zbMATH — the first resource for mathematics

Differential formal groups of J. F. Ritt. (English) Zbl 0552.14010
In this paper, the authors study Lie algebras of formal groups. In the special case considered by Ritt, the Lie algebras correspond biunivoquely to formal groups. The basic object is the algebra K[P] of differential operators where K is a field and P is a Lie ring acting as derivations of K. It turns out that K[P] has a twisted Hopf algebra structure - it is a Hopf K/k-algebra, where k is a subfield of K on which P acts trivially. The authors introduce first in general the Hopf K/k-algebra B, B-algebra, B-coalgebra and B-bialgebra. Thus one can define a formal Ritt B-group to be a complete topological B-bialgebra A which both has a unique maximal ideal and is finitely generated as a topological B-algebra. Under some additional finiteness conditions on B and A, the Lie algebra of the formal group is linearly compact. This allows the application of the methods and results of V. Guillemin [J. Differ. Geom. 2, 313-345 (1968; Zbl 0183.261)] and R. J. Blattner [Trans. Am. Math. Soc. 144(1969), 457-474 (1970; Zbl 0295.17002)]. The authors use these techniques to classify these K-algebras which admit a simple linearly compact K[P]-algebra structure. Then, they study the set of non- isomorphic K[P]-structures on each such K-algebra as follows. Fix one easily constructed structure as a reference point and each additional structure is described by a kind of 1-cocycle - a flat connection with values in the Lie algebra. Finally, the authors look at the question: Can formal Ritt groups be considered to be formalizations (completions of the local ring at identity) of some algebraic structures? The answer is ”yes” if their Lie algebras are finite dimensional and ”no” if their Lie algebras are simple of Cartan type.
Reviewer: E.Abe

MSC:
 14L05 Formal groups, $$p$$-divisible groups 13N05 Modules of differentials 17B45 Lie algebras of linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields
Full Text: