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How to prove that SU(3) is compact?I have no idea how to do this . And What is the significance of The compactness of SU(3) on the quark model???
Can't you just show that its closed and bounded in R^9?
Jamma has exactly the right plan. Here are hints.
Bounded: What do you know about the magnitudes of entries in a unitary matrix?
Closed: Determinant is a polynomial function whose variables are matrix entries.
How to prove that SU(3) is compact?I have no idea how to do this . And What is the significance of The compactness of SU(3) on the quark model???
Unfortunately no , My knowledge in Topology is rudimentary .
I may be wrong, but isn't the condition for determinant =1 enough to prove that the group's manifold is homemorphic to a sphere in R^(2n+1), thus compact ? And this should be valid for al su(n), regardless of n ?
Thanks jamma , Regarding the first question I have Computed the sum of the moduli of the elements of the unitary matrix I found it to be 3 . So I guess if the SU(3) matrices are living in R^9 they occupy the Surface of an 8-sphere whose radius is 3. So I conclude that it is bounded in R^9 . Is this right?
Yes, I believe that is correct (although I'd need to check- so there is an obvious action of SU(3) on the 5 sphere embedded in C^3, right? And the stabilizer of a point will locally look like a way of rotating the 5 dimensional space, so will correspond to SU(2) which is a 3 sphere?) Could you explain to me more simply why this is true?
It certainly isn't the 8 sphere though- that's not even paralizable (non of the even dimensional spheres are). In fact, you can't put a group structure on any of the spheres except for dimension 0,1,3 and 7.
Oops sorry I have seen it as SO(3).
If I find a continous map that maps SU(3) to a single point such that it is also the inverse image of that point than it is closed because that point is closed.
The proof is a verbatim repeat of what I said above for the SU(3) case.
Level sets can be defined as the inverse image of any fixed point in any space not just reals.