Complete reducibility in algebraic groups

代数群的完全可约性

• 批准号: EP/W000466/1
• 负责人: Adam Thomas
• 金额: $ 25.29万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW000466%2F1
• 关键词: Complete; reducibility; algebraic; groups

This research studies `algebraic groups', mathematical objects which live at the intersection of algebra and geometry. They are simultaneously `groups' and `varieties'. A group is the mathematical abstraction of the idea of symmetry and a variety is the mathematical collection of solutions to a set of equations. Let us start with groups. Consider a square and its symmetries, geometric operations we can do to it that leave it looking like the square we started with. We can rotate it by 0, 90, 180 or 270 degrees. We can also reflect it in 4 straight lines (2 joining corners to corners and 2 joining sides to sides). And these 8 symmetries are all of them. The mathematical abstraction is to see that to understand what is going on we want to consider the symmetries all together, not just on their own. Group theory is about studying these sets of symmetries and the key idea is that when we apply one symmetry and then apply another one, we get back one of our 8 original symmetries. This is the notion of composition. It enjoys special properties. For example, rotating by 0 degrees before or after another symmetry leaves that symmetry unchanged. We have all interacted with this idea from an early age: the integers form a group where composition is just addition of numbers. There, the identity is 0. If we add 0 to any number, it stays the same!The study of solutions to equations is a vast area of study and has been a focal point since the inception of mathematics. It is the foundation of modern areas like algebraic geometry and number theory. Fermat's Last Theorem is a very famous example of studying sets of solutions of equations and required incredibly deep techniques in algebraic geometry and number theory. Many of us encounter Pythagoras' Theorem and the equation a^2 + b^2 = c^2 at school. There are many integer solutions to this, known as Pythagorean triples, like (3,4,5) and (5,12,13). As with group theory, mathematicians have found that a good way to study these problems is to look at all solutions at once. A variety is a set of solutions to especially nice equations called polynomials.The study of algebraic groups is therefore highly intradisciplinary. Indeed, their introduction came from generalising Lie groups, which were an analytic invention to study continuous solutions of differential equations. This research will concentrate on the group theoretic side of the story. Algebraic groups have been classified into families and a fundamental open problem is to understand the `simple' ones. These simple groups are the building blocks of all algebraic groups and the word simple obscures the true nature of them. They are incredibly complicated and have a very rich and deep structure. They deserve studying in their own right, let alone due to the applications to other fields of mathematics.We will study the structure of these simple algebraic groups, where many open problems remain. Our main focus is studying the close relationship with another key area of modern mathematics, representation theory. To study an object, like an algebraic group, we consider how it acts on something more straightforward, in this case a linear space V. Linear spaces are nice, the world we live in is a linear space and as mathematicians we understand them. To study how a group acts on such a linear space, we consider it as a subgroup of the full group of symmetries of the linear space, called GL(V). The crucial part for this research is that GL(V) is itself an algebraic group. And so the structure of algebraic groups and representation theory are intimately related. This was made precise by J.-P. Serre when he introduced the concept of complete reducibility. We will tackle open problems about the subgroup structure of algebraic groups, especially certain classes of subgroups related, through this link, to representations that are not completely reducible, meaning they have a more complicated structure built up from the simpler representations.

这项研究研究“代数群”，生活在代数和几何的交叉点的数学对象。它们同时是“群体”和“品种”。群是对称性概念的数学抽象，而簇是一组方程的解的数学集合。让我们从群体开始。考虑一个正方形和它的对称性，我们可以对它做几何运算，使它看起来像我们开始时的正方形。我们可以将它旋转0度、90度、180度或270度。我们也可以用4条直线来反映它（2条连接角到角，2条连接边到边）。这八种对称性都是。数学的抽象是，为了理解发生了什么，我们要考虑所有的对称性，而不仅仅是它们自己。群论是关于研究这些对称性的，关键的思想是，当我们应用一种对称性，然后应用另一种对称性时，我们会得到原来的8种对称性中的一种。这就是组成的概念。它具有特殊的属性。例如，在另一个对称之前或之后旋转0度，该对称将保持不变。我们从小就与这个想法互动：整数形成一个组，其中组合只是数字的加法。在那里，身份是0。如果我们给任何数字加0，它保持不变！方程解的研究是一个广阔的研究领域，自数学诞生以来一直是一个焦点。它是代数几何和数论等现代领域的基础。费马大定理是研究方程组解的一个非常著名的例子，它需要代数几何和数论中令人难以置信的深刻技术。我们中的许多人在学校都会遇到毕达哥拉斯定理和方程a^2 + b^2 = c^2。有许多整数解，称为毕达哥拉斯三元组，如（3，4，5）和（5，12，13）。与群论一样，数学家们发现研究这些问题的一个好方法是同时研究所有的解。一个群是一组被称为多项式的特别好的方程的解。因此，代数群的研究是高度学科内的。事实上，他们的介绍来自推广李群，这是一个分析发明研究连续解微分方程。这项研究将集中在群论方面的故事。代数群已被划分为不同的族，一个基本的开放性问题是理解“简单”的代数群。这些简单的群是所有代数群的基石，简单这个词掩盖了它们的真实性质。它们非常复杂，具有非常丰富和深刻的结构。它们本身就值得研究，更不用说应用于其他数学领域了。我们将研究这些简单代数群的结构，其中仍然存在许多悬而未决的问题。我们的主要重点是研究与现代数学的另一个关键领域，表示论的密切关系。为了研究一个对象，比如一个代数群，我们考虑它如何作用于更直接的东西，在这种情况下是一个线性空间V.线性空间很好，我们生活的世界是一个线性空间，作为数学家，我们理解它们。为了研究一个群如何作用在这样的线性空间上，我们把它看作线性空间的对称全群的子群，称为GL（V）。研究的关键部分是GL（V）本身是一个代数群。因此代数群的结构和表示论是密切相关的。这是由J精确。P. Serre在他提出完全还原的概念时。我们将解决关于代数群的子群结构的公开问题，特别是通过这种联系与不完全可约的表示相关的某些子群类，这意味着它们具有从简单表示建立起来的更复杂的结构。

项目成果

Normalisers of maximal tori and a conjecture of Vdovin

最大环面的归一化器和 Vdovin 猜想

• DOI: 10.1016/j.jalgebra.2022.12.013
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Burness T

The classical topological invariants of homogeneous spaces

齐次空间的经典拓扑不变量

• DOI: 10.48550/arxiv.2310.14365
• 发表时间: 2023
• 作者: Jones J