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Katsunori Saito

Publications and source records attributed to Katsunori Saito.

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.

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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.

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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.

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arXiv2012

On the definition of the Galois group of linear differential equations

Let us consider a linear differential equation over a differential field K. For a differential field extension L/K generated by a fundamental system of the equation, we show that Galois group according to the general Galois theory of Umemura coincides with the Picard-Vessiot Galois group. This conclusion generalized the comparision theorem of Umemura and Casale.

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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.

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