Non Complete Affine Connections on Filiform Lie Algebras

We give a example of non nilpotent faithful representation of a filiform Lie algebra. This gives one counter-example of the conjecture saying that every affine connection on a filiform Lie group is complete. 1. Affine connection on a nilpotent Lie algebra 1.1. Affine connection on nilpotent Lie algebras. Definition 1. Let g be a n-dimensional Lie algebra over R. It is called affine if there is a bilinear mapping ∇ : g × g → g satisfying { 1) ∇ (X,Y )−∇ (Y,X) = [X,Y ] 2) ∇ (X,∇ (Y, Z))−∇ (Y,∇ (X,Z)) = ∇ ([X,Y ] , Z) for all X,Y, Z ∈ g. If g is affine, then the corresponding connected Lie group G is an affine manifold such that every left translation is an affine isomorphism of G. In this case, the operator ∇ is nothing that the connection operator of the affine connection on G. Let g be an affine Lie algebra. Then the mapping