{"title":"The Radon–Penrose Transformation for Quaternionic k-Regular Functions on Right-Type Groups","authors":"Qianqian Kang, Guangzhen Ren, Yun Shi","doi":"10.1007/s00006-024-01360-9","DOIUrl":null,"url":null,"abstract":"<div><p>The right-type groups are nilpotent Lie groups of step two having a pair of anticommutative operators, and many aspects of quaternionic analysis can be generalized to this kind of groups. In this paper, we use the twistor transformation to study the tangential <i>k</i>-Cauchy–Fueter equations and quaternionic <i>k</i>-regular functions on these groups. We introduce the twistor space over the <span>\\((4n+r)\\)</span>-dimensional complex right-type groups and use twistor transformation to construct an explicit Radon–Penrose type integral formula to solve the holomorphic tangential <i>k</i>-Cauchy–Fueter equation on these groups. When restricted to the real right-type group, this formula provides solutions to tangential <i>k</i>-Cauchy–Fueter equations. In particular, it gives us many <i>k</i>-regular polynomials.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Clifford Algebras","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00006-024-01360-9","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The right-type groups are nilpotent Lie groups of step two having a pair of anticommutative operators, and many aspects of quaternionic analysis can be generalized to this kind of groups. In this paper, we use the twistor transformation to study the tangential k-Cauchy–Fueter equations and quaternionic k-regular functions on these groups. We introduce the twistor space over the \((4n+r)\)-dimensional complex right-type groups and use twistor transformation to construct an explicit Radon–Penrose type integral formula to solve the holomorphic tangential k-Cauchy–Fueter equation on these groups. When restricted to the real right-type group, this formula provides solutions to tangential k-Cauchy–Fueter equations. In particular, it gives us many k-regular polynomials.
期刊介绍：
Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.
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