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Non-Relativistic Four Dimensional p-Brane Supersymmetric Theories and Lie Algebra Expansion

We apply the Lie algebra expansion method to the $\mathcal{N}=1$ super-Poincaré algerba in four dimensions. We define a set of p-brane projectors that induce a decomposition of the super-Poincaré algebra preparatory for the expansion. We show that starting from the $\mathcal{N}=1$ supergravity action in four dimensions it is possible to obtain two non-relativistic supersymmetric theories, one describing strings, the other membranes.