Z₂× Z₂-graded Lie Symmetries of the Lévy-Leblond Equations

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The first-order differential Lévy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of (1+d)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schrödinger equations for free particles or in the presence of a harmonic potential), admit a natural Z₂-graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric PDE's, a natural Z₂×Z₂-graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the (1+1)-dimensional Lévy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a Z₂×Z₂-graded Lie superalgebra, realized by first and second-order differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the Z₂- and Z₂×Z₂- graded extensions are no longer allowed. The construction of the Z₂× Z₂-graded Lie symmetry of the (1+2)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra.