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

TL;DR

This paper develops a rigorous discrete analogue of the Magnus expansion and explores its connections to quantum algebras, pre-Lie, and tridendriform structures, revealing new terms and algebraic links in the discrete setting.

Contribution

It introduces a discrete Magnus expansion, derives its pre-Lie analogue, and links these to quantum algebra structures like the Yangian through tridendriform and pre-Lie algebras.

Findings

Discrete Magnus expansion includes new significant terms absent in continuous cases.

Discrete Dyson series can be expressed using tridendriform algebra operations.

Quantum group realizations relate to tridendriform and pre-Lie algebra structures.

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.