Maximal compact subgroup

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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.

A maximal compact subgroup does not always exist for a general Lie group, but does exist for a semisimple Lie group, and they play an important role in their theory. Maximal compact subgroups are not in general unique, but are unique up to conjugation – they are essentially unique.

Contents 1 Example 2 Definition 3 Existence and uniqueness 3.1 Existence 3.2 Uniqueness 4 Applications 4.1 Representation theory 4.2 Topology 5 See also 6 Notes 7 References

[edit] 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.

[edit] 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).

[edit] Existence and uniqueness

[edit] Existence

Maximal compact subgroups may not exist for a general Lie group G, but always exist for a semisimple Lie group; this is a consequence of the Iwasawa decomposition, which is a stronger result. It is also true that any complex reductive group has a maximal compact subgroup.

[edit] Uniqueness

Maximal compact subgroups are not unique unless the group G is a semidirect product of a compact group and a contractible group, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element such that^[1] gKg ^{− 1} = L – hence a maximal compact subgroup is essentially unique, and people often speak of "the" maximal compact subgroup.

This non-uniqueness is because given any element that is not in K, conjugation by h gives a subgroup hKh ^{− 1} which is also a maximal compact subgroup. Thus K is unique if and only if hKh ^{− 1} = K, in which case K is normal. By the Iwasawa decomposition, K has a transversal, and thus the group splits.

For the example of the orthogonal group, this corresponds to the fact that any inner product defines a (compact) orthogonal group – and that conjugating by an element of the general linear group that is not orthogonal transforms the inner product.

[edit] Applications

[edit] 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.

[edit] Topology

The algebraic topology of the semisimple groups is also largely carried by a maximal compact subgroup K. To be precise, a semisimple Lie group is a product of a maximal compact K and a contractible space – – 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 and the deformation retraction are homotopy equivalences.

For the orthogonal group as a maximal compact in 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.

[edit] See also

• Hyperspecial subgroup

[edit] Notes

[edit] References

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

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