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In the paper, we first investigate symmetries of isospectral and non-isospectral four-potential Ablowitz-Ladik hierarchies. We express these hierarchies in the form of u n,t = L m H (0) , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero-curvature representations of the isospectral and non-isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four-potential and two-potential Ablowitz-Ladik hierarchies. The even order members in the four-potential Ablowitz-Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two-potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L 2 .

In the paper, we continue to consider symmetries related to the Ablowitz-Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non-isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac-Moody-Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.