Maximal compact subgroup

In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.

Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are not in general unique, but are unique up to conjugation – they are essentially unique.

Example

An example would be the subgroup O(2), the orthogonal group, inside the general linear group GL(2, R). A related example is the circle group SO(2) inside SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as any inner product has an associated orthogonal group, and the essential uniqueness corresponds to the essential uniqueness of the inner product.

Definition

A maximal compact subgroup is a maximal subgroup amongst compact subgroups – a maximal (compact subgroup) – rather than being (alternate possible reading) a maximal subgroup that happens to be compact; which would probably be called a compact (maximal subgroup), but in any case is not the intended meaning (and in fact maximal proper subgroups are not in general compact).

Existence and uniqueness

Existence

The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected locally compact group) admits maximal compact subgroups and that they are all conjugate to one another. For a semisimple Lie group this is a consequence of the Cartan fixed point theorem.

Uniqueness

Maximal compact subgroups of connected Lie groups are usually not unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element $g \in G$ such that^[1] gKg ^{− 1} = L – hence a maximal compact subgroup is essentially unique, and people often speak of "the" maximal compact subgroup.

For the example of the general linear group GL(n,R), this corresponds to the fact that any inner product on R^n defines a (compact) orthogonal group (its isometry group) – and that it admits a orthonormal basis: the change of basis defines the conjugating element conjugating the isometry group to the classical orthogonal group O(n,R).

Applications

Representation theory

Maximal compact subgroups play a basic role in the representation theory when G is not compact. In that case a maximal compact subgroup K is a compact Lie group (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier.

The operations relating the representation theories of G and K are restricting representations from G to K, and inducing representations from K to G,, and these are quite well understood; their theory includes that of spherical functions.

Topology

The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup K. To be precise, a connected Lie group is a topological product (though not a group theoretic product!) of a maximal compact K and a Euclidean space – $G = K \times R^d$ – thus in particular K is a deformation retract of G, and is homotopy equivalent, and thus they have the same homotopy groups. Indeed, the inclusion $K \hookrightarrow G$ and the deformation retraction $G \twoheadrightarrow K$ are homotopy equivalences.

For the general linear group, this decomposition is the QR decomposition, and the deformation retraction is the Gram-Schmidt process. For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.

See also

• Hyperspecial subgroup

Notes

1. ^ Note that this element g is not unique – any element in the same coset gK would do as well.

References

• Helgason, Sigurdur (1978), Differential Geometry, Lie groups and Symmetric Spaces, Academic Press, ISBN 978-0-12-338460-7 

• Onischchik and Vinberg, Lie Groups and Algebraic Groups, Springer Verlag 

• Malcev, A., On the theory of the Lie groups in the large, Mat.Sbornik N.S. vol. 16 (1945) pp. 163-189. 

• Iwasawa, K., On some types of topological groups, Ann. of Math. vol.50 (1949) pp. 507-558.