Proceedings of the Winter School on Geometry and Physics (Srn, 1989). Rend. Circ. Mat. Palermo (2) Suppl. No 22, pp. 171–175 (1990)Google Scholar
  38. 38.
    Molnar, R.K.: Semi-direct products of Hopf algebras. J. Algebra 47, 29–51 (1977)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Natale, S.: Semisimple Hopf algebras of dimension 60. J. Algebra 324, 3017–3034 (2010)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Ng, S.-H.: Hopf algebras of dimension 2p. Proc. Am. Math. Soc. 133(8) 2237–2242 (2005)CrossRefMATHGoogle Scholar
  41. 41.
    Ore, O.: Structures and group theory. I. Duke Math. J. 3(2), 149–174 (1937)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Radford, D.E.: On Kauffmans Knot invariants arising from finite-dimensional Hopf algebras, advances in Hopf algebras. Lectures Notes in Pure and Applied Mathematics, vol. 158, pp. 205–266. Marcel Dekker, N.Y. (1994)Google Scholar
  43. 43.
    Rédei, L.: Zur Theorie der faktorisierbaren Gruppen I. Acta Math. Acad. Sci. Hung. 1, 74–98 (1950)CrossRefMATHGoogle Scholar
  44. 44.
    Rotman, J.: An introduction to the theory of groups. Graduate Texts in Mathematics, vol. 148, 4th edn. Springer-Verlag, New York (1995)CrossRefGoogle Scholar
  45. 45.
    Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras. Commun. Algebra 9, 841–882 (1981)CrossRefMATHGoogle Scholar
  46. 46.
    Vaes, S., Vainerman, L.: Extensions of locally compact quantum groups and the bicrossed product construction. Adv. Math. 175(1), 1–101 (2003)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Zhu, Y.: Hopf algebras of prime dimension. Int. Math. Res. Notes 1, 53–59 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania

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