| Maximal Compact Subgroup |
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| CATEGORIES ABOUT MAXIMAL COMPACT SUBGROUP | |
| topological groups | |
| lie groups | |
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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. Maximal compact subgroups may not exist, in a given ''G''. In the theory of Semisimple Lie Group s, maximal compact groups are shown to exist, and play a basic role in the Representation Theory when ''G'' is not compact. In that case a maximal compact subgroup ''K'' must be a Compact Lie Group , for which the theory is easier. Restricting Representations from ''G'' to ''K'', and Inducing Representations from ''K'' to ''G'', are basic operations, and quite well understood; their theory includes that of Spherical Function s. The Algebraic Topology of the semisimple groups is also largely carried by a maximal compact subgroup ''K''. This can be expressed by means of the Gram-Schmidt Process , concretely, or more abstractly by the Iwasawa Decomposition of ''G'', in which ''K'' occurs in a product with a Contractible subgroup. |
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