Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 037, 35 pages      arXiv:1806.10007      https://doi.org/10.3842/SIGMA.2020.037

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

Sarah Post a and Paul Terwilliger b
a)  Department of Mathematics, University of Hawai`i at Manoa, Honolulu, HI 96822, USA
b)  Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA

Received August 18, 2019, in final form April 19, 2020; Published online May 04, 2020

Abstract
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 $\xi\in V$ 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 $\mathfrak{sl}_2$.

Key words: quantum group; $q$-Serre relations; derivation; $q$-Onsager algebra.

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