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Title: Remarks on Sekine quantum groups (English)
Author: Chen, Jialei
Author: Yang, Shilin
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 72
Issue: 3
Year: 2022
Pages: 695-707
Summary lang: English
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Category: math
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Summary: We first describe the Sekine quantum groups $\mathcal {A}_{k}$ (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of $\mathcal {A}_{k}$ and describe their representation rings $r(\mathcal {A}_{k})$. Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of $r(\mathcal {A}_{k})$. (English)
Keyword: Sekine quantum group
Keyword: representation ring
Keyword: Casimir number
MSC: 16D70
MSC: 16G10
MSC: 16T05
idZBL: Zbl 07584096
idMR: MR4467936
DOI: 10.21136/CMJ.2022.0112-21
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Date available: 2022-08-22T08:17:57Z
Last updated: 2024-10-04
Stable URL: http://hdl.handle.net/10338.dmlcz/150611
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Reference: [3] Kac, G. I., Paljutkin, V. G.: Finite ring groups.Trans. Mosc. Math. Soc. 15 (1966), 251-294. Zbl 0218.43005, MR 0208401
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Reference: [5] Sekine, Y.: An example of finite-dimensional Kac algebras of Kac-Paljutkin type.Proc. Am. Math. Soc. 124 (1996), 1139-1147. Zbl 0845.46031, MR 1307564, 10.1090/S0002-9939-96-03199-1
Reference: [6] Vaes, S., Vainerman, L.: Extensions of locally compact quantum groups and the bicrossed product construction.Adv. Math. 175 (2003), 1-101. Zbl 1034.46068, MR 1970242, 10.1016/S0001-8708(02)00040-3
Reference: [7] Wang, Z., Li, L., Zhang, Y.: A criterion for the Jacobson semisimplicity of the Green ring of a finite tensor category.Glasg. Math. J. 60 (2018), 253-272. Zbl 1444.16025, MR 3733845, 10.1017/S0017089517000246
Reference: [8] Zhang, H.: Idempotent states on Sekine quantum groups.Commun. Algebra 47 (2019), 4095-4113. Zbl 07089356, MR 3975989, 10.1080/00927872.2019.1579335
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