The second part asserts the complete reducibility of unitary representations of " G ".
2.
Especially in algebra and representation theory, " semi-simplicity " is also called "'complete reducibility " '.
3.
For example, Weyl's theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple.
4.
For example, the category is semisimple if " G " is a semisimple compact Lie group ( Weyl's theorem on complete reducibility ).
5.
One implication here is trivial, and the other, starting from a finite-dimensional invariant subspace, follows from complete reducibility of representations of " T ".
6.
The complete reducibility of finite-dimensional linear representations of compact groups, or connected semisimple Lie groups and complex semisimple Lie algebras goes sometimes under the name of " Weyl's theorem ".
7.
If \ mathfrak { g } is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and " V " is finite-dimensional, then " V " is semisimple ( Weyl's complete reducibility theorem ).
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