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

  • #1

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.

Main Question or Discussion Point

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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Answers and Replies

  • #2
13,214
10,111
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}##.
 
  • #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
13,214
10,111
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
Infrared
Science Advisor
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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##.
 

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