Algebras with action and coaction of Hopf algebras

具有 Hopf 代数作用和相互作用的代数

• 批准号: RGPIN-2018-04883
• 负责人: Kotchetov, Mikhail
• 金额: $ 1.46万
• 依托单位: Memorial University of Newfoundland
• 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=759023
• 关键词: Algebras; action; coaction; Hopf; algebras

My research area is Lie theory and its generalizations in the context of Hopf algebras.Lie theory describes the interplay between groups of continuous transformations, called Lie groups, and their linearizations, called Lie algebras. The theory was pioneered by the 19th century Norwegian mathematician Sophus Lie and, since then, has become ubiquitous in mathematics and physics. Examples of Lie groups include the group of all rigid motions in 3d space and the group of all Lorentz transformations in the 4d space-time of relativity theory. They also include the groups of symmetries of various objects (geometric bodies, systems of equations of motion, elasticity tensors, molecules - to name a few), that is, the collections of those transformations that preserve the given object.Hopf algebras were discovered by a topologist Heinz Hopf in the 20th century in his study of cohomology of Lie groups. They were later found useful in many branches of mathematics and physics including algebra, functional analysis, statistical mechanics, quantum field theory and string theory. Roughly speaking, while groups describe classical symmetry, Hopf algebras describe quantum symmetry and, for this reason, are sometimes called quantum groups. In algebra, they provide a unified approach to many classical problems. The focus of the proposed research program will be on actions (or coactions) of Hopf algebras on other algebras. In particular, the theory of Hopf algebras gives a unified approach to the study of graded algebras, algebras with automorphisms, and algebras with derivations by viewing them as objects in an appropriate category associated to a Hopf algebra. The proposed research deals with classical objects of mathematics such as matrix algebras and finite-dimensional simple Lie (super)algebras and is expected to contribute to the advancement of knowledge in fundamental areas of algebra. It may also find applications in other areas of mathematics and in theoretical physics.

我的研究领域是李理论及其在霍普夫代数背景下的推广。李理论描述了连续变换组（称为李群）及其线性化（称为李代数）之间的相互作用。该理论由 19 世纪挪威数学家索弗斯·李 (Sophus Lie) 首创，从那时起，它就在数学和物理学中变得无处不在。李群的例子包括 3d 空间中所有刚性运动的群和相对论 4d 时空中所有洛伦兹变换的群。它们还包括各种物体的对称群（几何体、运动方程组、弹性张量、分子等），即保留给定物体的那些变换的集合。霍普夫代数是由20 世纪拓扑学家 Heinz Hopf 在研究李群上同调时提出了这一观点。后来发现它们在数学和物理学的许多分支中都很有用，包括代数、泛函分析、统计力学、量子场论和弦理论。粗略地说，群描述经典对称性，而霍普夫代数描述量子对称性，因此有时被称为量子群。在代数中，它们为许多经典问题提供了统一的方法。拟议研究计划的重点将是 Hopf 代数对其他代数的作用（或相互作用）。特别是，Hopf 代数理论通过将分级代数、自同构代数和导数代数视为与 Hopf 代数相关的适当类别中的对象，为研究它们提供了统一的方法。拟议的研究涉及矩阵代数和有限维简单李（超）代数等经典数学对象，预计将有助于代数基础领域知识的进步。它也可能在其他数学领域和理论物理中找到应用。