The main part of this project concerns representations of a semisimple Hopf algebra, particularly the Frobenius-Schur indicators of its representations. The values of the indicators should give information not only about the representation category of a Hopf algebra, but also about the structure of the Hopf algebra itself. A related useful invariant is the trace of the antipode of a Hopf algebra. The project also concerns actions of Hopf algebras on non-commutative algebras, especially when the Jacobson or prime radical is stable under such an action. A basic problem is the relationship of the spectrum of primitive ideals to its stable analog, and to the primitive ideals of a Hopf semidirect product. Progress here would help in doing "invariant theory" for Hopf algebras.
Hopf algebras are special algebraic objects which arose in topology and algebraic groups in the 1940's and 1950's. More recently they have appeared in other parts of mathematics, such as geometry (knot theory), and in mathematical physics (conformal field theory). Frequently Hopf algebras give invariants of these structures. Thus a greater understanding of Hopf algebras themselves may eventually be useful in these areas. This project will involve several women students as part of the Women in Science and Engineering (WiSE) program at USC.
This project studies algebraic objects called Hopf algebras. These objects have arisen in mathematical physics as well as other parts of mathematics, such as geometry. So understanding them better should help in understanding other parts of mathematics and physics. In particular, to each Hopf algebra we may attach a set of complex numbers, called ``indicators'', and these indicators give us information about the Hopf algebra itself. The results of this project are related to indicators in several ways. In a positive direction, we show that for certain kinds of Hopf algebras in characteristic p, their indicators may be computed over the complex numbers instead, and this is much more striaghtforward. We also considered the question of when all the indicators are integers (that is, whole numbers), and showed that this is frequently the case, although even for ``nice'' Hopf algebras, non-trivial complex numbers can appear. Some of the results were more theoretical, predicting what to expect for certain kinds of Hopf algebras and which are ``good'' in a certain sense. It is hoped that these results will have applications back to other subjects, from which various Hopf algebras arose. The project also involved graduate students and post-docs who contributed to this effort, computing values in some special cases.
