Using a $q$-shuffle algebra to describe the basic module $V(Λ_0)$ for the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$

Abstract

We consider the quantized enveloping algebra U q ( sl 2 ) and its basic module V (Λ 0 ). This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe V (Λ 0 ) using a q-shuffle algebra in the following way. Start with the free associative algebra V on two generators x, y. The standard basis for V consists of the words in x, y. In 1995 M. Rosso introduced an associative algebra structure on V, called a q-shuffle algebra. For u, v ∈ {x, y} their q-shuffle product is uLet U denote the subalgebra of the q-shuffle algebra V that is generated by x, y. Rosso showed that the algebra U is isomorphic to the positive part of U q ( sl 2 ). In our first main result, we turn U into a U q ( sl 2 )-module. Let U denote the U q ( sl 2 )-submodule of U generated by the empty word. In our second main result, we show that the U q ( sl 2 )-modules U and V (Λ 0 ) are isomorphic. Let V denote the subspace of V spanned by the words that do not begin with y or xx. In our third main result, we show that U = U ∩ V.