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

We consider the quantized enveloping algebra Uq(sl^2). and its basic module V(Λ0). This is infinite-dimensional, irreducible, integrable, highest-weight. describe V(Λ0) using a q-shuffle in following way. Start with free associative V on two generators x, y. The standard basis for consists of words In 1995 M. Rosso introduced an structure V, called algebra. For u, v ∈ {x, y} their product u ⋆ = uv + q(u,v)vu, where (u,v) 2 (resp. − 2) if ≠ v). Let U denote subalgebra that generated by showed isomorphic to positive part our first main result, we turn into Uq(sl^2).-module. Uq(sl^2).-submodule empty word. second show Uq(sl^2).-modules are isomorphic. subspace spanned do not begin y or xx. third ∩ V.