article

Title: Tanaka structures modeled on extended Poincaré algebras

Authors: Andrea Altomani and Andrea Santi

Issue: Volume 63 (2014), Issue 1, 91-117

Abstract:

$Let $(V,(\cdot,\cdot))$ be a pseudo-Euclidean vector space and $S$ an irreducible $C\ell(V)$-module. An extended translation algebra is a graded Lie algebra $\mathfrak{m}=\mathfrak{m}_{-2}+\mathfrak{m}_{-1}=V+S$ with bracket given by $([s,t],v)=b(v\cdot s,t)$ for some nondegenerate $\mathfrak{so}(V)$-invariant reflexive bilinear form $b$ on $S$. An extended Poincar\'e structure on a manifold $M$ is a regular distribution $\mathcal{D}$ of depth $2$ whose Levi form $\mathcal{L}_x:\mathcal{D}_x\wedge\mathcal{D}_x\to T_x M/\mathcal{D}_x$ at any point $x\in M$ is identifiable with the bracket $[\cdot,\cdot]\colon S\wedge S\to V$ of a fixed extended translation algebra $\mathfrak{m}$. 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.$