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

Abstract

Expansions of Lie algebras are the opposite process of contractions. Starting from a Lie algebra, the expansion process goes to another one, non-isomorphic and less abelian. We propose an expansion method based in the Casimir invariants of the initial and expanded algebras and where the free parameters involved in the expansion are the curvatures of their associated homogeneous spaces. This method is applied for expansions within the family of Lie algebras of 3d spaces and (2+1)d kinematical algebras. We show that these expansions are classed in two types. The first type makes different from zero the curvature of space or space-time (i.e., it introduces a space or universe radius), while the other has a similar interpretation for the curvature of the space of worldlines, which is non-positive and equal to $-1/c^2$ in the kinematical algebras. We get expansions which go from Galilei to either Newton--Hooke or Poincar\'e algebras, and from these ones to de Sitter algebras, as well as some other examples.