On the solutions of universal differential equation by noncommutative Picard-Vessiot theory
V. C. Bui, V. Hoang Ngoc Minh, V. Nguyen Dinh, Q. H. Ngo
TL;DR
This work develops a noncommutative Picard-Vessiot framework for solving the universal differential equation $dS=M_n S$ with $M_n=\sum_{1\le i<j\le n} \omega_{i,j} t_{i,j}$, leveraging Chen-series, diagonal factorizations, and Lazard–Schützenberger factorizations. By establishing flatness and complete integrability of the connection, it constructs convergent Picard iterations that yield grouplike solutions, and uses dévissage to relate $KZ_n$ to $KZ_{n-1}$ through noncommutative generating series of polylogarithms and hyperlogarithms. The paper provides explicit factorized solutions and dual-base representations (via Lyndon words) and applies them to Knizhnik-Zamolodchikov equations, recovering Drinfeld associators in the abelianized setting and detailing concrete $KZ_3$ and $KZ_4$ cases. The approach offers a structured, computable path to solving high-dimensional NCDEs with rich algebraic structure, with implications for representation theory, braid groups, and multiple zeta-value phenomena.
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 dévissage.