Unifying differential and difference Picard Vessiot theories by using Hopfalgebras

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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