Finite-dimensional differential algebraic groups and the Picard-Vessiot theory

Anand Pillay. "Finite-dimensional differential algebraic groups and the Picard-Vessiot theory." Banach Center Publications 58.1 (2002): 189-199. <http://eudml.org/doc/282107>.

@article{AnandPillay2002,
abstract = {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.},
author = {Anand Pillay},
journal = {Banach Center Publications},
keywords = {Picard-Vessiot extension; differential Galois extension},
language = {eng},
number = {1},
pages = {189-199},
title = {Finite-dimensional differential algebraic groups and the Picard-Vessiot theory},
url = {http://eudml.org/doc/282107},
volume = {58},
year = {2002},
}

TY - JOUR
AU - Anand Pillay
TI - Finite-dimensional differential algebraic groups and the Picard-Vessiot theory
JO - Banach Center Publications
PY - 2002
VL - 58
IS - 1
SP - 189
EP - 199
AB - 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.
LA - eng
KW - Picard-Vessiot extension; differential Galois extension
UR - http://eudml.org/doc/282107
ER -