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An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$

Let $\mathbb F$ denote a field, and pick a nonzero $q \in \mathbb F$ that is not a root of unity. Let $\mathbb Z_4=\mathbb Z/4 \mathbb Z$ denote the cyclic group of order 4. Define a unital associative ${\mathbb F}$-algebra $\square_q$ by generators $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$ and relations $$\frac{q x_i x_{i+1}-q^{-1}x_{i+1}x_i}{q-q^{-1}} = 1,\qquad x^3_i x_{i+2} - \lbrack 3 \rbrack_q x^2_i x_{i+2} x_i + \lbrack 3 \rbrack_q x_i x_{i+2} x^2_i -x_{i+2} x^3_i = 0,$$ where $\lbrack 3 \rbrack_q = \big(q^3-q^{-3}\big)/\big(q-q^{-1}\big)$. Let $V$ denote a $\square_q$-module. A vector $ξ\in V$ is called NIL whenever $x_1 ξ= 0 $ and $x_3 ξ=0$ and $ξ\not=0$. The $\square_q$-module $V$ is called NIL whenever $V$ is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL $\square_q$-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 $\mathfrak{sl}_2$.