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

1. Feb 12, 2012

### thanhsonsply

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!

2. Feb 12, 2012

### Matterwave

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: Feb 12, 2012
3. Feb 12, 2012

### thanhsonsply

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. Feb 12, 2012

### Matterwave

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

5. Feb 12, 2012

### fzero

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. Feb 13, 2012

### tom.stoer

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.