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Induced representations of the two-parameter Jordanian quantum algebra. (English) Zbl 1052.81540

Summary: We construct induced infinite-dimensional representations of the two-parameter quantum algebra \(U_{g,h}(gl(2))\) which is in duality with the deformation \(GL_{g,h}(2)\). 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 subalgebra \(U_{\tilde g}(sl(2))\), \(\tilde g=g+h\). The representations of the latter can be mapped to representations in one complex variable, which give a new deformation of the standard one-parameter vector-field realization of \(sl(2)\). These infinite-dimensional representations are reducible for some values of the representation parameters, and then we obtain canonically the finite-dimensional representations of \(U_{\tilde g}(sl(2))\).

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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