nLab compact Lie group

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 $T$ of a compact Lie group is called the rank of $G$.

The normalizer $N(T)$ of a maximal torus $T$ determines $G$.

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

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

The cardinality of $W(G)$ for a compact connected $G$, equals the Euler characteristic of the homogeneous space $G/T$ (“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

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

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

$$G_{\mathrm{e}} \,\simeq\, T^n$$

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 $G$ retracts onto $G_0$.

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

Therefore we have a split exact sequence of Lie groups and hence an isomorphism

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

By the assumption that $G$ was compact abelian, $A \,\coloneqq\, G/G_{\mathrm{e}}$ 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 $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every smooth action of $G$ on $X$ is proper.

(e.g. Lee 12, Corollary 21.6)

Equivariant triangulation theorem

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

Spaces of homomorphisms

• nearby homomorphisms from compact Lie groups are conjugate

Examples

• A discrete group is a compact Lie group iff it is a finite group.

• The classical Lie groups for definite inner products are compact, such as the orthogonal groups, the unitary groups, the quaternionic unitary groups, etc., but not the Lorentz group etc.

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 $G/H$ are themselves G-CW-complexes.)

In gauge theory

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

Related concepts

• locally compact topological group

• compact topological group

• maximal compact subgroup

• closed subgroup

• proper Lie groupoid

• p-compact group

• twisted ad-equivariant K-theory

References

Textbook accounts:

• Glen Bredon, Section 0.6 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)

• Frank Adams, Lectures on Lie groups, University of Chicago Press, 1982 (ISBN:9780226005300, gbooks)

• Hans Duistermaat, Johan A. C. Kolk, Chapter 3 of: Lie groups, Springer (2000) [doi:10.1007/978-3-642-56936-4]

In the broader context of smooth manifolds:

• John Lee, Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, Draft pdf of the 1st edition)

Dedicated lecture notes:

• Dietmar Salamon, Notes on compact Lie groups, 2021 (pdf, pdf)

Discussion in the context of representation theory:

• Tammo tom Dieck, Theodor Bröcker, Representations of compact Lie groups, Springer 1985 (doi:10.1007/978-3-662-12918-0)

For more see also the references at equivariant homotopy theory.