Finite-dimensional differential algebraic groups and the Picard-Vessiot theory
Volume 58 / 2002
Abstract
We make some observations relating the theory of finite-dimensional differential algebraic groups (the $\partial_{0}$-groups of \cite{Buium}) to the Galois theory of linear differential equations. Given a differential field $(K,\partial)$, we exhibit a surjective functor from (absolutely) split (in the sense of Buium) $\partial_{0}$-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" $\partial_{0}$-groups. We also point out that the “Katz group" (a certain linear algebraic group over $K$) associated to the linear differential equation $\partial Y = AY$ over $K$, when equipped with its natural connection $\partial - [A,-]$, is $K$-split just if it is commutative.
