Hopf Algebras

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9401324 Nichols This award supports fundamental work on the study of Hopf algebras. The main goal of the project is the solution of Kaplansky's sixth conjecture on Hopf algebras--for a finite dimensional cosemisimple Hopf algebra H defined over an algebraically closed field, the dimension of every simple right H-comodule divides the dimension of H. The principal investigator will also study some questions involving symbolic computation which have arisen in connection with the Hopf algebra work. 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.