The differential rational representation algebra on a linear differential algebraic group

Abstract

In [2], the author began the study of affine differential algebraic groups. The study of such groups is somewhat analogous to the study of algebraic groups. However, the non-Noetherian character of the differential rings associated with differential algebraic sets has made it necessary to develop new techniques for the study of algebraic differential equations. The reader is referred to Kolchin [7]. It is well-known, easy to prove, and central to the study of algebraic groups, that the ring of everywhere defined rational functions on an affine algebraic set is the coordinate ring. The differential ring of everywhere defined differential rational functions on an affine differential algebraic set is not the differential coordinate ring, and is not, in general, finitely generated as a differential algebra. For this reason, an affine differential algebraic group need not be linear, i.e., it need not be differentially rationally isomorphic to a differential algebraic matric group. If G is an affine differential algebraic group, the differential algebra, generated over the coefficient field by the coordinate functions of all the linear differential rational representations of G, is called the dz@rentiaZ rational representation algebra on G, and is denoted by W(G). The purpose of this note is to prove the following theorem and its corollaries: