Right SUq(2)SUq(2)- and left SUq−1(2)SUq−1(2)-invariances of the qq-Hilbert–Schmidt Scalar products for an adjoint representation of the quantum algebra Ŭq(su2)

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).