Symbolic Computation and Hopf Algebras

摘要，Nichols 97-05132 尼科尔斯教授计划继续研究有关霍普夫代数的基本问题。他将开发并运用符号计算技术来解决其中的一些问题。几个具体的问题，关于霍普夫代数教授尼科尔斯计划调查有关的连接与Yetter-Drinfel'd模块。随着目标证明Kaplansky的猜想，程度的绝对不可约表示一个半单的霍普夫代数划分的层面的霍普夫代数，教授尼科尔斯，在联合工作与贝蒂娜里士满，一直在研究Grothendieck代数的霍普夫代数。 关于某些Yetter-Drinfel模块如何相乘的信息应该有助于解决该程序的关键部分的猜想。一个完全不同的调查涉及计算的双代数是完全确定的Yetter-Drinfel'd模块的有限群。即使小组很小，这些也可能非常复杂。Yetter-Drinfel'd模中的代数/余代数相互作用比Hopf代数本身中的相应相互作用更容易理解，但它足够丰富，可以为Hopf代数结构提供大量的光。在这种情况下，它的解释应该是相当有启发性的。 霍普夫代数和符号计算都是越来越感兴趣的领域，在数学领域中交叉的领域，以及在数学之外非常重要的领域。霍普夫代数在分子化学家感兴趣的纽结理论中很有用。某些类型的霍普夫代数现在被称为“量子群”，以表彰其在物理学中的应用。 符号计算是科学和工程计算这一关键技术的一部分，它对科学、技术和社会都变得越来越重要。它导致了数学理论和实践的巨大变化。 有很好的理由期待在霍普夫代数和符号计算的调查之间的协同关系。霍普夫代数的论点往往涉及复杂的符号计算，这是非常类似的（在交换的情况下），现在可以做的非常有效的机器。在另一个方向，霍普夫代数的思想有助于解释和组织计算。霍普夫代数为发展计算方法提供了一个具有挑战性的环境，但是导致简化关系稀缺的结构有助于跟踪存在的关系。预计在利用这种霍普夫结构中获得的见解将有助于推进符号计算的更广泛的目标。

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.