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maximal compact subgroup

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Definition

For GG a topological group a compact subgroup is a topological subgroup KGK \subset G which is a compact group.

Definition

A compact subgroup KGK \hookrightarrow G is called maximal compact if every compact subgroup of GG is conjugate to a subgroup of KK.

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

Properties

Definition

A locally compact topological group GG is called almost connected if the quotient topological space G/G 0G/G_0 (of GG 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 GG be a locally compact almost connected topological group.

Then

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

Theorem

Let GG be a locally compact almost connected topological group.

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

(in which case, due to theorem 1, it is necessarily homeomorphic to a Euclidean space).

This is (Antonyan, theorem 1.2).

Remark

In particular, in the above situation the subgroup inclusion

KG 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. See also compact Lie group.

Lie groupmaximal compact subgroup
real general linear group GL(n,)GL(n, \mathbb{R})orthogonal group O(n)O(n)
its connected component GL(n,) 0GL(n,\mathbb{R})_0special orthogonal group SO(n)SO(n)
complex general linear group GL(n,)GL(n, \mathbb{C})unitary group U(n)U(n)
symplectic group Sp(2n,)Sp(2n,\mathbb{R})unitary group U(n)U(n)
complex symplectic group? Sp(2n,)Sp(2n,\mathbb{C})compact symplectic group Sp(n)Sp(n)
Narain group O(n,n)O(n,n)two copies of the orthogonal group O(n)×O(n)O(n) \times O(n)
unitary group U(p,q)U(p,q)U(p)×U(q)U(p) \times U(q)
special Lorentz/AdS etc. group SO(p,q)SO(p,q)SO(p)×SO(q)SO(p) \times SO(q)
Lorentz / AdS spin group Spin(q,p)Spin(q,p)Spin(q)×Spin(q)/{(1,1),(1,1)}Spin(q) \times Spin(q) / \{(1,1), (-1,-1)\}

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

nnreal form E n(n)E_{n(n)}maximal compact subgroup H nH_ndim(E n(n))dim(E_{n(n)})dim(E n(n)/H n)dim(E_{n(n)}/H_n )
2SL(2,)×SL(2, \mathbb{R}) \times \mathbb{R}SO(2)SO(2)43
3SL(3,)×SL(2,)SL(3,\mathbb{R}) \times SL(2,\mathbb{R})SO(3)×SO(2)SO(3) \times SO(2)117
4SL(5,)SL(5, \mathbb{R})SO(5)SO(5)2414
5Spin(5,5)Spin(5,5)(Sp(2)×Sp(2))/ 2(Sp(2) \times Sp(2))/\mathbb{Z}_24525
6E6(6)Sp(4)/ 2Sp(4)/\mathbb{Z}_27842
7E7(7)SU(8)/ 2SU(8)/\mathbb{Z}_213370
8E8(8)Spin(16)/ 2Spin(16)/\mathbb{Z}_2248128

Counterexamples

A maximal compact subgroup may not exist at all without the almost connectedness assumption. An example is the Prüfer group [1/p]/\mathbb{Z}[1/p]/\mathbb{Z} endowed with the discrete (00-dimensional) smooth structure. This is a union of an increasing sequence of finite cyclic groups, each obviously compact.

References

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)

Revised on August 14, 2013 08:31:55 by David Roberts (192.43.227.18)