Quantum Groups, Discrete Magnus Expansion, Pre-Lie and Tridendriform Algebras

Anastasia Doikou<sup>ab

a</sup> Maxwell Institute for Mathematical Sciences, Edinburgh EH8 9BT, UK
^b Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK

Abstract: 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 binary operation. 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. The continuous limit as expected provides the corresponding non-local charges of the Yangian as members of the pre-Lie Magnus expansion.

Keywords: Yangians, integrability, discrete evolution problem, Magnus expansion, pre-Lie algebras, (tri)dendriform algebras.

MSC: 16T05, 17D99, 81R50

Received: July 9, 2024; in final form December 1, 2025; Published online December 11, 2025

Language: English

DOI: 10.3842/SIGMA.2025.105

ArXiv: 2211.00451