{"title":"The cohomology and deformations of O-operators on BiHom-associative algebras","authors":"Danli Huang, Ling Liu, Jiafeng Lü","doi":"10.1016/j.jalgebra.2024.07.056","DOIUrl":null,"url":null,"abstract":"<div><p>We first generalize the cohomology of <span><math><mi>O</mi></math></span>-operators on BiHom-associative algebras by construct a graded Lie-algebra, in which the Maurer-Cartan elements are characterized by the given <span><math><mi>O</mi></math></span>-operator, and show that the cohomology represents the Hochschild cohomology of a certain BiHom-associative algebra with coefficients in a bimodule. Next, we study the linear and formal deformations of <span><math><mi>O</mi></math></span>-operators on BiHom-associative algebras, which are controlled by the Hochschild cohomology. Finally, as applications, we introduce the deformations of BiHom-associative <strong>r</strong>-matrices and infinitesimal BiHom-bialgebras on certain regular BiHom-associative algebras.</p></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324004642","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We first generalize the cohomology of -operators on BiHom-associative algebras by construct a graded Lie-algebra, in which the Maurer-Cartan elements are characterized by the given -operator, and show that the cohomology represents the Hochschild cohomology of a certain BiHom-associative algebra with coefficients in a bimodule. Next, we study the linear and formal deformations of -operators on BiHom-associative algebras, which are controlled by the Hochschild cohomology. Finally, as applications, we introduce the deformations of BiHom-associative r-matrices and infinitesimal BiHom-bialgebras on certain regular BiHom-associative algebras.
首先，我们通过构建一个分级李代数，其中的毛勒-卡尔坦元素由给定的 O 运算符表征，从而概括出 BiHom-associative 代数上 O 运算符的同调，并证明该同调代表了具有双模子系数的某个 BiHom-associative 代数的霍赫希尔德同调。接下来，我们研究了 O 操作数在 BiHom-associative 代数上的线性变形和形式变形，这些变形都受霍赫希尔德同调的控制。最后，作为应用，我们介绍了 BiHom-associative r 矩和无穷小 BiHom 双桥在某些正则 BiHom-associative 对象上的变形。
期刊介绍：
The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.
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