Is Every Finite Complex Representation of a Compact Lie Group Unitary?

AI Thread Summary
A Lie group G is compact if and only if every finite complex representation of it is unitary, which is a well-established theorem. The converse is also true: if every finite complex representation is unitary, then G must be compact. Additionally, a Lie group is compact if and only if every irreducible unitary representation is finite-dimensional. If a Lie group is not compact, it does not necessarily follow that all its irreducible unitary representations are infinite-dimensional, as some may still be finite-dimensional. Understanding these relationships is crucial in the study of group theory and representation theory.
Andre' Quanta
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I am studying Group Theory at the moment and i am not sure about a theorem.
Is it true that a Lie Group G is compact if and only if every finite complex representation of it is unitary?
I know that is true the if, but what about the viceversa?
Same question.
Is it true that a Lie group is compact if and only if every irreducible representaion unitary is finite?
 
If a Lie group is not compact is it true that all its irreducibile unitary representations are infinite dimensional?
 

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