Hopf Algebras, Their Actions on Rings, and Graded Algebras

9802086蒙哥马利该奖项支持霍普夫代数及其相关结构方面的工作。主要研究人员将比较Hopf交叉积的素理想和Hopf代数作用于其上的基础代数的素理想。更一般地，她将比较Hopf Galois扩张中的素数。她还将研究涉及Hopf代数和合适的对偶Hopf代数的作用的“对偶问题”，以便研究无限维Hopf代数的素理想。关于有限维Hopf代数的分类问题也将在Yetter-Drinfeld范畴的背景下被考虑。最后讨论了群分次代数的一些问题。这项研究涉及一种称为Hopf代数的代数对象。虽然这个领域是非常技术性的，但Hopf代数正变得越来越感兴趣，甚至对非数学家也是如此。Hopf代数对于研究分子化学中出现的纽结理论类型很有用。考虑到在物理学中的应用，某些类型的Hopf代数现在被称为量子群。Hopf代数被用来研究微分算子，并使组合学更像微积分。

9802086 Montgomery This award supports work on Hopf algebras and related structures. The principal investigator will compare the prime ideals of a Hopf crossed product with the prime ideals of the underlying algebra on which the Hopf algebra acts. More generally, she will compare the primes in a Hopf Galois extension. She will also study "duality problems" involving both the action of a Hopf algebra and of a suitable dual Hopf algebra, in order to look at the prime ideals for infinite dimensional Hopf algebras. Problems on classifying finite-dimensional Hopf algebras will also be considered in the context of Yetter-Drinfeld categories. Finally some problems concerning algebras graded by groups will be considered. This research is concerned with an algebraic object called a Hopf algebra. While the area is quite technical, Hopf algebras are becoming of increasing interest, even to nonmathematicians. Hopf algebras are useful for studying the type of knot theory that appears in molecular chemistry. Certain types of Hopf algebras are now referred to as quantum groups in recognition of the applications in physics. Hopf algebras are used to study differential operators and to make combinatorics more like calculus.