Algebras with action and coaction of Hopf algebras

我的研究领域是李理论及其在Hopf代数背景下的推广。李理论描述了连续变换组（称为李群）和线性化组（称为李代数）之间的相互作用。这个理论是由19世纪世纪挪威数学家索菲斯·李开创的，从那时起，它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中4d时空中所有洛伦兹变换的群。它们还包括各种对象（几何体，运动方程系统，弹性张量，分子-仅举几例）的对称性群，即保留给定对象的那些变换的集合。霍普夫代数是由拓扑学家海因茨·霍普夫在世纪研究李群的上同调时发现的。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，当群描述经典对称性时，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。拟议的研究计划的重点将是行动（或共同作用）的其他代数的霍普夫代数。特别是，理论的霍普夫代数给出了一个统一的方法来研究分次代数，代数与自同构，代数与导子，通过查看它们的对象在一个适当的类别相关联的一个霍普夫代数。拟议的研究涉及数学的经典对象，如矩阵代数和有限维简单李（超）代数，预计将有助于代数基础领域知识的进步。它也可以在数学和理论物理的其他领域中找到应用。

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.