Mathematical Sciences: Actions of Finite Groups and Finite Dimensional Hopf Algebras on Rings

罗伦兹摘要 该奖项支持马丁·洛伦兹教授在非交换环理论领域的研究，特别强调有限群和有限维Hopf代数的作用所产生的不变量环的结构。 主要目标是描述这些环的Grothendieck群G_0和K_0，继续发展有限维Hopf代数的表示理论，并详细分析Laurent多项式环上乘法群作用的不变环。 群的不变量理论和表示理论是在纯数学中普遍存在的经典代数主题，也是应用数学和物理学中不可缺少的部分。 两者都被正式纳入霍普夫代数理论，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。

LORENZ ABSTRACT This award supports the research of Professor Martin Lorenz in the area of noncommutative ring theory with particular emphasis on the structure of rings of invariants arising from actions of finite groups and finite dimensional Hopf algebras. The main objectives will be the description of the Grothendieck groups G_0 and K_0 of these rings, the continued development of the representation theory of finite dimensional Hopf algebras, and the detailed analysis of invariant rings of multiplicative group actions on Laurent polynomial rings. 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 and physics as well. Both 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.