The quantum algebra and its equitable presentation

We show that the quantum algebra has a presentation with generators x±1,y,z and relations xx−1=x−1x=1, We call this the equitable presentation. We show that y (respectively z) is not invertible in by displaying an infinite-dimensional -module that contains a nonzero null vector for y (respectively z). We consider finite-dimensional -modules under the assumption that q is not a root of 1 and , where is the underlying field. We show that y and z are invertible on each finite-dimensional -module. We display a linear operator Ω that acts on finite-dimensional -modules, and satisfies on these modules. We define Ω using the q-exponential function.