Classification of ten-dimensional kinematical groups with space isotropy

Abstract

All the abstract ten-dimensional real Lie algebras that contain as a subalgebra the algebra of the three-dimensional rotation group (generators J) and decompose under the rotation group into three three-vector representation spaces (J itself, K, and P) and a scalar (generator H) are classified. In all cases, the existence of a homogeneous space of dimension 4 is shown.

Newton–Hooke spacetimes, Hpp-waves and the cosmological constant

“…These are defined as the four-dimensional coset spaces exp(tH + q i P i ) of the Newton-Hooke groups, i.e as the quotients of N ± 10 by the subgroup SO(3) ⊗ L R 3 generated by the J i and the K i . This is analogous to defining Minskowski space as the coset E(3, 1)/SO(3, 1) ; in fact Bacry and Nuyts [15] have shown that all 10-dimensional kinematical groups have a four dimensional space-time interpretation.…”

“…The two Newton-Hooke groups N ± 10 appear to have first surfaced in the work of Lévy-Leblond and Bacry [1] (see also [15]) who classified the possible ten-dimensional kinematic Lie algebras. The commutation relations of their Lie algebras n ± 10 are…”

Thoughts on tachyon cosmology

“…Just as the Poincaré group contracts to the Galilei group or the Carroll group [1,2], the de-Sitter and Anti-de-Sitter groups have interesting contractions which have been classified in [3,4]. One may also obtain the Poincaré group as a contraction of the de-Sitter or Anti-de-Sitter groups and these are in fact the only groups for which this is possible [42].…”

Time-dependent multi-centre solutions from new metrics with holonomy Sim( n − 2)

“…The Carroll group, and the Galilei group are both kinematic groups of a spacetime in the sense of [39,40] and both may be regarded as the symmetry group of a structure in a Lorentzian spacetime with one higher dimensions.…”