JournalsjncgVol. 11, No. 3pp. 919–955

Orders of Nikshych's Hopf algebra

  • Juan Cuadra

    University of Almeria, Spain
  • Ehud Meir

    Universität Hamburg, Germany
Orders of Nikshych's Hopf algebra cover
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Abstract

Let pp be an odd prime number and KK a number field having a primitive ppth root of unity ζp\zeta_p. We prove that Nikshych's non group-theoretical Hopf algebra HpH_p, which is defined over Q(ζp)\mathbb Q(\zeta_p), admits a Hopf order over the ring of integers OK\mathcal O_K if and only if there is an ideal II of OK\mathcal O_K such that I2(p1)=(p)I^{2(p-1)} = (p). This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over OK\mathcal O_K exists, it is unique and we describe it explicitly.

Cite this article

Juan Cuadra, Ehud Meir, Orders of Nikshych's Hopf algebra. J. Noncommut. Geom. 11 (2017), no. 3, pp. 919–955

DOI 10.4171/JNCG/11-3-5