Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 11, pp 1767–1776 | Cite as

A Class of Double Crossed Biproducts

  • T. S. Ma
  • H. Y. Li
  • L. H. Dong
Article
  • 13 Downloads

Let H be a bialgebra, let A be an algebra and a left H-comodule coalgebra, and let B be an algebra and a right H-comodule coalgebra. Also let f : HHAH, R : HAAH, and T : BHHB be linear maps. We present necessary and sufficient conditions for the onesided Brzeziński’s crossed product algebra \( A{\#}_R^f{H}_T\#B \) and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which generalizes the main results from [“On Ranford biproduct,” Comm. Algebra, 43, No. 9, 3946–3966 (2015)]. It is clear that both the Majid double biproduct [“Doublebosonization of braided groups and the construction of Uq(g), ” Math. Proc. Cambr. Philos. Soc., 125, No. 1, 151–192 (1999)] and the Wang–Jiao–Zhao crossed product [“Hopf algebra structures on crossed products,” Comm. Algebra, 26, 1293–1303 (1998)] are obtained as special cases.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. L. Agore and G. Militaru, “Extending structures II: The quantum version,” J. Algebra, 336, 321–341 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Andruskiewitsch and H.-J. Schneider, “On the classification of finite-dimensional pointed Hopf algebras,” Ann. Math., 171, No. 1, 375–417 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. J. Blattner, M. Cohen, and S. Montgomery, “Crossed products and inner actions of Hopf algebras,” Trans. Amer. Math. Soc., 289, 671–711 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    T. Brzeziński, “Crossed products by a coalgebra,” Comm. Algebra, 25, 3551–3575 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. Caenepeel, B. Ion, G. Militaru, and S. L. Zhu, “The factorization problem and the smash biproduct of algebras and coalgebras,” Algebr. Represent. Theory, 3, 19–42 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    T. S. Ma, Z. M. Jiao, and Y. N. Song, “On crossed double biproduct,” J. Algebra Appl., 12, No. 5, 17 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    T. S. Ma and H. Y. Li, “On Radford biproduct,” Comm. Algebra, 43, No. 9, 3946–3966 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    T. S. Ma and S. H. Wang, “General double quantum groups,” Comm. Algebra, 38, No. 2, 645–672 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Majid, “Double-bosonization of braided groups and the construction of U q(g),Math. Proc. Cambr. Philos. Soc., 125, No. 1, 151–192 (1999).CrossRefzbMATHGoogle Scholar
  10. 10.
    S. Majid, “Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group,” Comm. Math. Phys., 156, 607–638 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. Majid, “Algebras and Hopf algebras in braided categories,” Adv. Hopf Algebras, Marcel Dekker Lecture Notes in Pure Appl. Math., 158, 55–105 (1994).MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. Montgomery, “Hopf algebras and their actions on rings,” CBMS Lect. Math., 82 (1993).Google Scholar
  13. 13.
    F. Panaite and F. Van Oystaeyen, “L-R-smash biproducts, double biproducts and a braided category of Yetter–Drinfeld–Long bimodules,” Rocky Mountain J. Math., 40, No. 6, 2013–2024 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. E. Radford, “The structure of Hopf algebra with a projection,” J. Algebra, 92, 322–347 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D. E. Radford, “Hopf algebras,” Ser. Knots Everything, 49 , World Scientific, New Jersey (2012).Google Scholar
  16. 16.
    S. H. Wang, Z. M. Jiao, and W. Z. Zhao, “Hopf algebra structures on crossed products,” Comm. Algebra, 26, 1293–1303 (1998).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • T. S. Ma
    • 1
  • H. Y. Li
    • 1
  • L. H. Dong
    • 1
  1. 1.School of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina

Personalised recommendations