Symmetries of the Discrete Nonlinear Schrödinger Equation

The Lie algebra $L(h)$ of point symmetries of a discrete analogue of the nonlinear Schrödinger equation (NLS) is described. In the continuous limit, the discrete equation is transformed into the NLS, while the structure of the Lie algebra changes: a contraction occurs with the lattice spacing $h$ as the contraction parameter. A five-dimensional subspace of $L(h)$, generated by both point and generalized symmetries, transforms into the five-dimensional point symmetry algebra of the NLS.