  In the first part we present the Weyl algebra and our results concerning its
finite-dimensional Lie subalgebras. The second part is devoted to a more exotic
algebraic structure, the Lie algebra of order 3. We set the basis of a theory
of deformations and contractions of these algebraic structures. We then
concentrate on a particular such Lie algebra of order 3 which extends in a
non-trivial way the Poincar\'e algebra, this extension being different of the
supersymmetric extension. We then focus on the construction of a field
theoretical model based on this algebra, the {\it cubic supersymmetry} ({\it
3SUSY}). For this purpose we obtain bosonic multiplets with whom we construct
invariant Lagrangians. We then study the compatibility between this new
symmetry and the abelian gauge symmetry. Furthermore, the analyse of possible
interactions shows that interactions terms are not allowed by the cubic
supersymmetry invariance. Finally we establish results regarding the extension
in arbitrary dimensions of our model.
