Invariant Theory of Finite Groups and Finite-Dimensional Hopf Algebras

该奖项支持研究教授马丁洛伦茨在该地区的代数，特别强调环的结构所产生的不变量的行动有限群和有限维的霍普夫代数。 主要目标是描述Grothendieck群的这些环，科恩-麦考利财产不变环下有限群行动，合理性问题的某些领域的不变量，并继续发展的代表性理论的有限维霍普夫代数。 主题和所采用的方法主要是代数（交换和非交换），但也有很强的联系与代数K理论和代数几何。 这个项目的目的是深化我们目前的知识不变环的具体，特别是相关类型的有限群行动，同时也扩大边界不变和代表性理论的有限群接近这些领域，只要适当的，不变量理论和群的表示理论是经典的代数主题，在纯数学中无处不在，在应用数学的某些部分，特别是编码理论和理论物理学中不可或缺。 这两种理论都被正式地归入到霍普夫代数理论中，霍普夫代数是最近的代数结构，有着广泛的应用。 为了认识到它们在物理学中的相关性，某些类型的霍普夫代数被称为量子群。 霍普夫代数进一步自然地出现在纽结理论中，这是分子化学家和物理学家特别感兴趣的一个领域。

This award supports the research of Professor Martin Lorenz in the area of algebra with particular emphasis on the structure of rings of invariants arising from actions of finite groups and finite dimensional Hopf algebras. The main objectives are the description of the Grothendieck groups of these rings, the Cohen-Macaulay property of invariant rings under finite group actions, rationality questions for certain fields of invariants, and the continued development of the representation theory of finite dimensional Hopf algebras. The subject and the methods employed are mainly algebraic (commutative and noncommutative), but there are strong connections with algebraic K-theory and algebraic geometry. This project aims at deepening our present knowledge of the invariant rings of specific, especially relevant types of finite group actions while also extending the boundaries of both invariant and representation theory of finite groups by approaching these fields, whenever appropriate, from the more general perspective of Hopf algebras.Invariant theory and representation theory of groups are classical algebraic themes that are ubiquitous in pure mathematics and are indispensable in parts of applied mathematics, notably coding theory, and in theoretical physics as well. Both theories are formally subsumed in the theory of Hopf algebras, more recent algebraic structures with a wide range of applications. In recognition of their relevance in physics, certain types of Hopf algebras are referred to as quantum groups. Hopf algebras further naturally arise in knot theory, an area of particular interest to molecular chemists as well as physicists.