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.
