Algebras and Representation Theory

, Volume 21, Issue 2, pp 259–276 | Cite as

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

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Abstract

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.

Keywords

Projective modules Nonsemisimple Hopf algebra Support varieties Smash coproduct 

Mathematics Subject Classification (2010)

16T05 18D10 

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References

  1. 1.
    Andruskiewitsch, N.: Notes on extensions of Hopf algebras. Can. J. Math. 48(1), 3–42 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andruskiewitsch, N., Devoto, J.: Extensions of Hopf algebras. Algebra i Analiz 7(1), 22–61 (1995)MathSciNetMATHGoogle Scholar
  3. 3.
    Bakalov, B., Kirillov A. Jr.: Lectures on Tensor Categories and Modular Functors. Volume 21 of University Lecture Series. American Mathematical Society, Providence, RI (2001)Google Scholar
  4. 4.
    Benson, D.: Representations and Cohomology II: Cohomology of Groups and Modules. volume 31 of Cambridge Studies in Advanced Mathematics Cambridge University Press (1991)Google Scholar
  5. 5.
    Benson, D.: Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras, volume 30 of Cambridge Studies in Advanced Mathematics Cambridge University Press (1995)Google Scholar
  6. 6.
    Benson, D., Witherspoon, S.: Examples of support varieties for Hopf algebras with noncommutative tensor products. Arch. Math. (Basel) 102(6), 513–520 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Erdmann, K., Holloway, M., Snashall, N., Solberg, Ø., Taillefer, R.: Support varieties for selfinjective algebras. K-Theory 33(1), 67–87 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, Volume 205 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2015)MATHGoogle Scholar
  9. 9.
    Friedlander, E., Pevtsova, J.: π-supports for modules for finite group schemes over a field. Duke Math. J. 139(2), 317–368 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Karpilovsky, G.: Group Representations. Vol. 2, volume 177 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1993)Google Scholar
  11. 11.
    Majid, S.: More examples of bicrossproduct and double cross product Hopf algebras. Israel J. Math. Hopf algebras 72(1–2), 133–148 (1990). doi: 10.1007/BF02764616
  12. 12.
    Molnar, R.K.: Semi-direct products of Hopf algebras. J. Algebra 47(1), 29–51 (1977)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Montgomery, S., Vega, M.D., Witherspoon, S.: Hopf automorphisms and twisted extensions. J. Algebra Appl. 15(6), 14 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Pevtsova, J., Witherspoon, S.: Varieties for modules of quantum elementary abelian groups. Algebr. Represent. Theory 12(6), 567–595 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pevtsova, J., Witherspoon, S.: Tensor ideals and varieties for modules of quantum elementary abelian groups. Proc. Amer. Math. Soc. 143(9), 3727–3741 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Siegel, S., Witherspoon, S.: The Hochschild cohomology ring of a group algebra. Proc. Lond. Math. Soc. 79, 131–157 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Snashall, N., Solberg, Ø.: Support varieties and Hochschild cohomology rings. Proc. Lond. Math. Soc. (3) 88(3), 705–732 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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