Relativity symmetries and Lie algebra contractions

We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n)$SO(m,n)$ symmetry as an isometry on an m+n$m+n$ dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n)$G(m,n)$ preserving a symmetry of the same type at dimension m+n−1$m+n-1$, e.g.   a G(m,n−1)$G(m,n-1)$, together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4)$SO(2,4)$, which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3)$G(1,3)$ may be relevant to real physics.