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

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 $L^2$.