Non-Commutative Formal Groups in Positive Characteristic

We describe geometric non-commutative formal groups in terms of a geometric commutative formal group with a Poisson structure on its splay algebra. We describe certain natural properties of such Poisson structures and show that any such Poisson structure gives rise to a non-commutative formal group. We describe geometric non-commutative formal groups in terms of a geometric commutative formal group with a Poisson structure on its splay algebra. We describe certain natural properties of such Poisson structures and show that any such Poisson structure gives rise to a non-commutative formal group. In characterizing non-commutative formal groups G, it is natural to consider their distribution algebras, Dist(G). Although we are interested mainly in the characteristic p > 0 setting, we make an effort to work by analogy to the characteristic zero version of our problem. Let G be a formal group with Lie algebra g. We recall that, in characteristic zero, we have a canonical isomorphism between Dist(G) and the universal enveloping algebra Ug. Then our problem reduces to describing the algebraic relationship between the symmetric Sg, Ug, and the tensor algebra Tg as in the following diagram: Tg Sg ∼ − PBW Ug. (1) Here, the diagonal arrows are algebra maps and the horizontal arrow is a vector space isomorphism given by the Poincaré-Birkhoff-Witt (PBW) theorem. In the positive characteristic setting, there are a few features that present difficulties which are absent in the zero characteristic setting. In a sense made precise below, the problem is non-linear and, unlike in the zero characteristic setting, the algebras of importance are not quadratic. Let us recall that in characteristic zero, the PBW theorem gives an identification between Sg and the associated graded algebra of Ug. We may regard Sg as the universal enveloping algebra for the trivial Lie algebra structure on g, so