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

Abstract

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.