Hopf Algebras, Their Actions on Rings, and Graded Algebras

9802086蒙哥马利 该奖项支持霍普夫代数和相关结构的工作。 主要研究人员将比较的主要理想的一个霍普夫交叉产品的主要理想的基础代数上的霍普夫代数的行为。 更一般地说，她将比较一个霍普夫伽罗瓦扩展中的素数。 她还将研究“对偶问题”，涉及双方的行动的一个霍普夫代数和一个合适的对偶霍普夫代数，以期待在素理想的无限维霍普夫代数。 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。 最后讨论了关于群分次代数的一些问题。 这项研究涉及一个代数对象称为一个霍普夫代数。 虽然该领域是相当技术性的，但霍普夫代数正变得越来越有兴趣，甚至对非数学家也是如此。 霍普夫代数对于研究出现在分子化学中的纽结理论是有用的。 某些类型的霍普夫代数现在被称为量子群，以表彰其在物理学中的应用。 霍普夫代数用于研究微分算子，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.