Orders of Nikshych's Hopf algebra

Juan Cuadra, Ehud Meir

Research output: Contribution to journalArticle

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Abstract

Let p be an odd prime number and K a number field having a primitive pth root of unity ζp. We prove that Nikshych's non group-theoretical Hopf algebra Hp, which is defined over Q(ζp), admits a Hopf order over the ring of integers OK if and only if there is an ideal I of OK such that I2(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 exists, it is unique and we describe it explicitly.
Original languageEnglish
Pages (from-to)919-955
Number of pages37
JournalJournal of Noncommutative Geometry
Volume11
Issue number3
Early online date26 Sep 2017
DOIs
Publication statusPublished - Sep 2017

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Hopf Algebra
Number field
Ring
Cyclotomic Fields
Primitive Roots
Integer
Cyclotomic
Odd number
Roots of Unity
Prime number
Semisimple
If and only if

Keywords

  • fusion categories
  • semisimple Hopf algebras
  • Hopf orders
  • group schemes
  • cyclotomic integers

Cite this

Orders of Nikshych's Hopf algebra. / Cuadra, Juan; Meir, Ehud.

In: Journal of Noncommutative Geometry, Vol. 11, No. 3, 09.2017, p. 919-955.

Research output: Contribution to journalArticle

Cuadra, Juan ; Meir, Ehud. / Orders of Nikshych's Hopf algebra. In: Journal of Noncommutative Geometry. 2017 ; Vol. 11, No. 3. pp. 919-955.
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abstract = "Let p be an odd prime number and K a number field having a primitive pth root of unity ζp. We prove that Nikshych's non group-theoretical Hopf algebra Hp, which is defined over Q(ζp), admits a Hopf order over the ring of integers OK if and only if there is an ideal I of OK such that I2(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 exists, it is unique and we describe it explicitly.",
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KW - fusion categories

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KW - group schemes

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