Rota-Baxter operators, differential operators, pre- and Novikov structures on groups and Lie algebras

TL;DR

This paper introduces a unified approach to Rota-Baxter and differential operators on groups and Lie algebras, revealing their connections to pre-structures, Novikov groups, and solutions to the Yang-Baxter equation.

Contribution

It develops a perturbative framework for relative Rota-Baxter and differential operators with limit-weight, leading to new structures like pre-groups and Novikov groups.

Findings

Operators induce pre-Lie and Novikov algebra structures on tangent spaces.

Limit-weight Rota-Baxter operators on groups produce skew left braces.

The framework links operators to solutions of the Yang-Baxter equation.

Abstract

Rota-Baxter operators on various structures have found important applications in diverse areas, from renormalization of quantum field theory to Yang-Baxter equations. Relative Rota-Baxter operators on Lie algebras are closely related to pre-Lie algebras and post-Lie algebras. Some of their group counterparts have been introduced to study post-groups, skew left braces and set-theoretic solutions of Yang-Baxter equations, but searching suitable notions of relative Rota-Baxter operators on groups with weight zero and pre-groups has been challenging and has been the focus of recent studies, by provisionally imposing an abelian condition. Arising from the works of Balinsky-Novikov and Gelfand-Dorfman, Novikov algebras and their constructions from differential commutative algebras have led to broad applications. Finding their suitable counterparts for groups and Lie algebras has also attracted quite much recent attention. This paper uses one-sided-inverse pairs of maps to give a perturbative approach to a general notion of relative Rota-Baxter operators and differential operators on a group and a Lie algebra with limit-weight. With the extra condition of limit-abelianess on the group or Lie algebra, we give an interpretation of relative Rota-Baxter and differential operators with weight zero. These operators motivate us to define pre-groups and Novikov groups respectively as the induced structures. The tangent maps of these operators are shown to give Rota-Baxter and differential operators with weight zero on Lie algebras. The tangent spaces of the pre-Lie and Novikov Lie groups are pre-Lie algebras and Novikov Lie algebras, fulfilling the expected property. Furthermore, limit-weight relative Rota-Baxter operators on groups give rise to skew left braces and then set-theoretic solutions of the Yang-Baxter equation.