Cohomology and Extensions
topology (point-set topology)
see also algebraic topology, functional analysis and homotopy theory
nice topological space
Kolmogorov space, Hausdorff space, regular space, normal space
compact space, paracompact space, compactly generated space
connected space, locally connected space, contractible space, locally contractible space
topological vector space, Banach space, Hilbert space
empty space, point space
discrete space, codiscrete space
circle, torus, annulus
loop space, path space
Cantor space, Sierpinski space
long line, line with two origins
For a topological group a compact subgroup is a topological subgroup which is a compact group.
A compact subgroup is called maximal compact if every compact subgroup of is conjugate to a subgroup of .
If it exists then, by definition, it is unique up to conjugation.
See for instance (Hofmann-Morris, def. 4.24).
This is due to (Malcev) and (Iwasawa). See for instance (Stroppel, theorem 32.5).
This is (Antonyan, theorem 1.2).
For Lie groups
The following table lists some Lie groups and their maximal compact Lie subgroups. See also compact Lie group.
|Lie group||maximal compact subgroup|
|real general linear group ||orthogonal group |
|its connected component ||special orthogonal group |
|complex general linear group ||unitary group |
|complex special linear group ||special unitary group |
|symplectic group ||unitary group |
|complex symplectic group? ||compact symplectic group |
|Narain group ||two copies of the orthogonal group |
|unitary group |
|special Lorentz/AdS etc. group |
|Lorentz / AdS spin group |
The following table lists specifically the maximal compact subgroups of the “-series” of Lie groups culminating in the exceptional Lie groups .
|real form ||maximal compact subgroup |
A maximal compact subgroup may not exist at all without the almost connectedness assumption. An example is the Prüfer group endowed with the discrete (-dimensional) smooth structure. This is a union of an increasing sequence of finite cyclic groups, each obviously compact.
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)