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The Structure of Hopf Algebras Acting on Dihedral Extensions

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Advances in Algebra (SRAC 2017)

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Abstract

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group \(D_p\), \(p\ge 3\) prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case \(p=3\) and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras.

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Acknowledgements

The authors would like to thank the referee for comments and suggestions which improved the exposition and content of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Agnes Scott College, 141 E. College Ave., Decatur, GA, 30030, USA

    Alan Koch

  2. Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    Timothy Kohl

  3. School of Computing and Mathematics, Keele University, Staffordshire, ST5 5BG, UK

    Paul J. Truman

  4. Department of Mathematics and Computer Science, Auburn University at Montgomery, Montgomery, AL, 36124, USA

    Robert Underwood

Authors
  1. Alan Koch
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  2. Timothy Kohl
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  3. Paul J. Truman
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  4. Robert Underwood
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Corresponding author

Correspondence to Robert Underwood .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Jörg Feldvoss

  2. Department of Mathematics, Spring Hill College, Mobile, AL, USA

    Lauren Grimley

  3. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Drew Lewis

  4. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Andrei Pavelescu

  5. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Cornelius Pillen

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Koch, A., Kohl, T., Truman, P.J., Underwood, R. (2019). The Structure of Hopf Algebras Acting on Dihedral Extensions. In: Feldvoss, J., Grimley, L., Lewis, D., Pavelescu, A., Pillen, C. (eds) Advances in Algebra. SRAC 2017. Springer Proceedings in Mathematics & Statistics, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-030-11521-0_10

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