Cohomology and extensions of relative Rota–Baxter groups

IF 1.6 3区数学 Q1 MATHEMATICS Journal of Geometry and Physics Pub Date : 2024-10-30 DOI:10.1016/j.geomphys.2024.105353

Pragya Belwal, Nishant Rathee, Mahender Singh

引用次数: 0

Abstract

Relative Rota–Baxter groups are generalisations of Rota–Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang–Baxter equation. In this paper, we develop an extension theory of relative Rota–Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota–Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota–Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota–Baxter groups, the two cohomologies are isomorphic in dimension two.

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相对罗塔-巴克斯特群的同调与扩展

相对罗塔-巴克斯特群是罗塔-巴克斯特群的广义，最近被证明与斜左括号密切相关，众所周知，斜左括号能产生杨-巴克斯特方程的双射非退化解。在本文中，我们发展了相对 Rota-Baxter 群的扩展理论，并引入了它们的低维同调群，这些同调群不同于在 Lie 群上的 Rota-Baxter 算子中已知的同调群。我们在相对罗塔-巴克斯特群的扩展等价类集合和它们的第二同调之间建立了明确的双射关系。此外，我们还深入研究了该同调与相关斜左括号同调之间的联系。我们证明，对于双射相对罗塔-巴克斯特群，这两个同调在维数二中是同构的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity

期刊最新文献

Editorial Board On conformal collineation and almost Ricci solitons Cohomology and extensions of relative Rota–Baxter groups Direct linearization of the SU(2) anti-self-dual Yang-Mills equation in various spaces Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one