Picard-Vessiot theory in general Galois theory. (English) Zbl 1252.12007

Crespo, Teresa (ed.) et al., Algebraic methods in dynamical systems. Proceedings of the conference, Będlewo, Poland, May 16–22, 2010. Dedicated to Michael Singer on his 60th birthday. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-13-3/pbk). Banach Center Publications 94, 263-293 (2011).
It was proved in [S. Morikawa, “On a general difference Galois theory. I”, Ann. Inst. Fourier 59, No. 7, 2709–2732 (2009; Zbl 1194.12005)] that difference Picard-Vessiot theory is a part of general difference Galois theory [S. Morikawa and H. Umemura, “On a general difference Galois theory. II”, Ann. Inst. Fourier 59, No. 7, 2733–2771 (2009; Zbl 1194.12006)]. Assuming characteristic zero, a new proof of this result, depending on few fundamental principles, is given in the paper under review, so that it also works for other Picard-Vessiot theories. It is shown how this proof also works for the Picard-Vessiot theory of differential equations and the theory of iterative \(q\)-difference fields.
In addition, it is proved that, however non-commutative the ring of operators may be, the Galois hull is a commutative algebra, showing that the quantum groupoid cannot be encountered in studying linear difference-differential equations.
For the entire collection see [Zbl 1230.00043].


12H05 Differential algebra
12H10 Difference algebra
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