Symbolic Computation and Hopf Algebras

摘要，Nichols97-05132教授计划继续研究Hopf代数的基本问题。他将开发和运用符号计算技术来解决其中的一些问题。Nichols教授计划研究的关于Hopf代数的几个具体问题与它们与Yetter-Drinfel‘d模的联系有关。为了证明Kaplansky的猜想，即半单Hopf代数的绝对不可约表示的次数除以Hopf代数的维度，Nichols教授与Bettina Richmond合作，一直在研究Hopf代数的Grothendieck代数。关于某些Yetter-Drinfel模块如何倍增的信息应该有助于解决一个猜想，这是该计划的关键部分。一个完全不同的研究涉及双代数的计算，它完全由有限群的Yetter-Drinfel‘d模确定。即使小组规模很小，这些问题也可能非常复杂。Yetter-Drinfel‘d模中的代数/余代数相互作用比Hopf代数本身中相应的相互作用容易理解得多，但它足够丰富，足以揭示Hopf代数结构。它在这种情况下的阐明应该是很有启发性的。Hopf代数和符号计算都是人们越来越感兴趣的领域，都是数学中的交叉领域，也是数学之外非常重要的领域。Hopf代数在分子化学家感兴趣的纽结理论中很有用。某些类型的Hopf代数现在被称为“量子群”，以承认它们在物理学中的应用。符号计算是一项关键技术的一部分，即科学和工程计算，它对科学、技术和社会越来越重要。它正在导致数学理论和实践的戏剧性变化。有充分的理由期待Hopf代数的研究和符号计算之间的协同关系。Hopf代数论证经常涉及复杂的符号计算，这与(在交换情况下)现在可以非常有效地由机器完成的符号计算非常相似。另一方面，Hopf代数思想有助于解释和组织计算。Hopf代数为开发计算方法提供了一个具有挑战性的环境，但这种导致简化关系稀缺的结构有助于跟踪现有的关系。预计利用这种Hopf结构获得的洞察力将有助于推进符号计算的更广泛目标。

ABSTRACT, NICHOLS 97-05132 Professor Nichols plans to continue to study fundamental questions about Hopf algebras. He will develop and bring to bear techniques of symbolic computation to attack some of these questions. Several of the specific questions about Hopf algebras that Professor Nichols plans to investigate are related by their connection with Yetter-Drinfel'd modules. With the goal of proving Kaplansky's conjecture that the degree of an absolutely irreducible representation of a semisimple Hopf algebra divides the dimension of the Hopf algebra, Professor Nichols, in joint work with Bettina Richmond, has been studying the Grothendieck algebra of a Hopf algebra. Information about how certain Yetter-Drinfel'd modules multiply should help resolve a conjecture which is a key part of that program. A quite different investigation involves computations of bialgebras which are completely determined by Yetter-Drinfel'd modules of finite groups. Even when the groups are small, these can be extremely complicated. The algebra/coalgebra interaction in Yetter-Drinfel'd modules is considerably easier to understand than is the corresponding interaction in the Hopf algebra itself, yet it is rich enough to shed a great deal of light on the Hopf algebra structure. Its elucidation in this situation should be quite instructive. Hopf algebras and symbolic computation are both areas of growing interest, areas which cross fields in mathematics, and areas which are very important outside mathematics as well. Hopf algebras are useful in the type of knot theory that is of interest to molecular chemists. Certain types of Hopf algebras are now referred to as "quantum groups", in recognition of their applications in physics. Symbolic computation is part of a key technology, namely, scientific and engineering computation, that is becoming increasingly important to science, technology, and society. It is leading to dramatic changes in the theory and practice of mathematics. There are good reasons to expect a synergistic relationship between investigations in Hopf algebras and symbolic computation. Hopf algebra arguments often involve complicated symbolic calculations which are quite similar to ones which (in the commutative case) can now be done very effectively by machine. In the other direction, Hopf algebra ideas help to explain and organize computations. Hopf algebras provide a challenging setting in which to develop computational methods, but the very structure that causes the scarcity of simplifying relations helps to keep track of the ones that exist. It is expected that the insights gained in exploiting this Hopf structure will contribute to the wider goal of advancing symbolic computation.