Tanaka structures modeled on extended Poincaré algebras

Let (V,(·,·)) be a pseudo-Euclidean vector space and S an irreducible Cℓ(V)-module. An extended translation algebra is a graded Lie algebra m=m_{−2}+m_{−1} =V+S with bracket given by ([s,t], v)=b(v·s, t) for some nondegenerate so(V)-invariant reflexive bilinear form b on S. An extended Poincaré structure on a manifold M is a regular distribution D of depth 2 whose Levi form L_x:D_x∧D_x → T_xM/D_x at any point x∈M is identifiable with the bracket [·,·]:S∧S→V of a fixed extended translation algebra m. The classification of the standard maximally homogeneous manifolds with an extended Poincaré structure is given, in terms of Tanaka prolongations of extended translation algebras and of appropriate gradations of real simple Lie algebras.