Using a q-shuffle algebra to describe the basic module V(Λ_0) for the quantized enveloping algebra Uq(sl^2)

Abstract

We consider the quantized enveloping algebra Uq(sl^2). and its basic module V(Λ0). This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe V(Λ0) 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 u ⋆ v = uv + q(u,v)vu, where (u,v) = 2 (resp. (u,v) = − 2) if u = v (resp. u ≠ v). Let 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 Uq(sl^2). In our first main result, we turn U into a Uq(sl^2).-module. Let U denote the Uq(sl^2).-submodule of U generated by the empty word. In our second main result, we show that the Uq(sl^2).-modules U and V(Λ0) 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.