  In the paper we continue to consider symmetries related to the Ablowitz-Ladik
hierarchy. We derive symmetries for the integrable discrete nonlinear
Schr\"odinger hierarchy and discrete AKNS hierarchy. The integrable discrete
nonlinear Schr\"odinger hierarchy are 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 operartor. These discrete
AKNS flows form a Lie algebra which plays a key role in constructing symmetries
and their algebraic structures for both the integrable discrete nonlinear
Schr\"odinger 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 \textit{degree} in terms of lattice
spacing parameter $h$.
