Computing Galois cohomology of a real linear algebraic group

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-01 DOI:10.1112/jlms.12906

Mikhail Borovoi, Willem A. de Graaf

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引用次数: 0

Abstract

Let $G$ be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers $R$ . We describe a method, implemented on computer, to find the first Galois cohomology set $H^{1} (R, G)$ . The output is a list of 1-cocycles in $G$ . Moreover, we describe an implemented algorithm that, given a 1-cocycle $z \in Z^{1} (R, G)$ , finds the cocycle in the computed list to which $z$ is equivalent, together with an element of $G (C)$ realizing the equivalence.

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计算实线性代数群的伽罗瓦同调

设 G ${\bf G}$ 是实数域 R ${\mathbb {R}}$ 上的线性代数群，不一定是连通的或还原的。我们描述了一种在计算机上实现的寻找第一个伽罗瓦同调集 H 1 ( R , G ) ${rm H}^1({\mathbb {R}},{\bf G})$ 的方法。输出结果是 G ${\bf G}$ 中的 1 循环列表。此外，我们还描述了一种实现算法，当给定{\rm Z}^1({\mathbb {R}},{\bf G})$中的一个单循环 z ∈ Z 1 ( R , G ) $z\ 时，在计算出的列表中找到与 z $z$ 等价的单循环，以及 G ( C ) ${\bf G}({\mathbb {C}})$中实现等价的元素。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.