Wed 8 Feb 2006
I became interested in describing noncommutative formal groups in positive charactersitic. In:
Noncommutative Formal Groups In Positive Charactersitic
I describe some basic structure theory for such groups.
The zero characteristic analouge of the results in this paper are essentialy those that describe, for a Lie algebra , the relationship between its universal enveloping algebra and its symmetric algebra by the Poincare-Birkhoff-Witt (PBW) theorem, as well as the relationship with the free algebra and the Kostant-Kirilov Poisson bracket. In the positive characterisitc setting there are additional complications due to the many possibilities of non-isomorphic commutative formal groups. For this reason, in order to have an analagous theory, one needs to introduce the notion of a splay for “geometric formal groups.” It is on the splay which one measures the non-commutativity of the formal group via a Poisson bracket. These Poisson brackets satisfy certain conditions natural conditions, automatic in the zero charactersitic setting, and given such a Poisson bracket one obtains a non-commutative formal group by making use of a generalization of the PBW theorem in the setting of splays.



