2 00 5 The quantum algebra U q ( sl 2 ) and its equitable presentation ∗

We show that the quantum algebra Uq(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 (resp. z) is not invertible in Uq(sl2) by displaying an infinite dimensional Uq(sl2)-module that contains a nonzero null vector for y (resp. z). We consider finite dimensional Uq(sl2)-modules under the assumption that q is not a root of 1 and char(K) 6= 2, where K is the underlying field. We show that y and z are invertible on each finite dimensional Uq(sl2)-module. We display a linear operator Ω that acts on finite dimensional Uq(sl2)-modules, and satisfies ΩxΩ = y, ΩyΩ = z, ΩzΩ = x on these modules. We define Ω using the q-exponential function. 1 The algebra Uq(sl2) Let K denote a field and let q denote a nonzero scalar in K such that q 6= 1. For an integer n we define [n] = q − q q − q−1 and for n ≥ 0 we define [n] = [n][n− 1] · · · [2][1]. We interpret [0] = 1. We now recall the quantum algebra Uq(sl2). ∗