Maximal compact subgroup

2020 Mathematics Subject Classification: Primary: 22A05 [MSN][ZBL]

of a topological group $G$

A compact subgroup (cf. Compact group) $K \subset G$ which is not contained as a proper subgroup in any compact subgroup of $G$. For example, $K = \text{SO}(n)$ for $G = \text{GL}(n,\mathbf{R})$; $K=\{e\}$ for a solvable simply-connected Lie group $G$.

In an arbitrary group $G$ maximal compact subgroups need not exist (for example, if $G = \text{GL}(V)$, where $V$ is an infinite-dimensional Hilbert space), and if they do exist there may be non-isomorphic ones among them.

Maximal compact subgroups of Lie groups have been studied most. If $G$ is a connected Lie group, then any compact subgroup of $G$ is contained in some maximal compact subgroup (in particular, maximal compact subgroups must exist) and all maximal compact subgroups of $G$ are connected and conjugate to each other. The space of the group $G$ is diffeomorphic to $K \times \mathbf{R}^n$, therefore most of the topological questions about Lie groups reduce to the corresponding questions for compact Lie groups (cf. Lie group, compact).

References

[1]	E. Cartan, "La géométrie des groupes de transformations" J. Math. Pures Appl. , 6 (1927) pp. 1–119 Zbl 53.0388.01
[2]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) Zbl 0451.53038