Group Theory sub algebra of unitary group of U(6) group.

So ##O(3)## is a subgroup of ##U(3)## though, which in turn is a subgroup of ##SU(3)##.But it is hard to say what the rest is about without context.
  • #1
Vikas Katoch
3
0
TL;DR Summary
three sub algebra of Unitary group (6) as 1. U(5).
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.
three sub algebra of Unitary group (6) as 1. U(5) .
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.
 

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  • #2
Vikas Katoch said:
Summary:: three sub algebra of Unitary group (6) as 1. U(5).
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.

three sub algebra of Unitary group (6) as 1. U(5) .
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.
Sub groups, not sub algebras. Of course we need to specify each inclusion separately. And it is not really an inclusion in the sense of subsets, they are embeddings in the sense of monomorphisms, injective group homomorphisms.

E.g. ##O(n) \hookrightarrow O(n+1)## can be done by ##A\longmapsto \begin{bmatrix}A&0\\0&1\end{bmatrix}##.
 
  • #3
Unitary group of order six U(6) having three sub groups.
How these chains are produced. sheet attached.
 

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  • #5
Unitary group of order six U(6) having three sub groups.

Type 1. U(6)⊃ U(5) ⊃ O(5)⊃ O(3) ⊃ O(2)

Type 2. U(6) ⊃SU(3) ⊃ O(3) ⊃ O(2)

Type 3. U(6) ⊃O(6) ⊃ O(5) ⊃ O(3) ⊃ O(2)

How these chains are produced.
 
  • #6
Vikas Katoch said:
Unitary group of order six U(6) having three sub groups.

Type 1. U(6)⊃ U(5) ⊃ O(5)⊃ O(3) ⊃ O(2)

Type 2. U(6) ⊃SU(3) ⊃ O(3) ⊃ O(2)

Type 3. U(6) ⊃O(6) ⊃ O(5) ⊃ O(3) ⊃ O(2)

How these chains are produced.
I already told you in post #2.
 
  • #7
I don't think that ##O(3)## is a subgroup of ##SU(3)##. The former has elements of determinant ##\pm 1##, but ##SU(3)## only has elements of determinant ##1##.
 

1. What is a sub algebra of the unitary group of U(6) group?

A sub algebra of the unitary group of U(6) group is a subset of the unitary group that forms a closed algebraic structure under multiplication and contains the identity element. In simpler terms, it is a smaller group within the larger unitary group that shares similar properties.

2. How is a sub algebra of the unitary group of U(6) group related to group theory?

Group theory is the mathematical study of groups, which are sets of elements that follow certain algebraic rules. A sub algebra of the unitary group of U(6) group is a specific example of a group, and studying it can provide insights into the properties and behavior of larger groups.

3. What are the applications of studying a sub algebra of the unitary group of U(6) group?

Studying a sub algebra of the unitary group of U(6) group can have various applications in fields such as physics, chemistry, and computer science. It can be used to understand symmetries in physical systems, analyze molecular structures, and develop efficient algorithms for solving complex problems.

4. How is a sub algebra of the unitary group of U(6) group different from the unitary group itself?

A sub algebra of the unitary group of U(6) group is a smaller group that is contained within the unitary group. While the unitary group is infinite, the sub algebra is finite and has a specific structure and set of elements. The sub algebra inherits some properties from the unitary group, but it also has its own unique properties.

5. Is there a way to visualize a sub algebra of the unitary group of U(6) group?

Yes, a sub algebra of the unitary group of U(6) group can be visualized using group diagrams or Cayley diagrams. These diagrams represent the elements of the sub algebra and their relationships to each other through multiplication. They can help in understanding the structure and properties of the sub algebra.

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