(1) The PI will study the prime ideals in a Hopf Galois extension; such extensions include crossed products of Hopf algebras, as well as certain extensions of quantum groups obtained from quotients. She will also consider representations of finite-dimensional semisimple Hopf algebras, with particular interest in the Frobenius-Schur indicator of an irreducible representation. Progress on such problems might be of help in the classification program. The PI will also study the possible actions of the Taft Hopf algebras on quantum polynomial rings. Finally the PI will study what happens to a Hopf algebra and an algebra on which it acts when the Hopf algebra is twisted by a cocycle.
(2) Hopf algebras are algebraic objects which have arisen in other subjects, both in mathmatics itself (such as in knot theory and geometry) and in mathematical physics (such in statistical mechanics and conformal field theory). If one could classify certain Hopf algebras, it would help in describing these other structures. Moreover Hopf algebras provide a unifying framework for groups, Lie algebras, and their recent quantum analogs. Thus studying Hopf algebras themselves as well as their actions on other structures is an important problem.