Hopf Algebras and Their Actions on Rings

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This research concerns the action of a Hopf algebra H on an algebra R. These actions generalize the more familiar actions of groups as automorphisms, of Lie algebras as derivations, and of coactions of groups which give group-graded algebras. The principal investigator will study crossed products; duality problems involving the action of a Hopf algebra; the structure of general pointed Hopf algebras; the generalized Lie structure of associative algebras, where this Lie structure means that the Lie bracket is taken in a category of comodules of a suitable bialgebra; and extensions of Hopf algebras in categories. 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 non mathematicians. Hopf algebras are useful in the type of knot theory that is of interest to molecular chemists. Certain types of Hopf algebras are now referred to as quantum groups in recognition of their applications in physics. Hopf algebras are used to study differential operators and to make combinatorics more like calculus.