An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
Let denote a field, and pick a nonzero q ∈ that is not a root of unity. Let ℤ₄ = ℤ/4ℤ denote the cyclic group of order 4. Define a unital associative -algebra □q by generators {xᵢ}ᵢ∈ℤ4 and relations (qxᵢxᵢ₊₁ − q⁻¹xᵢ₊₁xᵢ)/(q−q⁻¹) = 1, x³ᵢxᵢ₊₂ − [3]qx²ᵢxᵢ + ₂xᵢ + [3]qxᵢxᵢ₊₂x²ᵢ − xᵢ₊₂x³ᵢ=0, where [3]...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210713 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂. Sarah Post and Paul Terwilliger. SIGMA 16 (2020), 037, 35 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862619500912836608 |
|---|---|
| author | Post, Sarah Terwilliger, Paul |
| author_facet | Post, Sarah Terwilliger, Paul |
| citation_txt | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂. Sarah Post and Paul Terwilliger. SIGMA 16 (2020), 037, 35 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let denote a field, and pick a nonzero q ∈ that is not a root of unity. Let ℤ₄ = ℤ/4ℤ denote the cyclic group of order 4. Define a unital associative -algebra □q by generators {xᵢ}ᵢ∈ℤ4 and relations (qxᵢxᵢ₊₁ − q⁻¹xᵢ₊₁xᵢ)/(q−q⁻¹) = 1, x³ᵢxᵢ₊₂ − [3]qx²ᵢxᵢ + ₂xᵢ + [3]qxᵢxᵢ₊₂x²ᵢ − xᵢ₊₂x³ᵢ=0, where [3]q=(q³−q⁻³)/(q−q⁻¹). Let V denote a □q-module. A vector ξ ∈ V is called NIL whenever x₁ξ = 0 and x₃ξ = 0, and ξ≠0. The □q-module V is called NIL whenever V is generated by a NIL vector. We show that up to isomorphism, there exists a unique NIL □q-module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description, an important role is played by the q-shuffle algebra for affine sl₂.
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| first_indexed | 2025-12-17T12:04:33Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210713 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:33Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Post, Sarah Terwilliger, Paul 2025-12-15T15:29:22Z 2020 An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂. Sarah Post and Paul Terwilliger. SIGMA 16 (2020), 037, 35 pages 1815-0659 2020 Mathematics Subject Classification: 17B37 arXiv:1806.10007 https://nasplib.isofts.kiev.ua/handle/123456789/210713 https://doi.org/10.3842/SIGMA.2020.037 Let denote a field, and pick a nonzero q ∈ that is not a root of unity. Let ℤ₄ = ℤ/4ℤ denote the cyclic group of order 4. Define a unital associative -algebra □q by generators {xᵢ}ᵢ∈ℤ4 and relations (qxᵢxᵢ₊₁ − q⁻¹xᵢ₊₁xᵢ)/(q−q⁻¹) = 1, x³ᵢxᵢ₊₂ − [3]qx²ᵢxᵢ + ₂xᵢ + [3]qxᵢxᵢ₊₂x²ᵢ − xᵢ₊₂x³ᵢ=0, where [3]q=(q³−q⁻³)/(q−q⁻¹). Let V denote a □q-module. A vector ξ ∈ V is called NIL whenever x₁ξ = 0 and x₃ξ = 0, and ξ≠0. The □q-module V is called NIL whenever V is generated by a NIL vector. We show that up to isomorphism, there exists a unique NIL □q-module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description, an important role is played by the q-shuffle algebra for affine sl₂. The first author acknowledges support by the Simons Foundation Collaboration Grant 3192112. The second author thanks Marc Rosso and Xin Fang for helpful comments about q-shuffle algebras. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ Article published earlier |
| spellingShingle | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ Post, Sarah Terwilliger, Paul |
| title | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ |
| title_full | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ |
| title_fullStr | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ |
| title_full_unstemmed | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ |
| title_short | An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ |
| title_sort | infinite-dimensional □q-module obtained from the q-shuffle algebra for affine sl₂ |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210713 |
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