The quantum algebra U-q(sl(2)) and its equitable presentation

Abstract

We show that the quantum algebra U-q(sl(2)) has a presentation with generators x(+/- 1), y, Z and relations xx(-1) = x(-1)x = 1, [GRAPHICS] We call this the equitable presentation. We show that y (respectively z) is not invertible in U-q (sl(2)) by displaying an infinite-dimensional U-q (sl(2))-module that contains a nonzero null vector for y (respectively z). We consider finite-dimensional Uq (sl(2))-modules under the assumption that q is not a root of 1 and char(K) not equal 2, where K is the underlying field. We show that y and z are invertible on each finite-dimensional Uq(sl(2))-module. We display a linear operator Omega that acts on finite-dimensional U-q(sl(2))-modules, and satisfies Omega(-1) x Omega = y, Omega(-1) y Omega = z, Omega(-1) z Omega = x on these modules. We define Omega using the q-exponential function. (c) 2005 Elsevier Inc. All rights reserved.