What U(n) or SU(n) or their multiplication isomorphic with SO(8)?

In summary, the conversation discusses the possibility of SO(8) being isomorphic to another group such as U(5)XU(2). However, the dimensions do not match and the Dynkin diagrams do not align, leading to the conclusion that SO(8) is not isomorphic to any other classical Lie group.
  • #1
thanhsonsply
3
0
I'm researching a problem relatived to group SO(8). I have searched many book of theory Group but I did'n find SO(8) isomorphic with what unitary group or their multiplication (SU(n), U(n) or SU(m)*U(n)). Please help me!
 
Physics news on Phys.org
  • #2
Are you sure there is one? Wikipedia's list of accidental isomorphisms only extends up to spin(6) (universal covering of SO(6)), and that's it.

A quick google search results that Spin(8) has SU(4) and SP(2) as subgroups, but that's about all I could find. Since SU(4) has dimension 15, and SO(8) has dimension 28, I can't see a way of making this work out on dimensional grounds. Perhaps someone more knowledgeable than me can answer this.

EDIT: After looking at Wikipedia more closely, I don't think this can be done. Wikipedia says SO(8) is simple, and so it can only be isomorphic to one factor of SU(n), and since SU(4) has dim 15, and SU(5) has dim 24, and SU(6) has dim 35, it doesn't seem like the dimensions match up.
 
Last edited:
  • #3
Thanks Matterwave! I have just read carefully on Wikipedia but I didn't find the answer. I thinks that:Maybe SO(8) isomorphic with U(5)*SU(2), they have the same dim 28. But I not sure. I haven't read it before.
 
  • #4
If it's isomorphic to U(5)xSU(2), then it wouldn't be a simple group by any means.
 
  • #5
thanhsonsply said:
Thanks Matterwave! I have just read carefully on Wikipedia but I didn't find the answer. I thinks that:Maybe SO(8) isomorphic with U(5)*SU(2), they have the same dim 28. But I not sure. I haven't read it before.

SO(8) is not isomorphic to U(5)XU(2). You might want to learn a bit about Lie groups and algebras, at least at the level of roots and weights.

The isomorphisms you are talking about all occur at low dimensions. One way to explain this is via the Dynkin diagrams: http://en.wikipedia.org/wiki/Dynkin_diagram. For example, the isomorphism between SO(6) and SU(4) is explained by the fact that they have the same Dynkin diagram. SO(8) = D4 has a unique Dynkin diagram and is not isomorphic to any other classical Lie group.
 
  • #6
SO(8) is simple, so the only chance for an isomorphism is another simple group; b/c the dimensions do not match there is none.
 

1. What is the difference between U(n), SU(n), and SO(8)?

U(n) is the unitary group, which consists of all n x n unitary matrices. SU(n) is the special unitary group, which consists of all n x n unitary matrices with determinant equal to 1. SO(8) is the special orthogonal group in 8 dimensions, which consists of all 8 x 8 orthogonal matrices with determinant equal to 1.

2. What does it mean for U(n) or SU(n) to be multiplication isomorphic with SO(8)?

Multiplication isomorphism refers to the fact that the group operation, or multiplication, of U(n) or SU(n) is equivalent to that of SO(8). This means that the same mathematical rules apply for combining elements within each group.

3. How are U(n) and SU(n) related to SO(8) in terms of their structure?

U(n) and SU(n) are both related to SO(8) in terms of their Lie algebra, which is a mathematical structure that describes the properties of a group. Specifically, the Lie algebra of U(n) and SU(n) are isomorphic to the Lie algebra of SO(8), meaning they have the same underlying structure.

4. What is the significance of the isomorphism between U(n) or SU(n) and SO(8)?

The isomorphism between U(n) or SU(n) and SO(8) has important implications in physics and mathematics. It allows for the application of techniques and concepts from one group to another, making it easier to study and understand their properties.

5. Is the isomorphism between U(n) or SU(n) and SO(8) a general rule for all groups?

No, the isomorphism between U(n) or SU(n) and SO(8) is a special case and does not hold for all groups. It is a result of the specific structure and properties of these groups and their Lie algebras.

Similar threads

A What is the difference between groups SU(n) and SO(n,C)?
    • Linear and Abstract Algebra
Replies
12
Views
3K
What is SU*(N)? Definition and Explanation
    • Linear and Abstract Algebra
Replies
1
Views
2K
I Spin group SU(2) and SO(3)
    • Linear and Abstract Algebra
Replies
11
Views
2K
MHB Prove that CnXCm isomorphic to Cgcd(m,n)XClcm(m,n)
    • Linear and Abstract Algebra
Replies
3
Views
1K
A Infinite-Dimensional Lie Algebra
    • Linear and Abstract Algebra
Replies
4
Views
936
I Adjoint Representation Confusion
    • Linear and Abstract Algebra
Replies
3
Views
2K
I Representation Theory clarification
    • Linear and Abstract Algebra
Replies
1
Views
1K
A Similarity transformation changing the determinant to 1
    • Linear and Abstract Algebra
Replies
20
Views
968
A Structure of modulo-integer matrix group GL(r,Z(n))?
    • Linear and Abstract Algebra
Replies
17
Views
4K
Connection between SU(2) and SO(3)
    • Linear and Abstract Algebra
Replies
8
Views
6K

Hot Threads

Recent Insights

Back
Top