Unifying differential and difference Picard Vessiot theories by using Hopfalgebras

使用 Hopfalgebras 统一微分和差分 Picard Vessiot 理论

基本信息

批准号：
18540009
负责人：
MASUOKA Akira
金额：
$ 1.68万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

项目摘要

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

微分方程的伽罗瓦理论称为 Picard-Vessiot 理论。van der Put 和 Singer 给出了类似的差分方程理论。另一方面，上述研究人员之一的 Mitsuhiro Takeuchi 在其 1989 年发表的 Hopf 论文中提出了这一理论。 - 理论的代数方法。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类型的动作。共交换 Hopf 代数，它还涉及另一种使用正特征的高级微分算子的类似理论，并且（2）使用群方案而不是代数群，我们可以处理不一定是代数封闭的任意基域。研究人员与其中一位研究人员 Amano 一起推出了这种方法，并证明 Takeuchi 的结果在“artinian 简单模代数”的广义背景下是正确的，它作用于更广泛的类别共交换 Ho …比 (1) 中引用的代数更多，可能包含群类。这项联合工作使得以统一的方式处理微分和差分 Picard-Vessiot 理论成为可能，并改进了 van der Put 和 Singer 的一些结果。目前的研究项目是为了推广我们的这项工作，我已经将我们迄今为止所取得的成果写在与天野和等人合作的论文《皮卡德-维西奥特理论的霍普夫代数方法》中。 Takeuchi，该书将出现在 Elsevier Hazewinkel 编辑的 Handbook of Algebra Vol. 5 中，2008 年。作为 Hopf-Galois 理论的另一个应用，我提出了一种使用余循环构造包含量化包络代数的特定类 Hopf 代数的方法。这个想法非常简单：给定一个 Hopf 代数，比如 H，感兴趣的是，我们通过变形来构造它cocycle，一个更简单的 Hopfalgebra，我们希望从中得出有关 H 的有用信息。我实际上专门针对量化包络代数执行了此操作，并证明了李代数上同调的怀特海德引理的量子模拟。通过我们的方法，我们可以简化。即使在广义情况下，三角分解和量子双结构也在很大程度上：我们可以避免检查复杂的定义关系结果包含在几个预印本“阿贝尔”中。和量化包络代数的非阿贝尔第二上同调”，“通过余循环变形构造量化包络代数”。 Less