On the solutions of universal differential equation by noncommutative Picard-Vessiot theory

TL;DR

This paper develops recursive methods using noncommutative Picard-Vessiot theory and algebraic combinatorics to construct solutions for universal differential equations, including the Knizhnik-Zamolodchikov equations.

Contribution

It introduces new recursive constructions of solutions to noncommutative differential equations using monoidal factorizations and algebraic combinatorics.

Findings

Constructed sequences of grouplike series converging to solutions.

Provided the unique asymptotic solution to Knizhnik-Zamolodchikov equations.

Applied noncommutative algebraic methods to differential equations.

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.