Two-Parameter Quantum Groups and -Matrices: Classical Types
We construct finite -matrices for the first fundamental representation of two-parameter quantum groups ᵣ,ₛ() for classical , both through the decomposition of ⊗ into irreducibles ᵣ,ₛ()-submodules as well as by evaluating the universal R-matrix. The latter is crucially based on the construction of...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/214100 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Two-Parameter Quantum Groups and -Matrices: Classical Types. Ian Martin and Alexander Tsymbaliuk. SIGMA 21 (2025), 064, 54 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862533582249000960 |
|---|---|
| author | Martin, Ian Tsymbaliuk, Alexander |
| author_facet | Martin, Ian Tsymbaliuk, Alexander |
| citation_txt | Two-Parameter Quantum Groups and -Matrices: Classical Types. Ian Martin and Alexander Tsymbaliuk. SIGMA 21 (2025), 064, 54 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We construct finite -matrices for the first fundamental representation of two-parameter quantum groups ᵣ,ₛ() for classical , both through the decomposition of ⊗ into irreducibles ᵣ,ₛ()-submodules as well as by evaluating the universal R-matrix. The latter is crucially based on the construction of dual PBW-type bases of ⁺⁻ᵣ,ₛ() consisting of the ordered products of quantum root vectors defined via (, )-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine -matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of (u) and (v), viewed as modules over two-parameter quantum affine algebras ᵣ,ₛ(ˆ) for classical . The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras.
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| first_indexed | 2026-03-21T11:32:40Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214100 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:32:40Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Martin, Ian Tsymbaliuk, Alexander 2026-02-19T11:14:51Z 2025 Two-Parameter Quantum Groups and -Matrices: Classical Types. Ian Martin and Alexander Tsymbaliuk. SIGMA 21 (2025), 064, 54 pages 1815-0659 2020 Mathematics Subject Classification: 17B37; 16T25 arXiv:2407.01450 https://nasplib.isofts.kiev.ua/handle/123456789/214100 https://doi.org/10.3842/SIGMA.2025.064 We construct finite -matrices for the first fundamental representation of two-parameter quantum groups ᵣ,ₛ() for classical , both through the decomposition of ⊗ into irreducibles ᵣ,ₛ()-submodules as well as by evaluating the universal R-matrix. The latter is crucially based on the construction of dual PBW-type bases of ⁺⁻ᵣ,ₛ() consisting of the ordered products of quantum root vectors defined via (, )-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine -matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of (u) and (v), viewed as modules over two-parameter quantum affine algebras ᵣ,ₛ(ˆ) for classical . The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras. This note represents a part (followed up by [35, 36]) of the year-long REU project at Purdue University. We are grateful to Purdue University for its support and for the opportunity to present these results at the REU math conference in April 2024. A.T. is deeply indebted to Andrei Negut¸ for numerous stimulating discussions over the years and for sharing the beautiful combinatorial features of quantum groups, to Sarah Witherspoon for a correspondence on two-parameter quantum groups, to Curtis Wendlandt for bringing attention to [16], to Rouven Fraseek and Daniele Valeri for invitations for research visits in Italy during the summer of 2024. A.T. is grateful to INdAM-GNSAGA and the FAR UNIMORE project CUP-E93C2300204000 for the support and wonderful working conditions during his visit to Italy, where the final version of the paper was completed; to IHES for the hospitality and wonderful working conditions in the summer of 2025, where the journal version of this paper was prepared. We are very grateful to the referees for their useful suggestions that improved the exposition. The work of both authors was partially supported by an NSF Grant DMS-2302661. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Two-Parameter Quantum Groups and -Matrices: Classical Types Article published earlier |
| spellingShingle | Two-Parameter Quantum Groups and -Matrices: Classical Types Martin, Ian Tsymbaliuk, Alexander |
| title | Two-Parameter Quantum Groups and -Matrices: Classical Types |
| title_full | Two-Parameter Quantum Groups and -Matrices: Classical Types |
| title_fullStr | Two-Parameter Quantum Groups and -Matrices: Classical Types |
| title_full_unstemmed | Two-Parameter Quantum Groups and -Matrices: Classical Types |
| title_short | Two-Parameter Quantum Groups and -Matrices: Classical Types |
| title_sort | two-parameter quantum groups and -matrices: classical types |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214100 |
| work_keys_str_mv | AT martinian twoparameterquantumgroupsandmatricesclassicaltypes AT tsymbaliukalexander twoparameterquantumgroupsandmatricesclassicaltypes |