nLab
Lie group

Idea

A Lie group is a group with smooth structure.

Definition

Definition

A Lie group is a smooth manifold whose underlying set of points is equipped with a structure of a group and the multiplication and inverse maps for the group are smooth maps.

In other words, it is a group object internal to Diff.

Usually the manifold is assumed to be over the real numbers or the complex numbers and of finite dimension (f.d.), but extensions to some other ground fields and infinite-dimensional setting 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 a compact Lie group (or connected, simply connected Lie group, etc) if its underlying space is compact (or connected, simply connected, etc).

Every connected finite dimensiona 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 $T^n=S^1\times S^1\times \dots \times S^1$).

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.

Examples

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

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

The classical Lie groups include

• the general linear group $\mathrm{GL}(n)$

• the orthogonal group $O(n)$ and special orthogonal group $\mathrm{SO}(n)$;

• the unitary group $U(n)$ and special unitary group $\mathrm{SU}(n)$;

• the symplectic group $\mathrm{Sp}(2n)$.

The exceptional Lie groups incude

• … E8

Properties

Lie’s three theorems

Sophus Lie has proved several theorems – Lie's three theorems – on the relationship between Lie algebras and Lie groups. What is called Lie's third theorem is about the equivalence of categories of f.d. real Lie algebras and local Lie groups. Élie Cartan has extended this to a global integrability theorem called the Cartan-Lie theorem, nowadays after Serre also called Lie’s third theorem.

Classification

Prposition

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 $T^n=S^1\times S^1\times \dots \times S^1$).

The simple Lie groups have a classification into infinite series of

• classical Lie groups

and a finite snumber o

• exceptional Lie groups

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 ${ℝ}^n$ 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.

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.

Related concepts

• semisimple Lie group, simple Lie group, exceptional Lie group

• real form

• compact Lie group, maximal compact subgroup

• Hamiltonian dynamics on 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, Innite dimensional Lie Theory from the point of view of Functional Analysis (pdf)