Representations of classical Lie groups

In mathematics, the finite-dimensional representations of the complex classical Lie groups G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , O ( n , C ) {\displaystyle O(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} , can be constructed using the general representation theory of semisimple Lie algebras. The groups S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} are indeed simple Lie groups, and their finite-dimensional representations coincide with those of their maximal compact subgroups, respectively S U ( n ) {\displaystyle SU(n)} , S O ( n ) {\displaystyle SO(n)} , S p ( n ) {\displaystyle Sp(n)} .

Source: Wikipedia — Representations of classical Lie groups (CC BY-SA 4.0)

Representations of classical Lie groups

In mathematics, the finite-dimensional representations of the complex classical Lie groups G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , O ( n , C ) {\displaystyle O(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} , can be constructed using the general representation theory of semisimple Lie algebras. The groups S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} are indeed simple Lie groups, and their finite-dimensional representations coincide with those of their maximal compact subgroups, respectively S U ( n ) {\displaystyle SU(n)} , S O ( n ) {\displaystyle SO(n)} , S p ( n ) {\displaystyle Sp(n)} .

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Source: Wikipedia "Representations of classical Lie groups" · CC BY-SA 4.0

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