Tensor operators and Clebsch–Gordan coefficients of the quantum algebra suq(1,1)

Abstract

Two kinds of suq(1,1) tensor operators are considered. One of them carries a nonunitary finite dimensional representation of suq(1,1), and the other carries a unitary infinite dimensional representation. Explicit formulas of the former tensor operators are constructed and a useful formula to calculate the matrix elements of rank 1 tensors is derived. By making use of the q analog of the Wigner–Eckart’s theorem, the Clebsch–Gordan coefficient can be extracted from the matrix element of a tensor operator. Explicit formulas of three kinds of the suq(1,1) Clebsch–Gordan coefficients are given, that is, the Clebsch–Gordan coefficient which couples two (non) unitary representations to get the third (non) unitary representation, and the one which couples a nonunitary and a unitary representations to get a new unitary representation. It is shown that these Clebsch–Gordan coefficients and the one of suq(2) can transmute one another by the appropriate replacement of their variables. It is further shown that, by using this property, various recoupling coefficients of suq(1,1), such as 6-j and 9-j symbols, can be directly obtained from the corresponding ones of suq(2).