Hopf Algebras and their Actions on Rings

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The investigator will study the relationship between the prime ideals of a Hopf crossed product and the prime ideals of the underlying algebra on which the Hopf algebra acts. A more general problem is to compare the primes in a Hopf Galois extension; such extensions include some well-known extensions of quantum groups. Finite-dimensional semisimple Hopf algebras and their representations will also be considered, in particular questions on the Schur indicator of an irreducible representation, as well as other invariants of the representation. Finally some problems about generalized Lie algebras will be studied, such as when can they be twisted via a Hopf cocycle to an ordinary Lie superalgebra. These generalized Lie algebras are Lie algebras in the category of comodules over a cotriangular Hopf algebra, and include, for example, Lie color algebras, the case when the Hopf algebra is a group algebra. It is desirable in many areas of mathematics to try to classify certain objects,or at least to know enough about them to distinguish them from similar objects. Thus one tries to attach "invariants" to these objects. Hopf algebras appear as such invariants both in mathematical physics (in areas such as conformal field theory) and in several areas of mathematics itself (such as studying knots in geometry). Thus to describe these structures, it would be useful to understand Hopf algebras better and to begin to classify them, as well as to understand how Hopf algebras act on other objects. Moreover within algebra itself, Hopf algebras provide a unifying framework for other algebraic objects, such as groups, Lie algebras, and their recent generalizations coming from quantum mechanics. Thus studying them should give new insights within algebra as well.