Tanaka structures modeled on extended Poincaré algebras

Details Tanaka structures modeled on extended Poincaré algebras Tanaka structures modeled on extended Poincaré algebras

2012-01-03

arxiv.org/abs/1201.0555v2

Mathematics

Differential Geometry

22 pages, 4 tables v2: minor revision

Scientific paper

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 Poincar\'e structure on a manifold M is a regular distribution D of depth 2 whose Levi form L_x: D_x\wedge D_x\rightarrow T_xM/D_x at any point x\in M is identifiable with the bracket [.,.]: S\wedge S\rightarrow V of a fixed extended translation algebra m. The classification of the standard maximally homogeneous manifolds with an extended Poincar\'e structure is given, in terms of Tanaka prolongations of extended translation algebras and of appropriate gradations of real simple Lie algebras.

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