An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2020
• Hauptverfasser: Post, Sarah, Terwilliger, Paul
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2020
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/210713
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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

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.