We make some observations relating the theory of finite-dimensional differential algebraic groups (the ∂₀-groups of [2]) to the Galois theory of linear differential equations. Given a differential field (K,∂), we exhibit a surjective functor from (absolutely) split (in the sense of Buium) ∂₀-groups G over K to Picard-Vessiot extensions L of K, such that G is K-split iff L = K. In fact we give a generalization to "K-good" ∂₀-groups. We also point out that the "Katz group" (a certain linear algebraic group over K) associated to the linear differential equation ∂Y = AY over K, when equipped with its natural connection ∂ - [A,-], is K-split just if it is commutative.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc58-0-14,
author = {Anand Pillay},
title = {Finite-dimensional differential algebraic groups and the Picard-Vessiot theory},
journal = {Banach Center Publications},
volume = {58},
year = {2002},
pages = {189-199},
zbl = {1036.12006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc58-0-14}
}
Anand Pillay. Finite-dimensional differential algebraic groups and the Picard-Vessiot theory. Banach Center Publications, Tome 58 (2002) pp. 189-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc58-0-14/