Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1898396 | Journal of Geometry and Physics | 2016 | 11 Pages |
The Jordan–Schwinger realization is used to construct tensor operators as the even and odd dimensional irreducible submodules of an adjoint representation of the quantum algebra Ŭq(su2). All Ŭq(su2)-submodules are equipped with the so-called left and right qq-Hilbert–Schmidt scalar products by using the Wigner–Eckart theorem. The bases of all irreducible submodules of the adjoint representation are orthonormal with respect to the left qq-Hilbert–Schmidt scalar product, and are orthogonal, but not normalized, with respect to the right one. Consequently, only with respect to the left qq-Hilbert–Schmidt scalar product, the adjoint representation of the quantum algebra Ŭq(su2) on the tensor operators is a ∗∗-representation. We show that both left and right qq-Hilbert–Schmidt scalar products are right SUq(2)SUq(2)-invariant and left SUq−1(2)SUq−1(2)-invariant. Moreover, every irreducible submodule of the adjoint representation of the quantum algebra Ŭq(su2) as an associative algebra with unit, is a left quantum space for O(SUq−1(2))O(SUq−1(2)) and a right quantum space for O(SUq(2))O(SUq(2)). Finally, it is shown that there is a natural compatibility between the coproducts and the Haar measures of the quantum groups O(SUq−1(2))O(SUq−1(2)) and O(SUq(2))O(SUq(2)) and the definitions of the left and right qq-Hilbert–Schmidt scalar products on the tensor operators of the Hopf algebra Ŭq(su2).