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.
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 Fi(x1, x2, ..., xn, y1, y2, ..., yn) in 2n variables, such that
where we write F for (F1, ..., Fn), x for (x1,..., xn), and so on.
The formal group law is called commutative if F(x,y) = F(y,x).
There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0.
A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that
A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x)= x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".
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