Models ofQ-Algebra Representations: The Group of Plane Motions

This paper continues a study of one- and two-variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebra considered is the Lie algebra m (2) of the group of plane motions. It is shown that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group, and the corresponding matrix elements of the group operators are computed on these representation spaces. This local approach applies to more general families of special functions, e. g. , those with complex arguments and parameters, than does the quantum group approach. A simple one-variable model of the infinite-dimensional irreducible representations is used to compute the Clebsch–Gordan coefficients for m (2) considered as a true quantum algebra. The authors derive a generalization of Koelink’s addition formula for Hahn–Exton q-Bessel functions. It is interpreted here as the expansion of the matrix elements of a group operator in a tensor product basis in terms of the matrix elements in a reduced basis.