On covariant realizations of the Poincaré group \(P(1,3)\) (Q1396669)

The author describes realizations of the Poincaré Lie algebra \(p = so(1,3) +\mathbb R^{1,3}\) in the space \(\mathbb R^{1,3} \times \mathbb C^n\) which are extensions of the standard action of \(p\) on the Minkowski space \(\mathbb R^{1,3}\) with the trivial action of the translation subalgebra \(\mathbb R^{1,3} \) on \(\mathbb C^n\). The problem reduces to the construction of realisations of the Lorentz Lie algebra \(so(1,3)\) by holomorphic vector fields on \(\mathbb C^n\) up to a bi-holomorphic transformations. All such realisations are described, using the fact that the complexification of the Lorentz Lie algebra is a direct sum of two copies of \(sl(2,\mathbb C)\).