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

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 $\{x_i{\}}_{i\in {\mathbb{Z}}_4}$ and relations $$\frac{q{x}_i{x}_{i+1}-q^{-1}{x}_{i+1}{x}_i}{q-q^{-1}}=1,x_i^3{x}_{i+2}-[3{]}_q{x}_i^2{x}_{i+2}{x}_i+[3{]}_q{x}_i{x}_{i+2}{x}_i^2-x_{i+2}{x}_i^3=0,$$ where $[3{]}_q=(q^3-q^{-3})/(q-q^{-1})$. Let $V$ denote a ${\square }_q$-module. A vector $\hat{I}¾\in V$ is called NIL whenever $x_1\hat{I}¾=0$ and $x_3\hat{I}¾=0$ and $\hat{I}¾\ne 0$. 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 ${\mathfrak{sl}}_2$.

42 pages