Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn]
We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra Tm²(q) and the group Hopf algebra K[Cn]: they are nm²-dimensional quantum groups Tωnm²(q) associated to an n-th root of unity ω. Furthermore, using Dirichlet's prime number...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209437 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] / A.-L. Agore // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 23 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862724547774513152 |
|---|---|
| author | Agore, A.-L. |
| author_facet | Agore, A.-L. |
| citation_txt | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] / A.-L. Agore // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra Tm²(q) and the group Hopf algebra K[Cn]: they are nm²-dimensional quantum groups Tωnm²(q) associated to an n-th root of unity ω. Furthermore, using Dirichlet's prime number theorem, we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if d=gcd(m,ν(n)) and ν(n)/d=p₁α₁⋯prαr is the prime decomposition of ν(n)/d then the number of types of Hopf algebras that factorize through Tm²(q) and K[Cn] is equal to (α1+1)(α2+1)⋯(αr+1), where ν(n) is the order of the group of n-th roots of unity in K. As a consequence of our approach, the automorphism groups of these Hopf algebras are described as well.
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| first_indexed | 2025-12-07T18:48:15Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209437 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:48:15Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Agore, A.-L. 2025-11-21T18:51:06Z 2018 Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] / A.-L. Agore // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16T10; 16T05; 16S40 arXiv: 1611.05674 https://nasplib.isofts.kiev.ua/handle/123456789/209437 https://doi.org/10.3842/SIGMA.2018.027 We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra Tm²(q) and the group Hopf algebra K[Cn]: they are nm²-dimensional quantum groups Tωnm²(q) associated to an n-th root of unity ω. Furthermore, using Dirichlet's prime number theorem, we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if d=gcd(m,ν(n)) and ν(n)/d=p₁α₁⋯prαr is the prime decomposition of ν(n)/d then the number of types of Hopf algebras that factorize through Tm²(q) and K[Cn] is equal to (α1+1)(α2+1)⋯(αr+1), where ν(n) is the order of the group of n-th roots of unity in K. As a consequence of our approach, the automorphism groups of these Hopf algebras are described as well. Parts of this work were undertaken while the author was visiting the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France. Their hospitality and the financial support offered by the Jean-Paul Gimon Chair are gratefully acknowledged. Also, we thank the referees for their comments and suggestions that substantially improved the first version of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] Article published earlier |
| spellingShingle | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] Agore, A.-L. |
| title | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] |
| title_full | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] |
| title_fullStr | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] |
| title_full_unstemmed | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] |
| title_short | Hopf Algebras which Factorize through the Taft Algebra Tm²(q) and the Group Hopf Algebra K[Cn] |
| title_sort | hopf algebras which factorize through the taft algebra tm²(q) and the group hopf algebra k[cn] |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209437 |
| work_keys_str_mv | AT agoreal hopfalgebraswhichfactorizethroughthetaftalgebratm2qandthegrouphopfalgebrakcn |