From Lorentz to SIM(2): contraction, four-dimensional algebraic relations and projective representations

We present a comprehensive study on [Formula: see text] and [Formula: see text] groups, their representations and algebraic aspects. These groups, together with [Formula: see text], arise as the symmetry groups of Very Special Relativity (VSR), where full Lorentz invariance is reduced while retaining many relativistic consequences. After obtaining [Formula: see text] through the Inönü–Wigner contraction procedure, a complete four-dimensional algebraic representation is shown for [Formula: see text] and [Formula: see text]. Besides that, we apply Bargmann’s formalism to investigate the (projective) representations for both cases, keeping track of the source of phase factors. We complete the study by presenting a particularly simple analysis to probe the existence of local phase factors, which is useful when dealing with non-abelian groups.