Noncommutative Symmetric Functions and the Inversion Problem

TL;DR

This paper explores the algebraic structure of noncommutative symmetric functions, deriving identities and formulas related to automorphisms, their inverses, and the Jacobian conjecture, using differential operators and Hopf algebra homomorphisms.

Contribution

It introduces new identities between differential operators in the noncommutative symmetric function system and applies them to automorphism expansions and the Jacobian conjecture.

Findings

Derived identities between differential operators in the NCS system.

Formulas for Taylor series expansions of automorphisms and their inverses.

Established a connection between the Jacobian conjecture and NCSF's.

Abstract

Let $K$ be any unital commutative $\bQ$-algebra and $z=(z_1, z_2, ..., z_n)$ commutative or noncommutative variables. Let $t$ be a formal central parameter and $\kttzz$ the formal power series algebra of $z$ over $K[[t]]$. In \cite{GTS-II}, for each automorphism $F_t(z)=z-H_t(z)$ of $\kttzz$ with $H_{t=0}(z)=0$ and $o(H(z))\geq 1$, a \cNcs (noncommutative symmetric) system (\cite{GTS-I}) $\Oft$ has been constructed. Consequently, we get a Hopf algebra homomorphism $\cSft: \cNsf \to \cDzz$ from the Hopf algebra $\cNsf$ (\cite{G-T}) of NCSF's (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the \cNcs system $\Oft$ by using some identities of NCSF's derived in \cite{G-T} and the homomorphism $\cSft$. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system $\Oft$ for the Taylor series expansions of $u(F_t)$ and $u(F_t^{-1})$ $(u(z)\in \kttzz)$; the D-Log and the formal flow of $F_t$ and inversion formulas for the inverse map of $F_t$. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSF's.