nLab
compact Lie group

Contents

Context

Group Theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

A real Lie group is compact if its underlying topological group is a compact topological group.

Properties

Maximal tori

All maximal tori of a compact Lie group are conjugate by inner automorphisms. The dimension of a maximal torus TT of a compact Lie group is called the rank of GG.

The normalizer N(T)N(T) of a maximal torus TT determines GG.

The Weyl group W(G)=W(G,T)W(G)=W(G,T) of GG with respect to a choice of a maximal torus TT is the group of automorphisms of TT which are restrictions of inner automorphisms of GG.

The maximal torus is of finite index in its normalizer; the quotient N(T)/TN(T)/T is isomorphic to W(G)W(G).

The cardinality of W(G)W(G) for a compact connected GG, equals the Euler characteristic of the homogeneous space G/TG/T (“flag variety”).

See also at relation between compact Lie groups and reductive algebraic groups

Abelian compact Lie groups

Proposition

The maximal torus of a connected compact Lie group is also the maximal abelian subgroup.

(e.g. Salamon 2021, Lem. 6.5)

In particular:

Proposition

All connected compact abelian Lie groups are tori, up to isomorphism.

(Adams 1982, Thm. 2.19, Cor. 2.20)

Proposition

(Classification of compact abelian Lie groups)
Assuming the axiom of choice, every abelian compact Lie group is isomorphic to the direct product group of an n-torus with a finite abelian group.

Proof

Write

G eG G_{\mathrm{e}} \xhookrightarrow{\;\;} G

for the subgroup which is the connected component of the neutral element in the given compact abelian Lie group GG. By Prop. this is a torus

G eT n G_{\mathrm{e}} \,\simeq\, T^n

hence its underlying abelian group is a divisible group and therefore, by this Prop., an injective object in the category Ab of abelian group. This implies that the dashed extension in the following diagram in Ab exists:

hence that GG retracts onto G 0G_0.

While this is, a priori, a diagram in abelian discrete groups not it abelian Lie group, the fact that the dashed morphism p:GG ep \colon G \to G_{\mathrm{e}} restricts to the identity morphism on G eG_{\mathrm{e}}, together with the assumption that GG is a disjoint union of copies of this connected component and using the homomorphism property implies that pp is a continuous homomorphism. But continuous homomorphisms of Lie groups are smooth, so that pp is in fact smooth.

Therefore we have a split exact sequence of Lie groups

and hence an isomorphism

GG e×G/G eT n×A. G \;\simeq\; G_{\mathrm{e}} \times G/G_{\mathrm{e}} \;\simeq\; T^n \times A \,.

By the assumption that GG was compact abelian, AG/G eA \,\coloneqq\, G/G_{\mathrm{e}} is finite abelian.

Invariant metric

Proposition

(compact Lie groups admit bi-invariant Riemannian metrics)
Every compact Lie group admits a bi-invariant Riemannian metric.

(Milnor 76, Cor. 1.4)

Smooth actions

Proposition

Let XX be a smooth manifold and let GG be a compact Lie group. Then every smooth action of GG on XX is proper.

(e.g. Lee 12, Corollary 21.6)

Equivariant triangulation theorem

The equivariant triangulation theorem (Illman 78, Illman 83) says that for GG a compact Lie group and XX a compact smooth manifold equipped with a smooth GG-action, there exists a GG-equivariant triangulation of XX.

Spaces of homomorphisms

Examples

Applications

In equivariant homotopy theory

Compact Lie groups make a somewhat unexpected appearance as equivariance groups in equivariant homotopy theory, where the compact Lie condition on the equivariance group is needed in order for (the available proofs of) the equivariant Whitehead theorem to hold.

(Namely, the equivariant triangulation theorem above is used in these proofs to guaratee that Cartesian products of coset spaces G/HG/H are themselves G-CW-complexes.)

In gauge theory

In gauge theory (Yang-Mills theory/Chern-Simons theory, …) …

References

Textbook accounts:

In the broader context of smooth manifolds:

Dedicated lecture notes:

Discussion in the context of representation theory:

For more see also the references at equivariant homotopy theory.

Last revised on November 4, 2021 at 14:05:36. See the history of this page for a list of all contributions to it.