On the solutions of universal differential equation by noncommutative Picard-Vessiot theory
Mathematical Physics
2025-12-23 v12 math.MP
Abstract
Basing on Picard-Vessiot theory of noncommutative differential equations and algebraic combinatorics on noncommutative formal series with holomorphic coefficients, various recursive constructions of sequences of grouplike series converging to solutions of universal differential equation are proposed. Basing on monoidal factorizations, these constructions intensively use diagonal series and various pairs of bases in duality, in concatenation-shuffle bialgebra and in a Loday's generalized bialgebra. As applications, the unique solution, satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is provided by d\'evissage.
Keywords
Cite
@article{arxiv.2209.08596,
title = {On the solutions of universal differential equation by noncommutative Picard-Vessiot theory},
author = {V. C. Bui and V. Hoang Ngoc Minh and V. Nguyen Dinh and Q. H. Ngo},
journal= {arXiv preprint arXiv:2209.08596},
year = {2025}
}