Validity of the Gell-Mann formula for \(\mathrm{su}(n,\mathbb R)\) and \(\mathrm{su}(n)\) algebras (Q2836548)

It was shown more than 40 years ago that the Gell-Mann formula describes in general the inverse of a simple Inönü-Wigner contraction ONLY in the case of \(\mathrm{so}(m,n)\). In addition, some sufficient conditions for irreducible representations were given under which the Gell-Mann formula works for other Lie algebras, too. Now, the authors prove that for \(\mathrm{sl}(n,\mathbb R)\) when contracted with respect to the maximal compact subalgebra \(so(n)\) the old sufficient condition that Wigner's little group is represented trivially is in fact necessary. Since the special case \(\mathrm{sl}(2,\mathbb R)\) corresponds to \(\mathrm{so}(2,1)\) it does not have to be treated separately. The presentation should be less vague, statements like ``The validity constraints are clarified'' in the abstract should be made more precise.