2024/05/18 by Xing Gao, Li Guo, Gao, Xing +3
Mathematics · #16W99 #17B38 #17B40 #22E60 #45N05 #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2405.11288
openalex publication_date 2024/05/18 · openalex created_date 2024/05/22 · openalex updated_date 2026/07/28
Rota-Baxter groups with weights ± 1 have attracted quite much attention since their recent introduction, thanks to their connections with Rota-Baxter Lie algebras, factorizations of Lie groups, post- and pre-Lie algebras, braces and set-theoretic solutions of the Yang-Baxter equation. Despite their expected importance from integrals on groups to pre-groups and Yang-Baxter equations, Rota-Baxter groups with weight zero and other weights has been a challenge to define and their search has been the focus of several attempts. By composing an operator with a section map as a perturbation device, we first generalize the notion of a Rota-Baxter operator on a group from the existing case of weight ± 1 to the case where the weight is given by a pair of maps and then a sequence limit of such pairs. From there, two candidates of Rota-Baxter operators with weight zero are given. One of them is the Rota-Baxter operator with limit-weight zero detailed here, with the other candidate introduced in a companion work. This operator is shown to have its tangent map the Rota-Baxter operator with weight zero on Lie algebras. It also gives concrete applications in integrals of maps with values in a class of topological groups called \RR-groups, satisfying a multiplicative version of the integration-by-parts formula. In parallel, differential groups in this framework is also developed and a group formulation of the First Fundamental Theorem of Calculus is obtained.