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zahero_2007
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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?
Jamma said:Can't you just show that its closed and bounded in R^9?
Tinyboss said: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.
zahero_2007 said: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?
zahero_2007 said:Unfortunately no , My knowledge in Topology is rudimentary .
dextercioby said: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 ?
zahero_2007 said: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?
Jamma said: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.
Sina said:Oops sorry I have seen it as SO(3).
If I find a continuous 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.
Sina said:Level sets can be defined as the inverse image of any fixed point in any space not just reals.
SU(3) is a special unitary group of 3x3 complex matrices with determinant equal to 1. It is a fundamental group in mathematics and has important applications in physics and quantum mechanics.
Proving that SU(3) is compact is important because it is a necessary step in understanding its properties and applications. Compact groups have well-defined topological and algebraic structures, which make them easier to study and apply in various fields of mathematics and science.
There are several ways to prove that SU(3) is compact. One approach is to show that every sequence in SU(3) has a convergent subsequence. Another approach is to use the fact that SU(3) is a closed and bounded set in the space of 3x3 complex matrices, and therefore must be compact.
SU(3) has important applications in physics, specifically in the study of quantum mechanics and the Standard Model of particle physics. It is also used in other branches of mathematics, such as representation theory and Lie groups.
Yes, an example of an SU(3) matrix is the Gell-Mann matrix, which is commonly used in the study of quarks and the strong nuclear force. It is a 3x3 complex matrix with special properties that make it a member of the SU(3) group.