Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
Assume that 𝔽 is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative 𝔽-algebra defined by generators and relations in the following way. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D and that each of [A, D]+AC−B...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
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| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210592 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero. Hau-Wen Huang and Sarah Bockting-Conrad. SIGMA 16 (2020), 018, 17 pages |
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Huang, Hau-Wen Bockting-Conrad, Sarah 2025-12-12T10:33:31Z 2020 Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero. Hau-Wen Huang and Sarah Bockting-Conrad. SIGMA 16 (2020), 018, 17 pages 1815-0659 2020 Mathematics Subject Classification: 81R10; 16S37 arXiv:1910.11446 https://nasplib.isofts.kiev.ua/handle/123456789/210592 https://doi.org/10.3842/SIGMA.2020.018 Assume that 𝔽 is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative 𝔽-algebra defined by generators and relations in the following way. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D and that each of [A, D]+AC−BA, [B, D]+BA−CB, [C, D]+CB−AC is central in ℜ. In this paper, we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules. The research of the first author is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| spellingShingle |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero Huang, Hau-Wen Bockting-Conrad, Sarah |
| title_short |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_full |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_fullStr |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_full_unstemmed |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_sort |
finite-dimensional irreducible modules of the racah algebra at characteristic zero |
| author |
Huang, Hau-Wen Bockting-Conrad, Sarah |
| author_facet |
Huang, Hau-Wen Bockting-Conrad, Sarah |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Assume that 𝔽 is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative 𝔽-algebra defined by generators and relations in the following way. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D and that each of [A, D]+AC−BA, [B, D]+BA−CB, [C, D]+CB−AC is central in ℜ. In this paper, we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210592 |
| citation_txt |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero. Hau-Wen Huang and Sarah Bockting-Conrad. SIGMA 16 (2020), 018, 17 pages |
| work_keys_str_mv |
AT huanghauwen finitedimensionalirreduciblemodulesoftheracahalgebraatcharacteristiczero AT bocktingconradsarah finitedimensionalirreduciblemodulesoftheracahalgebraatcharacteristiczero |
| first_indexed |
2025-12-17T12:04:17Z |
| last_indexed |
2025-12-17T12:04:17Z |
| _version_ |
1851756964561813504 |