Максимальна компактна підгрупа

Максимальна компактна підгрупа K топологічної групи G — це компактний простір з індукованою топологією, максимальний серед всіх підгруп. Максимальні компактні підгрупи грають важливу роль в класифікації груп Лі і, особливо, в класифікації напівпростих груп Лі. Максимальні компактні підгрупи груп Лі в загальному випадку не єдині, але єдині з точністю до спряженості — вони є суттєво сполученими.

Приклад

Як приклад використовуємо підгрупу O(2), ортогональну групу всередині загальної лінійної групи GL(2, R). Пов'язаним прикладом є група кола SO(2) всередині групи SL(2, R). Очевидно, що SO(2) всередині групи SL(2, R) є компактною і не максимальною. Неєдиність цих прикладів можна бачити з того, що будь-який скалярний добуток має асоційовану ортогональную групу і суттєва єдиність відповідає істотній єдиності скалярного добутку.

Джерела

  • Бурбакі Н. Загальна топологія: Топологічні групи. Числа і пов'язані з ними групи і простори. — М. : Наука, 1969. — С. 392. — (Елементи математики)(рос.)
  • Borel, Armand (1950), Sous-groupes compacts maximaux des groupes de Lie (Exposé No. 33), Séminaire Bourbaki, т. 1[недоступне посилання з квітня 2019]
  • Cartier, P. (1955), Structure topologique des groupes de Lie généraux (Exposé No. 22), Séminaire "Sophus Lie", т. 1[недоступне посилання з квітня 2019]
  • Dieudonné, J. (1977), Compact Lie groups and semisimple Lie groups, Chapter XXI, Treatise on analysis, т. 5, Academic Press, ISBN 012215505X
  • Helgason, Sigurdur (1978), Differential Geometry, Lie groups and Symmetric Spaces, Academic Press, ISBN 978-0-12-338460-7
  • Hilgert, Joachim; Neeb, Karl-Hermann (2012), Structure and geometry of Lie groups, Springer monographs in mathematics, Springer, ISBN 0387847944

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