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

In summary, Group Theory is a branch of mathematics that studies the structures and properties of groups, which are sets of elements that follow specific operations. The two theorems of Group Theory are Lagrange's Theorem and Cayley's Theorem, which have practical applications in fields such as physics, chemistry, and computer science. Key concepts in Group Theory include group operations, elements, identity, and isomorphism. While Group Theory can be challenging, with proper study and practice, anyone can understand its fundamentals and applications.
  • #1
Andre' Quanta
34
0
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?
 
  • #3
If a Lie group is not compact is it true that all its irreducibile unitary representations are infinite dimensional?
 

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

1. What is Group Theory?

Group Theory is a branch of mathematics that studies the properties and structures of groups, which are sets of elements that follow a specific set of operations. These operations can include addition, multiplication, and composition.

2. What are the two theorems of Group Theory?

The two theorems of Group Theory are the Lagrange's Theorem and the Cayley's Theorem. Lagrange's Theorem states that the order of a subgroup of a finite group must divide the order of the group. Cayley's Theorem states that every group can be represented as a subgroup of a symmetric group.

3. How are these theorems used in practical applications?

The theorems of Group Theory have many practical applications in different fields, such as physics, chemistry, and computer science. For example, they are used in cryptography to create secure communication systems and in quantum mechanics to understand the symmetries of particle interactions.

4. What are some key concepts in Group Theory?

Some key concepts in Group Theory include group operations, group elements, group identity, inverse elements, subgroup, and group isomorphism. These concepts help to define the properties and structures of groups and are essential in understanding the two theorems of Group Theory.

5. Is Group Theory a difficult subject to understand?

Group Theory can be a challenging subject for some people, as it involves abstract mathematical concepts and requires a strong foundation in algebra. However, with proper study and practice, anyone can understand the basics of Group Theory and its applications in various fields.

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