Hopf Algebras and their Actions on Rings

研究者将研究一个Hopf交叉积的素理想与该Hopf代数作用于其上的底层代数的素理想之间的关系。 一个更一般的问题是比较一个Hopf伽罗瓦扩张中的素数;这样的扩张包括一些著名的量子群的扩张。 也将考虑高维半单Hopf代数及其表示，特别是不可约表示的舒尔指标问题，以及表示的其他不变量。 最后研究了广义李代数的一些问题，如广义李代数何时可以通过一个Hopf上圈扭曲成一个普通的李超代数。 这些广义李代数是余三角Hopf代数上的余模范畴中的李代数，并且包括，例如，李色代数，当Hopf代数是群代数时的情况。 在数学的许多领域中，试图对某些对象进行分类是可取的，或者至少对它们有足够的了解，以便将它们与类似的对象区分开来。 因此，人们试图将“不变量”附加到这些对象上。霍普夫代数作为这样的不变量出现在数学物理（如共形场论）和数学本身的几个领域（如研究几何中的结）。因此，为了描述这些结构，更好地理解霍普夫代数并开始对它们进行分类以及理解霍普夫代数如何作用于其他对象将是有用的。 此外，在代数本身，霍普夫代数提供了一个统一的框架，其他代数对象，如群，李代数，和他们最近的推广来自量子力学。 因此，研究它们也应该在代数中提供新的见解。

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.