  We review the discrete evolution problem and the corresponding solution as a
discrete Dyson series in order to rigorously derive a generalized discrete
version of the Magnus expansion. We also systematically derive the discrete
analogue of the pre-Lie Magnus expansion and express the elements of the
discrete Dyson series in terms of a tridendriform algebra action. In the
generic discrete case extra significant terms that are absent in the continuous
or the linear discrete case appear in both Dyson and Magnus expansions. Based
on the rigorous discrete derivation key links between quantum algebras,
tridendriform and pre-Lie algebras are then established. This is achieved by
examining tensor realizations of quantum groups, such as the Yangian. We show
that these realizations can be expressed in terms of tridendriform and pre-Lie
algebras actions. The continuous limit as expected provides the corresponding
non-local charges of the Yangian as members of the pre-Lie Magnus expansion.
