{"title":"Quasi-twilled associative algebras, deformation maps and their governing algebras","authors":"Apurba Das, Ramkrishna Mandal","doi":"arxiv-2409.00443","DOIUrl":null,"url":null,"abstract":"A quasi-twilled associative algebra is an associative algebra $\\mathbb{A}$\nwhose underlying vector space has a decomposition $\\mathbb{A} = A \\oplus B$\nsuch that $B \\subset \\mathbb{A}$ is a subalgebra. In the first part of this\npaper, we give the Maurer-Cartan characterization and introduce the cohomology\nof a quasi-twilled associative algebra. In a quasi-twilled associative algebra $\\mathbb{A}$, a linear map $D: A\n\\rightarrow B$ is called a strong deformation map if $\\mathrm{Gr}(D) \\subset\n\\mathbb{A}$ is a subalgebra. Such a map generalizes associative algebra\nhomomorphisms, derivations, crossed homomorphisms and the associative analogue\nof modified {\\sf r}-matrices. We introduce the cohomology of a strong\ndeformation map $D$ unifying the cohomologies of all the operators mentioned\nabove. We also define the governing algebra for the pair $(\\mathbb{A}, D)$ to\nstudy simultaneous deformations of both $\\mathbb{A}$ and $D$. On the other hand, a linear map $r: B \\rightarrow A$ is called a weak\ndeformation map if $\\mathrm{Gr} (r) \\subset \\mathbb{A}$ is a subalgebra. Such a\nmap generalizes relative Rota-Baxter operators of any weight, twisted\nRota-Baxter operators, Reynolds operators, left-averaging operators and\nright-averaging operators. Here we define the cohomology and governing algebra\nof a weak deformation map $r$ (that unify the cohomologies of all the operators\nmentioned above) and also for the pair $(\\mathbb{A}, r)$ that govern\nsimultaneous deformations.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00443","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
A quasi-twilled associative algebra is an associative algebra $\mathbb{A}$
whose underlying vector space has a decomposition $\mathbb{A} = A \oplus B$
such that $B \subset \mathbb{A}$ is a subalgebra. In the first part of this
paper, we give the Maurer-Cartan characterization and introduce the cohomology
of a quasi-twilled associative algebra. In a quasi-twilled associative algebra $\mathbb{A}$, a linear map $D: A
\rightarrow B$ is called a strong deformation map if $\mathrm{Gr}(D) \subset
\mathbb{A}$ is a subalgebra. Such a map generalizes associative algebra
homomorphisms, derivations, crossed homomorphisms and the associative analogue
of modified {\sf r}-matrices. We introduce the cohomology of a strong
deformation map $D$ unifying the cohomologies of all the operators mentioned
above. We also define the governing algebra for the pair $(\mathbb{A}, D)$ to
study simultaneous deformations of both $\mathbb{A}$ and $D$. On the other hand, a linear map $r: B \rightarrow A$ is called a weak
deformation map if $\mathrm{Gr} (r) \subset \mathbb{A}$ is a subalgebra. Such a
map generalizes relative Rota-Baxter operators of any weight, twisted
Rota-Baxter operators, Reynolds operators, left-averaging operators and
right-averaging operators. Here we define the cohomology and governing algebra
of a weak deformation map $r$ (that unify the cohomologies of all the operators
mentioned above) and also for the pair $(\mathbb{A}, r)$ that govern
simultaneous deformations.