Cluster algebras through representation theory

通过表示论的簇代数

• 批准号: RGPIN-2018-04513
• 负责人: Paquette, Charles
• 金额: $ 1.82万
• 依托单位: Royal Military College of Canada
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758740
• 关键词: Cluster; algebras; through; representation; theory

Abstract algebra is the study of algebraic structures. For instance, the set of integers together with addition and multiplication is an algebraic structure that we are all familiar with. It is formally called a ring, because the two given operations, addition and multiplication, satisfy some very natural properties. A ring need not be constructed from integers, and the operations involved need not be similar to addition and multiplication as we know them. Abstract algebra often provides an algebraic setting in which one can study a problem arising in nature, or in another field of mathematics.My research lies in representation theory of algebras, which is a branch of abstract algebra. We are interested in studying some algebraic structures, similar to rings, that are called algebras. Representation theory refers to the idea of representing a complex algebraic object by one that is easier to understand. In representation theory of algebras, we are interested in studying some objects called modules, by using elementary methods from basic linear algebra.Representation theory of algebras arises in many branches of mathematics, and even of physics. For instance, string theory, which is a branch of theoretical physics, has recently been studied by using a class of algebras called cluster algebras. The latter are deeply connected to representation theory of algebras. Indeed, in the recent years, representation theory of algebras has developed powerful tools to better understand these cluster algebras. Being a branch of abstract algebra, representation theory is in close connection to category theory, algebraic geometry and homological algebra.My proposal consists of developing representation theory of algebras, through its interactions with category theory, algebraic geometry and homological algebra. As in most areas of pure mathematics, it is very hard to predict the immediate impacts of this research. Many ideas in mathematics that are crucial today were developed in an abstract setting hundreds years ago. As representation theory is becoming more and more useful as a tool in other research areas, it is important to develop it, in concert with the development of these other fields. As a secondary objective of this proposal, I would like to use the new theoretical methods developed to better understand the cluster algebras. This has the potential to bring new methods for studying problems in physics, including problems in string theory.

抽象代数是对代数结构的研究。例如，整数集合以及加法和乘法是我们都熟悉的代数结构。它的正式名称是环，因为两个给定的运算（加法和乘法）满足一些非常自然的性质。环不需要由整数构成，并且所涉及的运算不需要类似于我们所知的加法和乘法。抽象代数常常提供一种代数环境，人们可以在其中研究自然界或其他数学领域中出现的问题。我的研究方向是代数表示论，它是抽象代数的一个分支。我们有兴趣研究一些代数结构，类似于环，称为代数。表示论是指用更容易理解的代数对象来表示复杂代数对象的思想。在代数表示论中，我们感兴趣的是通过使用基本线性代数的初等方法来研究一些称为模的对象。代数表示论出现在数学甚至物理学的许多分支中。例如，弦理论是理论物理学的一个分支，最近通过使用一类称为簇代数的代数进行了研究。后者与代数表示论密切相关。事实上，近年来，代数表示论已经开发出了强大的工具来更好地理解这些簇代数。作为抽象代数的一个分支，表示论与范畴论、代数几何和同调代数密切相关。我的建议包括通过其与范畴论、代数几何和同调代数的相互作用来发展代数表示论。与纯数学的大多数领域一样，很难预测这项研究的直接影响。当今至关重要的许多数学思想都是在数百年前的抽象背景下发展起来的。随着表示论作为一种工具在其他研究领域变得越来越有用，因此将其与其他领域的发展相结合就显得非常重要。作为该提案的次要目标，我想使用新开发的理论方法来更好地理解簇代数。这有可能为研究物理学问题（包括弦理论问题）带来新的方法。