Tanaka structures modeled on extended Poincar\'e algebras

TL;DR

This paper classifies certain geometric structures called extended Poincaré structures on manifolds using algebraic methods involving Tanaka prolongations of extended translation algebras.

Contribution

It provides a classification of maximally homogeneous manifolds with extended Poincaré structures via Tanaka prolongations and gradations of real simple Lie algebras.

Findings

Classification of maximally homogeneous extended Poincaré manifolds

Connection between geometric structures and algebraic prolongations

Use of graded Lie algebras for structure analysis

Abstract

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.