An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
Let 𝔽 denote a field, and pick a nonzero q ∈ 𝔽 that is not a root of unity. Let ℤ₄ = ℤ/4ℤ denote the cyclic group of order 4. Define a unital associative 𝔽-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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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Інститут математики НАН України
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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 𝔽 denote a field, and pick a nonzero q ∈ 𝔽 that is not a root of unity. Let ℤ₄ = ℤ/4ℤ denote the cyclic group of order 4. Define a unital associative 𝔽-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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ |
| spellingShingle |
An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ Post, Sarah Terwilliger, Paul |
| title_short |
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_sort |
infinite-dimensional □q-module obtained from the q-shuffle algebra for affine sl₂ |
| author |
Post, Sarah Terwilliger, Paul |
| author_facet |
Post, Sarah Terwilliger, Paul |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Let 𝔽 denote a field, and pick a nonzero q ∈ 𝔽 that is not a root of unity. Let ℤ₄ = ℤ/4ℤ denote the cyclic group of order 4. Define a unital associative 𝔽-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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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210713 |
| 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 |
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2025-12-17T12:04:33Z |
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2025-12-17T12:04:33Z |
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