Hopf Algebras and their Actions on Rings

研究者将研究Hopf交叉积的素理想与Hopf代数所作用的基础代数的素理想之间的关系。一个更一般的问题是比较Hopf伽罗瓦扩展中的素数；这些扩展包括一些众所周知的量子群的扩展。有限维半简单Hopf代数及其表示也将被考虑，特别是关于不可约表示的Schur指示的问题，以及该表示的其他不变量。最后研究了广义李代数的一些问题，如什么时候可以通过Hopf环扭转到一个普通李超代数上。这些广义李代数是在共三角Hopf代数上的模范畴中的李代数，并且包括，例如，当Hopf代数是群代数时的李彩色代数。在数学的许多领域中，试图对某些对象进行分类，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，这是可取的。因此，人们试图将“不变量”附加到这些对象上。Hopf代数在数学物理（如共形场论）和数学本身的一些领域（如研究几何中的结）中都以不变量的形式出现。因此，为了描述这些结构，更好地理解Hopf代数并开始对它们进行分类，以及理解Hopf代数如何作用于其他对象，将是有用的。此外，在代数本身，霍普夫代数为其他代数对象提供了一个统一的框架，如群，李代数，以及它们最近来自量子力学的推广。因此，研究它们也会给代数带来新的见解。

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.