nLab maximal compact subgroup

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 it is not properly contained in another compact subgroup of $G$.

If $G$ is a Lie group, it is unique up to conjugation. In a $p$-adic group, there may be finitely many conjugacy classes of maximal compact subgroups.

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 Lie 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

$$K \hookrightarrow G$$

is a homotopy equivalence of topological spaces.

Examples

For Lie groups

The following table lists some Lie groups and their maximal compact Lie subgroups (e.g. Conrad). See also compact Lie group.

Lie group	maximal 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 $O(n,n)$	direct product group of two orthogonal groups $O(n) \times O(n)$
unitary group $U(p,q)$	$U(p) \times U(q)$
special Lorentz/AdS etc. group $SO(p,q)$	$S\big(O(p) \times O(q)\big)$
Lorentz / AdS pin group $Pin(q,p)$	$Pin(q) \times Pin(q) / \{(1,1), (-1,-1)\}$

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	$SL(2, \mathbb{R}) \times \mathbb{R}$	$SO(2)$	4	3
3	$SL(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$SO(3) \times SO(2)$	11	7
4	$SL(5, \mathbb{R})$	$SO(5)$	24	14
5	$Spin(5,5)$	$(Sp(2) \times Sp(2))/\mathbb{Z}_2$	45	25
6	E6(6)	$Sp(4)/\mathbb{Z}_2$	78	42
7	E7(7)	$SU(8)/\mathbb{Z}_2$	133	70
8	E8(8)	$Spin(16)/\mathbb{Z}_2$	248	128

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.

Related concepts

• closed subgroup

• locally compact topological group

• compact Lie group

• maximal subgroup

• maximal torus

• reduction and lift of structure groups

References

Textbooks accounts:

• 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)

See also

• Wikipedia, Maximal compact subgroup

Original articles:

• 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:pss/2046647)

• Karl Heinrich Hofmann, Christian Terp, Compact Subgroups of Lie Groups and Locally Compact Groups, Proceedings of the American Mathematical Society Vol. 120, No. 2 (Feb., 1994), pp. 623-634 (jstor:2159906, doi:10.2307/2159906)

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

The maximal compact subgroups inside the (indefinite) rotation groups

• Brian Conrad, Examples of maximal compact subgroups (pdf)