Hilbert-Schmidt Inner Product for an Adjoint Representation of the Quantum Algebra U⌣Q(SU2)

The Jordan-Schwinger realization of quantum algebra U⌣q(su2) is used to construct the irreducible submodule Tl of the adjoint representation in two different bases. The two bases are known as types of irreducible tensor operators of rank l which are related to each other by the involution map. The bases of the submodules are equipped with q-analogues of the Hilbert-Schmidt inner product and it is also shown that the adjoint representation corresponding to one of those submodules is a *-representation.