Induced Representations of the Two-Parameter Jordanian Quantum Algebra

Abstract

We construct induced infinite-dimensional representations of the two-parameter quantum algebraUg,h(gl(2)) which is in duality with the deformationGLg,h(2). The representations depend on two representation parameters, but split into one-parameter representations of a one-generator central subalgebra and the three-generator Jordanian quantum subalgebraU\(_{\tilde g} \) (sl(2)),\(\tilde g\) =g + h. The representations of the latter can be mapped to representations in one complex variable, which give anew deformation of the standard one-parameter vector-field realization ofsl(2). These infinite-dimensional representations are reducible for some values of the representation parameters, and then we obtain canonically the finite-dimensional representations ofU\({\tilde g}\) (sl(2)).