Formal group

In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner (1946). The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology.

Definitions

A one-dimensional formal group law over a commutative ring R is a power series F(x,y) with coefficients in R, such that

The simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.

More generally, an n-dimensional formal group law is a collection of n power series F_i(x₁, x₂, ..., x_n, y₁, y₂, ..., y_n) in 2n variables, such that