# nLab Lie group

Contents

group theory

### Cohomology and Extensions

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A Lie group is a group with smooth structure. Lie groups form a category, LieGrp.

## Definition

###### Definition

A Lie group is a smooth manifold whose underlying set of elements is equipped with the structure of a group such that the group multiplication and inverse-assigning functions are smooth functions.

In other words, a Lie group is a group object internal to the category SmthMfd of smooth manifolds.

Usually the smooth manifold is assumed to be defined over the real numbers and to be of finite dimension (f.d.), but extensions of the definition to some other ground fields or to -infinite-dimensional manifolds are also relevant, sometimes under other names (such as Fréchet Lie group when the underlying manifold is an infinite-dimensional Fréchet manifold).

A real Lie group is called a compact Lie group (or connected, simply connected Lie group, etc) if its underlying topological space is compact (or connected, simply connected, etc).

Every connected finite dimensional real Lie group is homeomorphic to a product of a compact Lie group (its maximal compact subgroup) and a Euclidean space.

Every abelian connected compact finite dimensional real Lie group is a torus (a product of circles $T^n = S^1\times S^1 \times \ldots \times S^1$).

There is an infinitesimal version of a Lie group, a so-called local Lie group, where the multiplication and the inverse are only partially defined, namely if the arguments of these operations are in a sufficiently small neighborhood of identity. There is a natural equivalence of local Lie groups by means of agreeing (topologically and algebraically) on a smaller neighborhood of the identity. The category of local Lie groups is equivalent to the category of connected and simply connected Lie groups.

The first order infinitesimal approximation to a Lie group is its Lie algebra.

## Properties

### Lie’s three theorems

Sophus Lie proved several theorems, known as Lie's three theorems, on the relationship between Lie algebras and Lie groups. Lie’s third theorem is about the equivalence of categories of finite-dimensional real Lie algebras and local Lie groups. Because Élie Cartan extended this to a global integrability theorem, Lie’s third theorem is also called the Cartan-Lie theorem.

### Lie subgroups

###### Proposition

(Cartan's closed subgroup theorem)

If $H \subset G$ is a closed subgroup of a (finite dimensional) Lie group, then $H$ is a sub-Lie group, hence a smooth submanifold such that its group operations are smooth functions with respect to the the submanifold smooth structure.

### Classification

###### Proposition

Every connected finite-dimensional real Lie group is homeomorphic to a product of a compact Lie group and a Euclidean space. Every abelian connected compact f.d. real Lie group is a torus (a product of circles $T^n = S^1\times S^1 \times \ldots \times S^1$).

The simple Lie groups have a classification into infinite series of

and a finite snumber o

### Different Lie group structures on a group

For $G$ a bare group (without smooth structure) there may be more than one way to equip it with the structure of a Lie group.

###### Example

As bare abelian groups, the Cartesian spaces $\mathbb{R}^n$ are, for all $n$, vector spaces over the rational numbers $\mathbb{Q}$ whose dimension is the cardinality of the continuum, $2^{\aleph_0}$.

Therefore these are all isomorphic as bare group. But equipped with their canonical Lie group structure (as in the Examples) they are of course not isomorphic.

### Different topologies on a Lie group

• Linus Kramer, The topology of a simple Lie group is essentially unique, (arXiv)

Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every ‘abstract’ isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.

### Homotopy groups

List of homotopy groups of the manifolds underlying the classical Lie groups are for instance in (Abanov 09).

## Applications

### In differential geometry

A central concept of differential geometry is that of a $G$-principal bundle $P \to X$ over a smooth manifold $X$ for $G$ a Lie group.

### In gauge theory

In the physics of gauge fields – gauge theory – Lie groups appear as local gauge groups parameterizing gauge transformations: notably the Yang-Mills field is modeled by a $G$-principal bundle with connection for some Lie group $G$. For models that describe experimental observations the group $G$ in question is a quotient of a product of special unitary groups and the circle group. For details see standard model of particle physics

## In higher category theory

The notion of group generalizes in higher category theory to that of 2-group, … ∞-group.

Accordingly, so does the notion of Lie group generalize to Lie 2-group, … ∞-Lie group. For details see ∞-Lie groupoid.

## Examples

### Basic examples

• The real line $\mathbb{R}$ with its standard smooth structure and the group operation being addition is a Lie group. So is every Cartesian space $\mathbb{R}^n$ with the componentwise addition of real numbers.

• The quotient of $\mathbb{R}$ by the subgroup of integers $\mathbb{Z} \hookrightarrow \mathbb{R}$ is the circle group $S^1 = \mathbb{R}/\mathbb{Z}$. The quotient $\mathbb{R}^n/\mathbb{Z}^n$ is the $n$-dimensional torus.

• The automorphism group of any Lie group is canonically itself a Lie group: the automorphism Lie group.

### Classical Lie groups

The classical Lie groups include

### Exceptional Lie groups

The exceptional Lie groups incude

### Infinite-dimensional examples

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 neighbourhoodgerm of a spaceopen 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

### General

In the generality of Lie semigroups:

In the generality of quantum groups:

### Homotopy groups

• Alexander Abanov, Homotopy groups of Lie groups 2009 (pdf)

### On infinite-dimensional Lie groups

References on infinite-dimensional Lie groups

• Andreas Kriegl, Peter Michor, Regular infinite dimensional Lie groups Journal of Lie Theory

Volume 7 (1997) 61-99 (pdf)

• Rudolf Schmid, Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics Advances in Mathematical Physics Volume 2010, (pdf)

• Josef Teichmann, Infinite dimensional Lie Theory from the point of view of Functional Analysis (pdf)

• Karl-Hermann Neeb, Monastir summer school: Infinite-dimensional Lie groups (pdf)

### Spaces of homomorphisms

for maps out of finitely generated discrete groups“:

for maps out of compact Lie groups and the fact that nearby homomorphisms from compact Lie groups are conjugate: