Using a $q$-shuffle algebra to describe the basic module $V({\mathrm{\Lambda }}_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$

We consider the quantized enveloping algebra $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ and its basic module $V({\mathrm{\Lambda }}_0)$. This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe $V({\mathrm{\Lambda }}_0)$ using a $q$-shuffle algebra in the following way. Start with the free associative algebra $\mathbb{V}$ on two generators $x,y$. The standard basis for $\mathbb{V}$ consists of the words in $x,y$. In 1995 M. Rosso introduced an associative algebra structure on $\mathbb{V}$, called a $q$-shuffle algebra. For $u,v\in \lbrace x,y\rbrace$their$q$-shuffleproductis$u\star v = uv+q^{(u,v)}vu$, where $(u,v)=2$ (resp. $(u,v)=-2$) if $u=v$ (resp. $u≠v$). Let $\mathbb{U}$ denote the subalgebra of the $q$-shuffle algebra $\mathbb{V}$ that is generated by $x,y$. Rosso showed that the algebra $\mathbb{U}$ is isomorphic to the positive part of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$. In our first main result, we turn $\mathbb{U}$ into a $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$-module. Let $\mathbf{U}$ denote the $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$-submodule of $\mathbb{U}$ generated by the empty word. In our second main result, we show that the $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$-modules $\mathbf{U}$ and $V({\mathrm{\Lambda }}_0)$ are isomorphic. Let $\mathbf{V}$ denote the subspace of $\mathbb{V}$ spanned by the words that do not begin with $y$ or $xx$. In our third main result, we show that $\bf U = \mathbb U \cap {\bf V}$.