The quantum algebra U_q(sl₂) and its equitable presentation

We show that the quantum algebra U_q(sl₂) has a presentation with generators x^±1,y, z and relations xx^-1 = x^-1x = 1, qxy - q^-1yx/q - q^-1 = 1, qyz - q^-1zy/q - q^-1 = 1, qzx - q^-1xz/q - q^-1 = 1. We call this the equitable presentation. We show that y (respectively z) is not invertible in U_q(sl₂) by displaying an infinite-dimensional U_q(sl₂)-module that contains a nonzero null vector for y(respectively z). We consider finite-dimensional U_q(sl₂)-modules under the assumption that q is not a root of 1 and char (K) ≠ 2, where K is the underlying field. We show that y and z are invertible on each finite-dimensional U_q(sl₂)-module. We display a linear operator Ω that acts on finite-dimensional U_q (sl₂)-modules, and satisfies Ω^-1xΩ = y, Ω^-1yΩ = z, Ω^-1zΩ = x on these modules. We define Ω using the q-exponential function.