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In the paper we first investigate symmetries of isospectral andnon-isospectral four-potential Ablowitz-Ladik hierarchies. We express thesehierarchies in the form of $u_{n,t}=L^m H^{(0)}$, where $m$ is an arbitraryinteger (instead of a nature number) and $L$ is the recursion operator. Then bymeans of the zero-curvature representations of the isospectral andnon-isospectral flows, we construct symmetries for the isospectral equationhierarchy as well as non-isospectral equation hierarchy, respectively. Thesymmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. Therecursion operator $L$ is proved to be hereditary and a strong symmetry forthis isospectral equation hierarchy. Besides, we make clear for the relationbetween four-potential and two-potential Ablowitz-Ladik hierarchies. The evenorder members in the four-potential Ablowitz-Ladik hierarchies together withtheir symmetries and algebraic structures can be reduced to two-potential case.The reduction keeps invariant for the algebraic structures and the recursionoperator for two potential case becomes $L^2$.

Symmetries for the Ablowitz-Ladik hierarchy: I. Four-potential case