A Magnus- and Fer-type formula in dendriform algebras
Combinatorics
2009-04-11 v3 Mathematical Physics
Classical Analysis and ODEs
math.MP
Rings and 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.
Cite
@article{arxiv.0707.0607,
title = {A Magnus- and Fer-type formula in dendriform algebras},
author = {Kurusch Ebrahimi-Fard and Dominique Manchon},
journal= {arXiv preprint arXiv:0707.0607},
year = {2009}
}
Comments
13 pages, LaTeX. Terminology issues fixed. Page setup problems fixed on pdf file. Final version as to appear in Foundations of Computational Mathematics