On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations

Details On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations

1996-10-25

arxiv.org/abs/q-alg/9610031v1

Int. J. Mod. Phys. A, Vol. 12, No. 13 (1997) 2301-2319

Mathematics

Quantum Algebra

19 pages, latex-file, version to be published in Int. Jour. Mod. phys

Scientific paper

10.1142/S0217751X97001341

By solving a set of recursion relations for the matrix elements of the ${\cal U}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\cal U}_h(sl(2))$ algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction, yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.

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