Geodesic
An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 \begin{gather*} \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, \end{gather*} 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 $\xi\in V$ is called NIL whenever $x_1 \xi = 0 $ and $x_3 \xi=0$ and $\xi \not=0$. 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 $\mathfrak{sl}_2$.
Keywords: quantum group, derivation
Mots-clés : $q$-Serre relations, $q$-Onsager algebra. 
@article{SIGMA_2020_16_a36,
     author = {Sarah Post and Paul Terwilliger},
     title = {An {Infinite-Dimensional} $\square_q${-Module} {Obtained} from the $q${-Shuffle} {Algebra} for {Affine} $\mathfrak{sl}_2$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a36/}
}
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Sarah Post; Paul Terwilliger. An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a36/

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