Wigner-Eckart theorem for tensor operators of Hopf algebras

TL;DR

This paper extends the Wigner-Eckart theorem to tensor operators within Hopf algebras, providing a proof based on irreducible representations and Schur's lemma, and calculates reduced matrix elements for specific cases.

Contribution

It generalizes the Wigner-Eckart theorem to arbitrary Hopf algebras and explicitly constructs elements of the center of U_t(su(2)).

Findings

Proved Wigner-Eckart theorem for Hopf algebra tensor operators.

Calculated reduced matrix elements for U_t(su(2)) tensor operators.

Constructed elements of the center of U_t(su(2)).

Abstract

We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra U$_{t}$(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of U$_{t}$(su(2)) is given.