arXiv subjects

Hiroshi Umemura

Publications and source records attributed to Hiroshi Umemura.

5 recordsLinked to original sources

arXiv2016

Toward quantization of Galois theory

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to a Hopf algebra, we arrived at a final form if the base field consists of constants. In this case, we have non-commutative Picard-Vessiot rings and asymmetric Tannaka theory. For non-linear equations there are examples that might make us optimistic.

math.QA↗

arXiv2013

Toward quantized Picard-Vessiot theory, Further observations on our previous example

It is quite natural to wonder whether there is a difference-differential equations, the Galois group of which is a quantum group that is neither commutative nor co-commutative. Believing that there was no such linear equations, we explored non-linear equations and discovered a such equation. We show that the example is related with a linear equation. We treat only one charming example. It is not, however, an isolated example. We open a window that allows us to have a look at a quantized Picard-Vessiot theory.

math.QA↗

arXiv2013

Can we quantize Galois theory ?

We present a non-linear difference-differential equation whoes Galois group is a non-commutative and non-co-comutative Hopf algebra.

math.QA↗

arXiv2012

Quantization of Galois theory, Examples and Observations

If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We show by Examples that if we consider non-linear equations Galois group is quantized. The results depend on general differential Galois theory for non-linear differential equations developed by the second author.

math.QA↗

arXiv2000

Painlevé equations and deformations of rational surfaces with rational double points

In this paper, we show that the Bäcklund transformations of Painlevé equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlevé equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlevé equations.

math.AG↗