Rota-Baxter groups with weight zero and integration on topological groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917670256902144 |
|---|---|
| author | Gao, Xing Guo, Li Han, Zongjian |
| author_facet | Gao, Xing Guo, Li Han, Zongjian |
| contents | Rota-Baxter groups with weights $\pm 1$ 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 $\pm 1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11288 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rota-Baxter groups with weight zero and integration on topological groups Gao, Xing Guo, Li Han, Zongjian Quantum Algebra Group Theory 17B38, 22E60, 17B40, 16W99, 45N05 Rota-Baxter groups with weights $\pm 1$ 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 $\pm 1$ 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. |
| title | Rota-Baxter groups with weight zero and integration on topological groups |
| topic | Quantum Algebra Group Theory 17B38, 22E60, 17B40, 16W99, 45N05 |
| url | https://arxiv.org/abs/2405.11288 |