Invariant Theory of Finite Groups and Finite-Dimensional Hopf Algebras

该奖项支持Martin Lorenz教授在代数领域的研究，特别强调由有限群和有限维Hopf代数的作用产生的不变量环的结构。主要目的是刻画这类环的Grothendieck群，有限群作用下不变环的Cohen-Macaulay性质，某些不变量域的合理性问题，以及有限维Hopf代数表示理论的继续发展。所使用的主题和方法主要是代数(交换和非交换)，但与代数K-理论和代数几何有很强的联系。这个项目的目的是加深我们对特定的，特别是相关类型的有限群作用的不变环的现有知识，同时也通过在适当的时候从更广泛的Hopf代数的角度来探讨有限群的不变和表示理论来扩展有限群的不变和表示理论的边界。群的不变理论和表示理论是在纯数学中普遍存在的经典代数主题，在应用数学的某些部分，特别是编码理论中，以及在理论物理中也是不可或缺的。这两个理论在形式上都包含在Hopf代数理论中，Hopf代数是一种较新的具有广泛应用范围的代数结构。为了认识到它们在物理学中的相关性，某些类型的Hopf代数被称为量子群。Hopf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.