||During the past 35 years the theory of algebraic groups has grown from infancy to vigorous maturity and has been widely applied to other fields of mathematics. Because algebraic equations form a special case of algebraic differential equations, it is natural to try to generalize the theory of algebraic groups to a theory of differential algebraic groups and to expect that it, too, will have widespread connections with other fields.
Such a theory has been under development for more than a decade. In a series of five papers (so far), P. J. Cassidy [3-7] defined the notion of affine differential algebraic group and proved many basic results, especially for those that are linear (that is, that can be suitably embedded in GL(n) for some natural number л). Although much remains to be done in this direction, a general theory of differential algebraic groups is overdue. Even for algebraic groups, linear ones do not tell the whole story; nontrivial Abelian varieties are not linear.
The purpose of the present book is to establish such a general theory. The reader is expected to be acquainted with certain concepts and results from differential algebra, for most of which references are given (mainly to my book ). A preliminary Chapter 0 contains some differential algebraic material that either is new or is not available elsewhere in the form used here or is presented simply for the convenience of the reader. An increasingly prevalent convention in differential algebra is to call a differential ring (or field) with set A of derivation operators a "A-ring" (or "A-field ") and, more generally, to use the prefix "A-" as a synonym for "differential" or "differentially." This convention is adopted here.