Contents

group theory

# Contents

## Definition

For $G$ a topological group a compact subgroup is a topological subgroup $K \subset G$ which is a compact group.

###### Definition

A compact subgroup $K \hookrightarrow G$ is called maximal compact if every compact subgroup of $G$ is conjugate to a subgroup of $K$.

If it exists then, by definition, it is unique up to conjugation.

## Properties

###### Definition

A locally compact topological group $G$ is called almost connected if the quotient topological space $G/G_0$ (of $G$ by the connected component of the neutral element) is compact.

See for instance (Hofmann-Morris, def. 4.24).

###### Example

Every compact and every connected topological group is almost connected.

Also every quotient of an almost connected group is almost connected.

###### Theorem

Let $G$ be a locally compact almost connected topological group.

Then

• $G$ has a maximal compact subgroup $K$;

• the coset space $G/K$ is homeomorphic to a Euclidean space.

This is due to (Malcev) and (Iwasawa). See for instance (Stroppel, theorem 32.5).

###### Theorem

Let $G$ be a locally compact almost connected topological group.

Then a compact subgroup $K \hookrightarrow G$ is maximal compact precisely if the coset space $G/K$ is contractible

(in which case, due to theorem , it is necessarily homeomorphic to a Euclidean space).

This is (Antonyan, theorem 1.2).

###### Remark

In particular, in the above situation the subgroup inclusion

$K \hookrightarrow G$

## Examples

### For Lie groups

The following table lists some Lie groups and their maximal compact Lie subgroups. See also compact Lie group.

Lie groupmaximal compact subgroup
real general linear group $GL(n, \mathbb{R})$orthogonal group $O(n)$
its connected component $GL(n,\mathbb{R})_0$special orthogonal group $SO(n)$
complex general linear group $GL(n, \mathbb{C})$unitary group $U(n)$
complex special linear group $SL(n, \mathbb{C})$special unitary group $SU(n)$
symplectic group $Sp(2n,\mathbb{R})$unitary group $U(n)$
complex symplectic group? $Sp(2n,\mathbb{C})$compact symplectic group $Sp(n)$
Narain group $O(n,n)$two copies of the orthogonal group $O(n) \times O(n)$
unitary group $U(p,q)$$U(p) \times U(q)$
special Lorentz/AdS etc. group $SO(p,q)$$SO(p) \times SO(q)$
Lorentz / AdS spin group $Spin(q,p)$$Spin(q) \times Spin(q) / \{(1,1), (-1,-1)\}$

The following table lists specifically the maximal compact subgroups of the “$E$-series” of Lie groups culminating in the exceptional Lie groups $E_n$.

$n$real form $E_{n(n)}$maximal compact subgroup $H_n$$dim(E_{n(n)})$$dim(E_{n(n)}/H_n )$
2$SL(2, \mathbb{R}) \times \mathbb{R}$$SO(2)$43
3$SL(3,\mathbb{R}) \times SL(2,\mathbb{R})$$SO(3) \times SO(2)$117
4$SL(5, \mathbb{R})$$SO(5)$2414
5$Spin(5,5)$$(Sp(2) \times Sp(2))/\mathbb{Z}_2$4525
6E6(6)$Sp(4)/\mathbb{Z}_2$7842
7E7(7)$SU(8)/\mathbb{Z}_2$13370
8E8(8)$Spin(16)/\mathbb{Z}_2$248128

### Counterexamples

A maximal compact subgroup may not exist at all without the almost connectedness assumption. An example is the Prüfer group $\mathbb{Z}[1/p]/\mathbb{Z}$ endowed with the discrete ($0$-dimensional) smooth structure. This is a union of an increasing sequence of finite cyclic groups, each obviously compact.

A general survey is given in

Textbooks with relevant material include

• M. Stroppel, Locally compact groups, European Math. Soc., (2006)

• Karl Hofmann, Sidney Morris, The Lie theory of connected pro-Lie groups, Tracts in Mathematics 2, European Mathematical Society, (2000)

Original articles include

• A. Malcev, On the theory of the Lie groups in the large, Mat.Sbornik N.S. vol. 16 (1945) pp. 163-189
• K. Iwasawa, , On some types of topological groups, Ann. of Math. vol.50 (1949) pp. 507-558.
• M. Peyrovian, Maximal compact normal subgroups, Proceedings of the American Mathematical Society, Vol. 99, No. 2, (1987) (jstor)

• Sergey A. Antonyan, Characterizing maximal compact subgroups (arXiv:1104.1820v1)