Non-existence of Hopf orders for a twist of the alternating and symmetric groups

Juan Cuadra; Ehud Meir

doi:10.1112/jlms.12203

Non-existence of Hopf orders for a twist of the alternating and symmetric groups

Juan Cuadra^* (Corresponding Author), Ehud Meir^* (Corresponding Author)

^*Corresponding author for this work

Mathematical Science

University of Almeria

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

17 Downloads (Pure)

Abstract

We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel’d twists of group algebras for the following groups: A_n, the alternating group on n elements, with n ≥ 5; and S_2m, the symmetric group on 2m elements, with m ≥ 4 even. The twist for A_n arises from a 2-cocycle on the Klein four-group contained in A₄. The twist for S_2m arises from a 2-cocycle on a subgroup generated by certain transpositions which is isomorphic to Z_m2 . This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.

Original language	English
Pages (from-to)	137-158
Number of pages	22
Journal	Journal of the London Mathematical Society
Volume	100
Issue number	1
Early online date	3 Jan 2019
DOIs	https://doi.org/10.1112/jlms.12203
Publication status	Published - Aug 2019

Bibliographical note

Juan Cuadra was supported by grant MTM2017-86987-P from MICINN and FEDER and by the research group FQM0211 from Junta de Andalucía. Ehud Meir was supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory”.
The authors would like to thank Sonia Natale for her comments on a first version
of this paper and, specially, for drawing their attention to [7]. The authors are
indebted to the attentive referee for his careful revision and apt comments, which
helped to improve the original manuscript.

Keywords

16T05 (primary)
16G30 (secondary)

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Accepted Manuscript
This is the accepted version of the following article: Cuadra, J. and Meir, E. (2019), Non‐existence of Hopf orders for a twist of the alternating and symmetric groups. J. London Math. Soc., which has been published in final form at https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.12203.
Accepted author manuscript, 560 KBLicence: Other

Cite this

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title = "Non-existence of Hopf orders for a twist of the alternating and symmetric groups",

abstract = "We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel{\textquoteright}d twists of group algebras for the following groups: An, the alternating group on n elements, with n ≥ 5; and S2m, the symmetric group on 2m elements, with m ≥ 4 even. The twist for An arises from a 2-cocycle on the Klein four-group contained in A4. The twist for S2m arises from a 2-cocycle on a subgroup generated by certain transpositions which is isomorphic to Zm2 . This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.",

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author = "Juan Cuadra and Ehud Meir",

note = "Juan Cuadra was supported by grant MTM2017-86987-P from MICINN and FEDER and by the research group FQM0211 from Junta de Andaluc{\'i}a. Ehud Meir was supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory”. The authors would like to thank Sonia Natale for her comments on a first version of this paper and, specially, for drawing their attention to [7]. The authors are indebted to the attentive referee for his careful revision and apt comments, which helped to improve the original manuscript.",

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AU - Meir, Ehud

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N2 - We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel’d twists of group algebras for the following groups: An, the alternating group on n elements, with n ≥ 5; and S2m, the symmetric group on 2m elements, with m ≥ 4 even. The twist for An arises from a 2-cocycle on the Klein four-group contained in A4. The twist for S2m arises from a 2-cocycle on a subgroup generated by certain transpositions which is isomorphic to Zm2 . This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.

AB - We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel’d twists of group algebras for the following groups: An, the alternating group on n elements, with n ≥ 5; and S2m, the symmetric group on 2m elements, with m ≥ 4 even. The twist for An arises from a 2-cocycle on the Klein four-group contained in A4. The twist for S2m arises from a 2-cocycle on a subgroup generated by certain transpositions which is isomorphic to Zm2 . This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.

KW - 16T05 (primary)

KW - 16G30 (secondary)

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UR - http://www.mendeley.com/research/nonexistence-hopf-orders-twist-alternating-symmetric-groups

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JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 1

ER -

Non-existence of Hopf orders for a twist of the alternating and symmetric groups

Abstract

Bibliographical note

Keywords

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