2+1 kinematical expansions: from Galilei to de Sitter algebras

Expansion of a Lie algebra is the opposite process to contraction. Starting from a Lie algebra, the expansion process goes to another algebra, which i s non-isomorphic and less Abelian. We propose an expansion method based on the Casimir invariants of the initial and expanded algebras and where the f ree parameters involved in the expansion are the curvatures of their associ ated homogeneous spaces. This method is applied for expansions within the f amily of Lie algebras of three-dimensional spaces and (2 + 1)D kinematical algebras. We show that these expansions are classed into two types. The fir st type makes the curvature of space or spacetime different from zero (i.e. it introduces a space or universe radius), while the other has a similar i nterpretation for the curvature of the space of worldlines, which is non-po sitive and equal to -1/c(2) in the kinematical algebras. We obtain expansio ns which go from Galilei to either Newton-Hooke or Poincare algebras, and f rom these to de Sitter algebras, as well as some other examples.