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Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 399–415 | Cite as

Double Frobenius algebras

  • Zhihua Wang
  • Libin Li
Research Article

Abstract

Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of (strict or non-strict) double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a (strict) double Frobenius algebra is given.

Keywords

Double Frobenius algebra bi-Frobenius algebra trivial extension 

MSC

16W10 

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Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11471282), the China Postdoctoral Science Foundation (Grant No. 2017M610316), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20170589).

References

  1. 1.
    Abrams L. Modules, comodules and cotensor products over Frobenius algebras. J Algebra, 1999, 219: 201–213MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen Q G, Wang S H. Radford’s formula for generalized weak biFrobenius algebras. Rocky Mountain J Math, 2014, 44(2): 419–433MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Doi Y. Substructures of bi-Frobenius algebras. J Algebra, 2002, 256: 568–582MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Doi Y. Group-like algebras and their representations. Comm Algebra, 2010, 38(7): 2635–2655MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Doi Y, Takeuchi M. BiFrobenius algebras. Contemp Math, 2000, 267: 67–98CrossRefzbMATHGoogle Scholar
  6. 6.
    Etingof P, Gelaki S, Nikshych D, Ostrik V. Tensor Categories. Math Surveys Monogr, Vol 205. Providence: AMS, 2015CrossRefzbMATHGoogle Scholar
  7. 7.
    Ferrer Santos W, Haim M. Radford’s formula for bi-Frobenius algebras and applications. Comm Algebra, 2008, 36(4): 1301–1310MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Haim M. Group-like algebras and Hadamard matrices. J Algebra, 2007, 308: 215–235MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Koppinen M. On algebras with two multiplications, including Hopf algebras and Bose-Mesner algebras. J Algebra, 1996, 182: 256–273MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lam T Y. Lectures on Modules and Rings. Grad Texts in Math, Vol 189. New York: Springer-Verlag, 1999CrossRefGoogle Scholar
  11. 11.
    Lorenz M. Some applications of Frobenius algebras to Hopf algebras. Contemp Math, 2011, 537: 269–289MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wang Y H, Chen X W. Construct non-graded bi-Frobenius algebras via quivers. Sci China Ser A, 2007, 50(3): 450–456MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang Y H, Zhang P. Construct bi-Frobenius algebras via quivers. Tsukuba J Math, 2004, 28(1): 215–227MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang Z H, Li L B. On realization of fusion rings from generalized Cartan matrices. Acta Math Sin (Engl Ser), 2017, 33(3): 362–376MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang Z H, Li L B, Zhang Y H. Green rings of pointed rank one Hopf algebras of nilpotent type. Algebr Represent Theory, 2014, 17(6): 1901–1924MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang Z H, Li L B, Zhang Y H. Green rings of pointed rank one Hopf algebras of non-nilpotent type. J Algebra, 2016, 449: 108–137MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Department of MathematicsTaizhou CollegeTaizhouChina
  3. 3.School of Mathematical ScienceYangzhou UniversityYangzhouChina

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