{"title":"零星Mathieu群块的1次Hochschild上同调的李代数结构","authors":"William Murphy","doi":"10.1515/jgth-2021-0176","DOIUrl":null,"url":null,"abstract":"Abstract Let 𝐺 be one of the sporadic simple Mathieu groups M 11 M_{11} , M 12 M_{12} , M 22 M_{22} , M 23 M_{23} or M 24 M_{24} , and suppose 𝑘 is an algebraically closed field of prime characteristic 𝑝, dividing the order of 𝐺. In this paper, we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the 𝑝-blocks of k G kG . In particular, we calculate the dimension of HH 1 ( B ) \\mathrm{HH}^{1}(B) for the 𝑝-blocks 𝐵 of k G kG , and in almost all cases, we determine whether HH 1 ( B ) \\mathrm{HH}^{1}(B) is a solvable Lie algebra.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"28 1","pages":"161 - 191"},"PeriodicalIF":0.4000,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups\",\"authors\":\"William Murphy\",\"doi\":\"10.1515/jgth-2021-0176\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let 𝐺 be one of the sporadic simple Mathieu groups M 11 M_{11} , M 12 M_{12} , M 22 M_{22} , M 23 M_{23} or M 24 M_{24} , and suppose 𝑘 is an algebraically closed field of prime characteristic 𝑝, dividing the order of 𝐺. In this paper, we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the 𝑝-blocks of k G kG . In particular, we calculate the dimension of HH 1 ( B ) \\\\mathrm{HH}^{1}(B) for the 𝑝-blocks 𝐵 of k G kG , and in almost all cases, we determine whether HH 1 ( B ) \\\\mathrm{HH}^{1}(B) is a solvable Lie algebra.\",\"PeriodicalId\":50188,\"journal\":{\"name\":\"Journal of Group Theory\",\"volume\":\"28 1\",\"pages\":\"161 - 191\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Group Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2021-0176\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Group Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2021-0176","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
摘要
摘要设𝐺为散在的简单Mathieu群M 11 M_{11}, M 12 M_{12}, M 22 M_{22}, M 23 M_{23}或M 24 M_{24}中的一个,设𝑘为素数特征的代数闭域𝑝,分𝐺的阶。本文描述了k ^ gkg的𝑝-blocks的第一Hochschild上同调群的一些李代数结构。特别是,我们计算的维数HH 1(B) \ mathrm {HH} ^ {1} (B)𝑝-blocks𝐵kG公斤,在几乎所有的情况下,我们决定HH 1(B) \ mathrm {HH} ^ {1} (B)是一个可解李代数。
The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups
Abstract Let 𝐺 be one of the sporadic simple Mathieu groups M 11 M_{11} , M 12 M_{12} , M 22 M_{22} , M 23 M_{23} or M 24 M_{24} , and suppose 𝑘 is an algebraically closed field of prime characteristic 𝑝, dividing the order of 𝐺. In this paper, we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the 𝑝-blocks of k G kG . In particular, we calculate the dimension of HH 1 ( B ) \mathrm{HH}^{1}(B) for the 𝑝-blocks 𝐵 of k G kG , and in almost all cases, we determine whether HH 1 ( B ) \mathrm{HH}^{1}(B) is a solvable Lie algebra.
期刊介绍:
The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.
Topics:
Group Theory-
Representation Theory of Groups-
Computational Aspects of Group Theory-
Combinatorics and Graph Theory-
Algebra and Number Theory