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Every affine structure on Lie algebra  defines arepresentation of  in aff(ℝn). If  is a nilpotent Lie algebra provided with acomplete affine structure then the corresponding representationis nilpotent. We describe noncomplete affinestructures on the filiform Lie algebra Ln. As a consequence we give a nonnilpotent faithful linear representation of the 3-dimensional Heisenberg algebra

Noncomplete affine structures on Lie algebras of maximal class