∞-Lie theory

group theory

# Contents

## Idea

A formal group is a group object internal to infinitesimal spaces. More general than Lie algebras, which are group objects in first order infinitesimal spaces, formal groups may be of arbitrary infinitesimal order. They sit between Lie algebras and finite Lie groups or algebraic groups.

Since infinitesimal spaces are typically modeled as formal duals to algebras, formal groups are typically conceived as group objects in formal duals to power series algebras.

### Formal group laws

One of the oldest formalisms is the formalism of formal group laws (early study by Bochner and Lazard), which are a version of representing a group operation in terms of coefficients of the formal power series rings. A formal group law of dimension $n$ is given by a set of $n$ power series $F_i$ of $2n$ variables $x_1,\ldots,x_n,y_1,\ldots,y_n$ such that (in notation $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, $F(x,y) = (F_1(x,y),\ldots,F_n(x,y))$)

$F(x,F(y,z))=F(F(x,y),z)$
$F_i(x,y) = x_i+y_i+\,\,higher\,\,order\,\,terms$

Formal group laws of dimension $1$ proved to be important in algebraic topology, especially in the study of cobordism, starting with the works of Novikov, Buchstaber and Quillen; among the generalized cohomology theories the complex cobordism is characterized by the so-called universal group law; moreover the usage is recently paralleled in the theory of algebraic cobordism of Morel and Levine in algebraic geometry. Formal groups are also useful in local class field theory; they can be used to explicitly construct the local Artin map according to Lubin and Tate.

### Formal group schemes

Much more general are formal group schemes from (Grothendieck)

Formal group schemes are simply the group objects in a category of formal schemes; however usually only the case of the formal spectra of complete $k$-algebras is considered; this category is equivalent to the category of complete cocommutative $k$-Hopf algebras.

### Formal groups over an operad

For a generalization over operads see (Fresse).

## Properties

### In characteristic 0

###### Proposition

The quotient moduli stack $\mathcal{M}_{FG} \times Spec \mathbb{Q}$ of formal group over the rational numbers is isomorphic to $\mathbf{B}\mathbb{G}_m$, the delooping of the multiplicative group (over $Spec \mathbb{Q}$). This means that in characteristic 0 every formal group is determined, up to unique isomorphism, by its Lie algebra.

For instance (Lurie 10, lecture 12, corollary 3).

## Examples

Formal geometry is closely related also to the rigid analytic geometry.

(nlab remark: we should explain connections to the Witt rings, Cartier/Dieudonné modules).

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

## References

### General

• Alexander Grothendieck et al. SGA III, vol. 1, Expose VIIB (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES (part B) 474-560
• Benoit Fresse, Lie theory of formal groups over an operad, J. Alg. 202, 455–511, 1998, doi

Quillen's theorem on MU is due to

### 1-Dimensional formal groups

A basuc introduction is in

• Carl Erickson, One-dimensional formal groups (pdf)

• Takeshi Torii, One dimensional formal group laws of height $N$ and $N-1$, PhD thesis 2001 (pdf)