On the structure of tensor operators in \(\mathrm{SU}(3)\) (Q1066291)

The authors achieve a global description of the structure of all \(\mathrm{SU}(3)\) tensor operators [\textit{L. Michel}, Group Represent. Math. Phys., Lect. Notes Phys. 6, 36--143 (1970; Zbl 0206.27001)]. They show that the complex Hilbert space on which the \(\mathrm{SU}(3)\) tensor operators act carries a single (topologically) irreducible skew-symmetric representation of the Lie algebra \(\mathfrak{so}(6,2)\) which contains exactly once every irreducible skew-symmetric representation of the Lie algebra \(\mathfrak{su}(3)\). Moreover, the set of all \(\mathrm{SU}(3)\) tensor operators can be embedded into a simple algebra \(\mathcal A\) isomorphic to a quotient of the universal enveloping algebra of the complex Lie algebra \(\mathfrak{so}(8,\mathbb C)\). The authors carry out the decomposition of \(\mathcal A\) under \(\mathrm{SU}(3)\) and use it to give a global formulation of \(\mathrm{SU}(3)\) tensor operators.