An Infinite-Dimensional Square(Q)-Module Obtained From The Q-Shuffle Algebra For Affine Sl(2)

Let F denote a field, and pick a nonzero q is an element of F that is not a root of unity. Let Z(4) = Z/4Z denote the cyclic group of order 4. Define a unital associative F-algebra square(q) by generators {x(i)}(i is an element of z4) and relationsqx(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 square(q) -module. A vector xi is an element of V is called NIL whenever x(1)xi = 0 and x(3)xi = 0 and xi not equal 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 sl(2).