Injective Rota-Baxter operators of weight zero on $F[x]$

TL;DR

This paper proves a conjecture characterizing injective Rota-Baxter operators of weight zero on polynomial algebras over fields of characteristic zero, revealing their structure and symmetries.

Contribution

It confirms Zheng, Guo, and Rosenkranz's conjecture and describes the moduli space structure and symmetries of these operators.

Findings

Confirmed the conjecture over any field of characteristic zero.

Established an ind-variety structure on the moduli space.

Described an infinitely transitive action on codimension one subsets.

Abstract

Rota-Baxter operators present a natural generalisation of integration by parts formula for the integral operator. In 2015, Zheng, Guo, and Rosenkranz conjectured that every injective Rota-Baxter operator of weight zero on the polynomial algebra $\mathbb{R}[x]$ is a composition of the multiplication by a nonzero polynomial and a formal integration at some point. We confirm this conjecture over any field of characteristic zero. Moreover, we establish a structure of an ind-variety on the moduli space of these operators and describe an additive structure of generic modality two on it. Finally, we provide an infinitely transitive action on codimension one subsets.