A remark on differential algebraic groups.

Abstract

Let U be a universal differential field of characteristic 0 with a set of commuting derivations. In [C] it is shown that if G is an almost simple differential algebraic group then G is isogeneous (as a differential algebraic group) to a linear algebraic group relative to a δclosed subfield of U. Here we show, using some model-theoretic facts, that isogeneous can be replaced by isomorphic. In differential algebra the underlying universe is taken to be a “universal” differential field U with a set of commuting derivations. Starting with subsets of Un defined by differential polynomial equations (theδ-closed subsets of Un), and maps defined by differential polynomials, a category of differential algebraic sets (varieties) is constructed [K]. A differential algebraic group is a group object in this category. If G is such a group then G is said to be A-connected if G has no proper δ-closed subgroup of finite index. G is said to be almost - simple (or, in the language of [C], δ-simple) if G has no proper nont...