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. 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. 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. 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). 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 uniqueness is a consequence of the Cartan fixed point theorem, which asserts that if a compact group acts by isometries on a complete simply connected negatively curved Riemannian manifold then it has a fixed point. 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 ∈ G such that 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 Rn defines a (compact) orthogonal group (its isometry group) – and that it admits an orthonormal basis: the change of basis defines the conjugating element conjugating the isometry group to the classical orthogonal group O(n, R). For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in Borel (1950) and Helgason (1978). Cartier (1955) and Hochschild (1965) discuss the extension to connected Lie groups and connected locally compact groups. For semisimple groups, existence is a consequence of the existence of a compact real form of the noncompact semisimple Lie group and the corresponding Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding Riemannian symmetric space G/K has negative curvature and Cartan's fixed point theorem. Mostow (1955) showed that the derivative of the exponential map at any point of G/K satisfies |d exp X| ≥ |X|. This implies that G/K is a Hadamard space, i.e. a complete metric space satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the Bruhat-Tits fixed point theorem. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its circumcenter. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.