Tensor operators of the Lie algebras \({\mathfrak U}(n)\) and \({\mathfrak o}(v)\) (Q912958)

Let \({\mathfrak g}\) be a semisimple Lie algebra and \(U({\mathfrak g})\) be its universal enveloping algebra. The adjoint representation of \({\mathfrak g}\) in \({\mathfrak g}\) defines a reducible representation of \({\mathfrak g}\) in \(U({\mathfrak g})\). This representation decomposes into a direct sum of finite dimensional irreducible representations which act in subspaces of \(U({\mathfrak g})\). Let \(e_ 1,...,e_ n\) be a basis of one of these subspaces. The corresponding representation will be denoted by R. The basis elements \(e_ j\) are elements of \(U({\mathfrak g})\). If T is a fixed (possibly, reducible) representation of \({\mathfrak g}\), then we can consider the operators \(T(e_ j)=E_ j\). The set of operators \(E_ j\), \(j=1,2,...,n\), is called tensor operator which transforms according to the irreducible representation R. The author derives simple formulas for matrix elements of tensor operators for the cases of the Lie algebras of the unitary and orthogonal groups. Recurrence relations and generating functions for these matrix elements are given. The author also gives an example of utilization of his result in nuclear physics.