Unifying differential and difference Picard Vessiot theories by using Hopfalgebras

微分方程的伽罗瓦理论被称为Picard-Vessiot理论。货车德普特和辛格给出了差分方程的类似理论。另一方面，Mitsuhiro Takeuchi，上面的研究者之一，在1989年发表的论文中提出了一种霍普夫代数理论。这种方法的优点是：（1）将微分算子推广到一类上交换的Hopf代数的作用上，它还涉及到另一个类似的理论，即用正特征的高阶微分算子;（2）用群模式代替代数群，我们可以在任意的基域上工作，这些基域不一定是代数闭的。首席研究员与其中一名研究员天野之弥一起推出了这一方法，并证明了竹内的结果在“artinian单模代数”的广义上下文中成立，在该代数上作用了更广泛的一类余交换Ho 关于我们 pf代数比（1）中引用的代数，可能包含群类。这一联合工作使得统一处理微分和差分Picard-Vessiot理论成为可能，并改进了货车der Put和Singer的一些结果。目前的研究计划就是要把我们的这项工作推向前进。我已经写了我们的结果到目前为止，在论文“霍普夫代数方法的皮卡德-Vessiot理论”联合与天野和竹内，这是出现在手册代数卷。作为Hopf-Galois理论的另一个应用，我提出了一种利用上循环变形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：给定一个感兴趣的Hopf代数，比如H，我们通过对它进行上圈变形来构造它，这是一个更简单的Hopf代数，我们希望从中得到关于H的有用信息。实际上，我特别对量子化包络代数进行了这种处理，并证明了李代数上同调的怀特黑德引理的量子模拟。通过我们的方法，我们可以在很大程度上简化，即使在广义的情况下，三角分解和量子二重结构：我们可以避免检查复杂的定义关系。结果包含在一对预印本中，“量子化包络代数的阿贝尔和非阿贝尔第二上同调”，“量子化包络代数的上循环变形构造”。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