×

Differential algebraic groups. (English) Zbl 0556.12006

This book gives the foundations of a theory of differential algebraic groups. It is intended that such a theory bears to algebraic groups the same relation that the theory of differential equations bears to the theory of algebraic equations. Now algebraic groups can be viewed as groups in the category of algebraic varieties, where the latter are taken to be locally given as sets of simultaneous solutions of algebraic equations. But in fact the foundational aspects of the theory are better served by an abstract view of algebraic varieties as sets with an additional structure, where the latter is usually a sheaf of integral domains, but could be taken as a collection of fields with specialization relations (the ring of sections over a set is replaced by its quotient field and the restriction maps indue the specializations). To lay the foundations of differential algebraic groups, the author adopts a similar formulation: the objects are sets with attached collections of fields and specialization maps, except now the fields are extensions, inside some fixed universal differential field U, of a fixed subfield F with set of derivation operators D, and the specializations must preserve the D-F structure.
With these objects the theory is developed. This requires defining the objects, which are termed D-F groups (Chapter I), describing what happens when F, D, and U change (Chapter II), endowing the objects with a topology (Chapter II) and rational functions (Chapter V), how the associated objects like D-F Lie algebras should be defined (Chapter VI), and what the differential analogue of Galois cohomology should be (Chapter VII). In this framework, the basic theory is produced: subgroups and quotients make sense (Chapter IV), and there is a suitable Lie theory (Chapter VIII).
This is a demanding book. The author requires familiarity with differential algebra as presented in his ”Differential algebra and algebraic groups” (1973; Zbl 0264.12102) and in his ”Constrained extensions of differential fields” [Adv. Math. 12, 141-170 (1974; Zbl 0279.12103)]. Although the reader familiar with the theory of algebraic groups will find that many of the results of the theory of differential algebraic groups are analogous to the algebraic theory, he should be warned that not all are: for example, D-F groups may not be locally affine, and even affine ones need not be linear. In other words, this is a genuinely different subject. By his careful presentation of the subject’s foundations in this volume, the author has prepared the way for its future development and applications.
Reviewer: A.R.Magid

MSC:

12H05 Differential algebra
12-02 Research exposition (monographs, survey articles) pertaining to field theory
14L40 Other algebraic groups (geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
13N05 Modules of differentials
17B45 Lie algebras of linear algebraic groups