We review the discrete evolution problem and the corresponding solution as a
    discrete Dyson series in order to rigorously derive a discrete version of
    Magnus expansion. We also systematically derive the discrete analogue of the
    pre-Lie Magnus expansion and show that the elements of the discrete Dyson
    series are expressed in terms of a tridendriform algebra action. 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.

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