# Formal group law

In mathematics, a **formal group law** is (roughly speaking) the formal power series analogue of a Lie group. Given a Lie group *G* and a chart around its identity element *e*, such that *e* is given coordinates (0, 0, ..., 0), the group operation in *G* is some function

*F(x*_{1},x_{2}, ..., x_{n},y_{1},y_{2}, ..., y_{n})

where *n* is the dimension of *G*, taking values in **R**^{n}. Suppose we take the power series expansion of *F* (formal Taylor series) and look at the identities it should satisfy on the basis of the group axioms - for example the associative law (for the inverse operation take also an expansion of the inverse mapping *I*). Then we can turn these power series identities into a fresh definition, of a **formal group law** (often just **formal group**) in *n* variables.

The nature of a 'formal group' is not therefore as a group in the set-theoretic sense: it is more like the bare definition

*F(x,y) = x + y + xy*

that comes from thinking about multiplication near 1. It is not either a group object in the sense of category theory (as a Lie group is). It does provide an actual group when the variable *x* is given values in a ring such as

**R**[*t*]/(*t*), for^{k}*k*= 1, 2, ...

for which convergence is guaranteed since *t* has been forced to be nilpotent. There is an abstract sense in which the formal group here is an inverse limit of group schemes; or a functor from artinian **R**-algebras to groups.

There was early work on formal groups of elliptic curves by Lutz; the idea later became important in the classification of abelian varieties in non-zero characteristic. The theory was also much used in algebraic topology in the 1960s, after it was seen that it played a role in homology theory. To be precise, cobordism theory needed the *universal* one-dimensional formal group over **Z**, the existence of which had been proved by Lazard.