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Representation theory of modular Lie algebras and superalgebras

模李代数和超代数的表示论

基本信息

批准号：
EP/R018952/1
负责人：
Simon Goodwin
金额：
$ 42.46万
依托单位：
University of Birmingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR018952%2F1
关键词：
Representation theory modular Lie algebras

项目摘要

Representation theory of Lie groups and Lie algebras has been a topic at the heart of mathematics for over 100 years with wide-ranging applications in mathematics and physics. This subject has origins in the view of Felix Klein in the 19th century that geometry of spacetime should be governed by its group of symmetries and the subsequent pioneering work of Sophus Lie to develop a theory of symmetries for differential equations.Lie groups can be viewed as continuous symmetries of geometric objects. For example, a circle has infinitely many symmetries, namely rotations and reflections, which we can vary in a continuous way. Taking a step back we are able to view a Lie group more abstractly, and then representation theory provides the language to understand the different ways that a Lie group can act as symmetries. The Lie algebra of a Lie group is a first order approximation of a Lie group, which is more accessible to study, but retains all the local structure of the group. The abundance of continuous symmetry in mathematics and physics explains the wide ranging applications of this theory.In the 1950s the "analytic theory" of Lie groups and Lie algebras was extended so that it can approached more algebraically, and this spurned a large area of mathematics now known as algebraic Lie theory. This is one of the most active areas of mathematics research today, which finds diverse applications across the physical sciences. An important area of algebraic Lie theory is the representation theory of modular Lie algebras. These Lie algebras can be thought of as versions of real or complex Lie algebras where usual arithmetic using real or complex numbers is replaced by modular arithmetic as is used in coding theory and cryptography.The aim of this project is to exploit exciting recent developments in algebraic Lie theory to give a new perspective of the representation theory of modular Lie algebras. In order to understand representations of Lie algebras, we want to associate numerical data, which governs the structure of the representations. The most important pieces of data are the dimension and characters, and the ambitious goal of this project is to develop a methods for determining formulae for these.

一百多年来，李群和李代数的表示论一直是数学的核心主题，在数学和物理领域有着广泛的应用。该学科起源于 19 世纪菲利克斯·克莱因 (Felix Klein) 的观点，即时空几何应受其对称群支配，以及随后索菲斯·李 (Sophus Lie) 发展微分方程对称理论的开创性工作。李群可以被视为几何对象的连续对称性。例如，圆有无限多种对称性，即旋转和反射，我们可以连续地改变它们。退一步，我们可以更抽象地看待李群，然后表示论提供了语言来理解李群作为对称性的不同方式。李群的李代数是李群的一阶近似，更容易研究，但保留了群的所有局部结构。数学和物理学中丰富的连续对称性解释了该理论的广泛应用。在 20 世纪 50 年代，李群和李代数的“解析理论”得到了扩展，使其可以更加代数地逼近，这抛弃了现在称为代数李理论的一大部分数学领域。这是当今数学研究最活跃的领域之一，在物理科学中有着多种应用。代数李理论的一个重要领域是模李代数的表示论。这些李代数可以被认为是实数或复数李代数的版本，其中使用实数或复数的常用算术被编码理论和密码学中使用的模算术所取代。该项目的目的是利用代数李理论中令人兴奋的最新发展，为模李代数的表示论提供新的视角。为了理解李代数的表示，我们想要关联控制表示结构的数值数据。最重要的数据是维度和特征，该项目的宏伟目标是开发一种确定这些数据公式的方法。