Formal group law

In mathematics, a formal group law is a formal power series behaving as if it were the product of a Lie group. They were introduced by. 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 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 with coefficients in R, such that

F = x + y + terms of higher degree
F = F .

The simplest example is the additive formal group law F = 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 in 2n variables, such that

F = x + y + terms of higher degree
F = F

where we write F for, x for, and so on.
The formal group law is called commutative if F = F. If R is torsionfree, then one can embed R into a Q-algebra and use the exponential and logarithm to write any one-dimensional formal group law F as F = exp + log), so F is necessarily commutative. More generally, we have:
There is no need for an axiom analogous to the existence of inverse elements 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 power series G such that F = 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 + 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".

Examples

The additive formal group law is given by
The multiplicative formal group law is given by

Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by. Over general commutative rings R there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic.

More generally, we can construct a formal group law of dimension n from any algebraic group or Lie group of dimension n, by taking coordinates at the identity and writing down the formal power series expansion of the product map. The additive and multiplicative formal group laws are obtained in this way from the additive and multiplicative algebraic groups. Another important special case of this is the formal group of an elliptic curve.
F = / is a formal group law coming from the addition formula for the hyperbolic tangent function: tanh = F, tanh), and is also the formula for addition of velocities in special relativity.
is a formal group law over Z found by Euler, in the form of the for an elliptic integral :
Lie algebras

Any n-dimensional formal group law gives an n-dimensional Lie algebra over the ring R, defined in terms of the quadratic part F₂ of the formal group law.
The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group:
Over fields of characteristic 0, formal group laws are essentially the same as finite-dimensional Lie algebras: more precisely, the functor from finite-dimensional formal group laws to finite-dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well-known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information. So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic p > 0.

The logarithm of a commutative formal group law

If F is a commutative n-dimensional formal group law over a commutative Q-algebra R, then it is strictly isomorphic to the additive formal group law. In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that
Examples:

The logarithm of F = x + y is f = x.
The logarithm of F = x + y + xy is f = log, because log = log + log.

If R does not contain the rationals, a map f can be constructed by extension of scalars to R ⊗ Q, but this will send everything to zero if R has positive characteristic. Formal group laws over a ring R are often constructed by writing down their logarithm as a power series with coefficients in R ⊗ Q, and then proving that the coefficients of the corresponding formal group over R ⊗ Q actually lie in R. When working in positive characteristic, one typically replaces R with a mixed characteristic ring that has a surjection to R, such as the ring W of Witt vectors, and reduces to R at the end.

The invariant differential

When F is one-dimensional, one can write its logarithm in terms of the invariant differential ω. Let where is the free -module of rank 1 on a symbol dt. Then ω is translation ''invariant in the sense that where if we write, then one has by definitionIf one then considers the expansion, the formuladefines the logarithm of F''.

The formal group ring of a formal group law

The formal group ring of a formal group law is a cocommutative Hopf algebra analogous to the group ring of a group and to the universal enveloping algebra of a Lie algebra, both of which are also cocommutative Hopf algebras. In general cocommutative Hopf algebras behave very much like groups.
For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomes more involved.
Suppose that F is a formal group law over R. Its formal group ring is a cocommutative Hopf algebra H constructed as follows.

As an R-module, H is free with a basis 1 = D, D, D,...
The coproduct Δ is given by ΔD = ΣD ⊗ D.
The counit η is given by the coefficient of D.
The identity is 1 = D.
The antipode S takes D to ⁿD.
The coefficient of D in the product DD is the coefficient of xⁱy^j in F.

Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.

Formal group laws as functors

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F whose underlying set is Nⁿ where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of Nⁿ; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.
This makes F into a functor from commutative R-algebras S to groups.
We can extend the definition of F to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F to be the inverse limit of the corresponding groups. For example, this allows us to define F with values in the p-adic numbers.
The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g is called group-like if Δg = g ⊗ g and εg = 1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form
for nilpotent elements x. In particular we can identify the group-like elements of H ⊗ S with the nilpotent elements of S, and the group structure on the group-like elements of H ⊗ S is then identified with the group structure on F.

Height

Suppose that f is a homomorphism between one-dimensional formal group laws over a field of characteristic p > 0. Then f is either zero, or the first nonzero term in its power series expansion is for some non-negative integer h, called the height of the homomorphism f. The height of the zero homomorphism is defined to be ∞.
The height of a one-dimensional formal group law over a field of characteristic p > 0 is defined to be the height of its multiplication by p map.
Two one-dimensional formal group laws over an algebraically closed field of characteristic p > 0 are isomorphic if and only if they have the same height, and the height can be any positive integer or ∞.
Examples:

The additive formal group law F = x + y has height ∞, as its pth power map is 0.
The multiplicative formal group law F = x + y + xy has height 1, as its pth power map is ^p − 1 = x^p.
The formal group law of an elliptic curve has height 1 if the curve is ordinary and height 2 if the curve is supersingular. Supersingularity can be detected by the vanishing of the Eisenstein series.