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

We present a comprehensive study on $SIM(2)$ and $ISIM(2)$ groups, their representations and algebraic aspects. After obtaining $SIM(2)$ through the Inönü-Wigner contraction procedure, a complete four-dimensional algebraic representation is shown for $\mathfrak{sim(2)}$ and $\mathfrak{isim(2)}$. 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.