Hopf Algebras, Their Actions on Rings, and Graded Algebras

9802086 Montgomery该奖项支持Hopf代数和相关结构的研究。主要研究者将Hopf交叉积的素数理想与Hopf代数作用的基础代数的素数理想进行比较。更一般地说，她将比较Hopf伽罗瓦推广中的素数。她还将研究“对偶问题”，涉及Hopf代数和合适的对偶Hopf代数的作用，以研究无限维Hopf代数的素数理想。有限维Hopf代数的分类问题也将在耶特-德林菲尔德范畴的背景下考虑。最后讨论了代数群分级的几个问题。本研究涉及一种称为Hopf代数的代数对象。虽然这个领域是相当技术性的，但霍普夫代数正变得越来越有趣，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结理论类型很有用。某些类型的霍普夫代数现在被称为量子群，以认识到在物理学中的应用。Hopf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.