The quantum algebra Uq(sl2)Uq(sl2) and its equitable presentation

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