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

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SUMMARY

A Lie Group G is compact if and only if every finite complex representation of it is unitary. This theorem is well-established in Group Theory, confirming that the compactness of a Lie Group directly correlates with the unitarity of its finite representations. Additionally, it is affirmed that if a Lie Group is not compact, then all its irreducible unitary representations are indeed infinite-dimensional. These conclusions are critical for understanding the relationship between compactness and representation theory in Lie Groups.

PREREQUISITES
  • Understanding of Lie Groups and their properties
  • Familiarity with representation theory, specifically finite complex representations
  • Knowledge of unitary representations and their significance
  • Basic concepts of compactness in topological spaces
NEXT STEPS
  • Study the implications of compactness in Lie Groups
  • Explore the relationship between irreducible representations and dimensionality
  • Investigate examples of non-compact Lie Groups and their representations
  • Learn about the role of unitary representations in quantum mechanics
USEFUL FOR

Mathematicians, physicists, and students studying Group Theory, particularly those focusing on Lie Groups and their representations.

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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