Let U be a universal differential field of characteristic 0 with a set[image omitted] 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 [image omitted] 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 nontrivial δ-closed δ-connected normal subgroup (so G may have a nontrivial centre). (Similarly one defines δ-semisimple etc.) U δ is the m-dimensional vector space over U with basis A. Any element of U A is a derivation of U, and if X is a subset of U δ then[image omitted] is a subfield of U. In fact these subfields CX are precisely the δ-closed subfields of U, and they are all algebraically closed. If G is an (ordinary) algebraic group with respect to some such Cx (namely G is the Cx-rational points of an algebraic group defined over Cx), then in particular G is a differential algebraic group. In [C] it is proved that if G is an almost simple differential algebraic group then there is some [image omitted] some linear algebraic group H with respect to Cx and a surjective differential rational homomorphism f from G to H with finite kernel. The purpose of this note is to show that one can replace the isogeny f by a differential rational isomorphism.
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