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

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Discussion Overview

The discussion revolves around the isomorphism of the group SO(8) with unitary groups such as U(n) or SU(n), as well as their potential products. Participants explore theoretical aspects of group theory, specifically focusing on the properties and dimensions of these groups.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the isomorphism of SO(8) with unitary groups, noting a lack of information in group theory literature.
  • Another participant expresses skepticism about the existence of such an isomorphism, referencing Wikipedia's list of accidental isomorphisms and pointing out dimensional discrepancies between SO(8) and potential candidates like SU(4) and SU(5).
  • A different participant suggests that SO(8) might be isomorphic to U(5) x SU(2) due to matching dimensions, but acknowledges uncertainty.
  • Another reply counters the suggestion of U(5) x SU(2), stating that if SO(8) were isomorphic to such a product, it would not be a simple group.
  • Further clarification is provided regarding the simplicity of SO(8) and the implications for isomorphisms, emphasizing that the dimensions do not align with other simple groups.
  • A participant mentions the relevance of Dynkin diagrams in understanding the isomorphisms of Lie groups, specifically noting that SO(8) has a unique Dynkin diagram.

Areas of Agreement / Disagreement

Participants express differing views on the possibility of an isomorphism involving SO(8), with some asserting that no such isomorphism exists due to dimensional and simplicity considerations, while others propose potential candidates without consensus.

Contextual Notes

Participants highlight limitations in the existing literature and the need for a deeper understanding of Lie groups and their properties, particularly regarding roots and weights, which may affect the discussion of isomorphisms.

thanhsonsply
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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!
 
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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.
 
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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.
 
If it's isomorphic to U(5)xSU(2), then it wouldn't be a simple group by any means.
 
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.
 
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.
 

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