Unifying differential and difference Picard Vessiot theories by using Hopfalgebras

微分方程式的伽罗瓦理论称为皮卡德-维西奥特理论。Van der Put和Singer给出了差分方程组的相似理论。另一方面，上述研究人员之一竹内光弘在1989年发表的论文中提出了一种Hopf-代数方法来解释这一理论。这种基于Hopf-Galois理论的方法的优点是：(1)将微分算子推广到某种余交换Hopf代数的作用上，它还涉及到另一种利用具有正特征的高阶微分算子的类似理论；(2)使用群方案而不是代数群，允许我们在不一定是代数闭的任意基域上工作。首席调查员和其中一名调查员天野提出了这一方法，并证明了武口的结果在作用于更广泛类别的余交换Ho…的广义Artin单模代数的上下文中成立比(1)中所引用的更多的PF代数，可能包含群类。这种联合工作使得统一地处理微分和差分Picard-Vessiot理论成为可能，并改进了van der Put和Singer的一些结果。本研究项目就是为了推动我们这项工作的开展。我已经在与Amano和Takeuchi联合发表的论文《Hopf-代数方法到Picard-Vessiot理论》中写下了我们到目前为止所获得的结果，这篇论文将出现在2008年由Elsevier的Hazewinkel编辑的《代数手册》第5卷中。作为Hopf-Galois理论的另一个应用，我提出了一种利用余圈变形来构造一类包含量子化包络代数的Hopf代数的方法。这个想法很简单：给定一个Hopf代数，比方说H，在感兴趣的情况下，我们通过上循环形变来构造它，一个更简单的Hopfagebra，我们有望从它得到关于H的有用信息。实际上，我特别为量子化的包络代数执行了这一点，并证明了李代数上同调的Whitehead引理的量子模拟。通过我们的方法，我们可以在很大程度上简化，即使在一般情况下，三角分解和量子双重结构：我们可以避免检查复杂的定义关系。结果包含在预印本《量子化包络代数的交换和非交换二次上同调》、《利用余圈形变构造量子化包络代数》中。较少

The Galois theory for differential equations is called Picard-Vessiot Theory. An analogous theory for difference equations was given by van der Put and Singer. On the other hand, Mitsuhiro Takeuchi, one of the investigators above, proposed in his paper published 1989 a Hopf-algebraic approach to the theory. Advantages of this approach which is based on the Hopf-Galois theory are: (1) generalizing differential operators to actions by a certain kind of cocommutative Hopf algebras, it involves as well, another analogous theory using higher differential operators in positive characteristic, and (2) using group schemes instead of algebraic groups, we are allowed to work over arbitrary base fields which are not necessarily algebraically closed. The head investigator together with one of the investigators, Amano, pushed out this approach, and proved that Takeuchi's results hold true in the generalized context of "artinian simple module algebras" on which acts a wider class of cocommutative Ho … More pf algebras than those cited in (1), possibly containing grouplikes. This joint work made it possible to treat with differential and difference Picard-Vessiot theories in a unified way, and improved some of results by van der Put and Singer. The present research project is to push out this work of ours. I have written up our results obtained so far, in the paper "Hopf-algebraic approach to the Picard-Vessiot theory" joint with Amano and Takeuchi, which is to appear in Handbook of Algebra Vol. 5 edited by Hazewinkel from Elsevier, 2008. As another application of Hopf-Galois theory, I proposed a way of constructing a certain class of Hopf algebras containing the quantized enveloping algebras by using cocycle deformation. The idea is quite simple: given a Hopf algebra, say H, in interest, we construct it by deforming by cocycle, a simpler Hopfalgebra, from which we can hopefully derive useful information on H. I actually performed this especially for the quantized enveloping algebras, and proved a quantum analogue of the Whitehead Lemma for Lie-algebra cohomology. By our method, we can simplify to a large extent even in generalized situations, the triangular decomposition and the quantum double construction: we can avoid checking complicated defining relations. The results are contained in a couple of preprint, "Abelian and non-abelian second cohomologies of quantized enveloping algebras", "Construction of quantized enveloping algebras by cocycle deformation". Less