{"title":"Novel non-involutive solutions of the Yang–Baxter equation from (skew) braces","authors":"Anastasia Doikou, Bernard Rybołowicz","doi":"10.1112/jlms.12999","DOIUrl":null,"url":null,"abstract":"<p>We produce novel non-involutive solutions of the Yang–Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces, they are not necessarily involutive. In the case of two-sided (skew) braces, one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non-involutive frame, and we identify the twisted <span></span><math>\n <semantics>\n <mi>r</mi>\n <annotation>$r$</annotation>\n </semantics></math>-matrices and twisted coproducts. We observe, as in the involutive case, that the underlying quantum algebra is not a quasi-triangular bialgebra, as one would expect, but a quasi-triangular quasi-bialgebra. The same applies to the quantum algebra of the twisted <span></span><math>\n <semantics>\n <mi>r</mi>\n <annotation>$r$</annotation>\n </semantics></math>-matrices, contrary to the involutive case.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12999","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12999","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We produce novel non-involutive solutions of the Yang–Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces, they are not necessarily involutive. In the case of two-sided (skew) braces, one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non-involutive frame, and we identify the twisted -matrices and twisted coproducts. We observe, as in the involutive case, that the underlying quantum algebra is not a quasi-triangular bialgebra, as one would expect, but a quasi-triangular quasi-bialgebra. The same applies to the quantum algebra of the twisted -matrices, contrary to the involutive case.
我们从(斜)括号中得到了杨-巴克斯特方程的新的非渐开线解。这些解是对来自括号和斜括号的已知解的概括,令人惊讶的是,在括号的情况下,这些解并不一定是非卷积解。在双面(倾斜)括号的情况下,我们可以为集合的每个元素分配这样的解。我们还引入了与逆解相关的新的双射映射。此外,我们还证明了最近得出的渐开线情况下的德林菲尔德扭转在非渐开线框架中仍然是可接受的,并且我们确定了扭转的 r $r$ -矩阵和扭转的协积。我们观察到,与非卷积情况一样,底层量子代数并不像我们所期望的那样是一个准三角形双代数,而是一个准三角形准双代数。这同样适用于扭曲 r $r$ -矩阵的量子代数,与非累加情况相反。
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.