Mathematical Sciences: Hopf Algebras and Their Actions on Rings

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The principal investigator will investigate the actions of Hopf algebras on rings, in particular crossed products and inner actions. The object is to extend earlier results for actions of groups as automorphisms and of Lie algebras as derivations. Other problems concern when the Weyl algebra can be a fixed ring, and studying the coalgebra structure of the quantum enveloping algebras. The postdoctoral associate will work on quantum groups and quasitriangular Hopf algebras. She will also consider general Hopf algebras and their actions on H-module algebras. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplica- tion defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.