Algebras with action and coaction of Hopf algebras

我的研究领域是Lie理论及其在Hopf代数中的推广。*Lie理论描述了称为李群的连续变换群与其线性化(称为李代数)之间的相互作用。这一理论是19世纪挪威数学家索菲斯·李首创的，从那时起，这一理论在数学和物理学中变得无处不在。李群的例子包括3D空间中的所有刚性运动的群和相对论的4D时空中的所有洛伦兹变换的群。它们还包括各种对象(几何体、运动方程组、弹性张量、分子--仅举几例)的对称性群，即保持给定对象的变换的集合。*Hopf代数是拓扑学家Heinz Hopf在20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，群描述经典对称性，而Hopf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一种统一的方法。*拟议的研究计划的重点将是Hopf代数在其他代数上的作用(或余作用)。特别是，Hopf代数理论通过将分次代数、具有自同构的代数和具有导子的代数视为与Hopf代数相关的适当范畴中的对象，给出了研究分次代数、具有自同构的代数和具有导子的代数的统一方法。这项研究涉及经典的数学对象，如矩阵代数和有限维单李(超)代数，有望促进代数基础领域知识的发展。它还可能在数学的其他领域和理论物理中得到应用。

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.