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

LORENZ 摘要 该奖项支持 Martin Lorenz 教授在非交换环理论领域的研究，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。 主要目标是描述这些环的格洛腾迪克群G_0和K_0，有限维Hopf代数表示论的继续发展，以及洛朗多项式环上乘法群作用的不变环的详细分析。 群的不变论和表示论是纯数学中普遍存在的经典代数主题，在应用数学和物理学的某些部分中也是不​​可或缺的。 两者都正式包含在霍普夫代数理论中，这是一种具有广泛应用的更新代数结构。考虑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.