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(«) for some
natural number «). 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.