Rota-Baxter systems and skew trusses

TL;DR

This paper introduces Rota-Baxter systems as a new generalization of Rota-Baxter groups and Lie algebras, exploring their structure, decompositions, and solutions to twisted Yang-Baxter equations, extending the theory of skew trusses.

Contribution

It defines Rota-Baxter systems, studies their properties, decompositions, and connections to Lie algebras and groups, and introduces twisted modified Yang-Baxter equations with solutions.

Findings

Rota-Baxter systems can be decomposed into a direct sum of two semigroups.

A factorization theorem for Rota-Baxter systems generalizes that of Rota-Baxter groups.

Solutions to twisted modified Yang-Baxter equations are provided via Rota-Baxter systems of Lie algebras.

Abstract

As a generalization of skew braces, the notion of skew trusses was introduced by T. Brzezinski. It was shown that every Rota-Baxter group has the structure of skew braces by V. G. Bardakov and V. Gubarev. To investigate an analogue of Rota-Baxter groups which has the structure of skew trusses, we define RotaBaxter systems. We study the relationship between Rota-Baxter systems and Rota-Baxter groups. Furthermore, we prove that a Rota-Baxter system can be decomposed as a direct sum of two semigroups. A factorization theorem is proved, generalizing the factorization theorem of Rota-Baxter groups. The notion of Rota-Baxter systems of Lie algebras was introduced, as a generalization of Rota-Baxter Lie algebras. The connection between Rota-Baxter systems of Lie algebras and Lie groups is studied. Finally, as a generalization of the modified Yang-Baxter equation, we define twisted modified Yang-Baxter equations. We give solutions of twisted modified Yang-Baxter equations by Rota-Baxter systems of Lie algebras.