A Magnus- and Fer-type formula in dendriform algebras

Abstract

We provide a refined approach to the classical Magnus and Fer expansion, unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X=1+ta<X and Y=1-tY> a in A[[t]] is provided, where (A,<,>) is a dendriform algebra. Then, we present the solutions to these equations as an infinite product expansion of exponentials. Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.

13 pages, LaTeX. Terminology issues fixed. Page setup problems fixed on pdf file. Final version as to appear in Foundations of Computational Mathematics

References in corpus (4)

• Loday-type Algebras and the Rota-Baxter Relation
• New identities in dendriform algebras
• Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
• Hopf algebras in dynamical systems theory

Cited by in corpus (17)

• The Magnus expansion and some of its applications
• Two interacting Hopf algebras of trees
• Dendriform Equations
• Cumulants, free cumulants and half-shuffles
• Post-Lie algebras and factorization theorems
• Twisted dendriform algebras and the pre-Lie Magnus expansion
• The Magnus expansion and Post-Lie algebras
• Time-ordering and a generalized Magnus expansion
• On expansions for nonlinear systems, error estimates and convergence issues
• The pre-Lie structure of the time-ordered exponential
• Post-symmetric braces and integration of post-Lie algebras
• From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras
• The combinatorics of Green's functions in planar field theories
• Cumulant-cumulant relations in free probability theory from Magnus' expansion
• The tridendriform structure of a Magnus expansion
• Dendriform-Tree Setting for Fully Non-commutative Fliess Operators
• Continuous changes of variables and the Magnus expansion