Lie algebra contractions and symmetries of the Toda hierarchy

The Lie algebra L(Delta) of generalized and point symmetries of the equations in the Toda hierarchy is shown to be a semidirect sum of two infinite-dimensional Lie algebras, one perfect, the other Abelian. In the continuous limit the structure of the Lie algebra changes: a contraction occurs with the lattice spacing as the contraction parameter. In particular, for the Toda equation itself, a set of five elements, involving both point symmetries and generalized ones, contracts to the point symmetry algebra of the potential KdV equation.