What is the center of SU(3) group

In summary, the center of SU(3) in the fundamental representation is generated by the matrix C = \alpha I_3, where \alpha = \exp(2\pi i/3) is a third root of unity. This means that the center consists of the matrices I_3 and \alpha I_3, which can be written as c1 = diag{1,1,1} and c2 = diag{\alpha, \alpha^2, \alpha^3}.
  • #1
Dear Every One,

In literatures on QCD confinement, I usually see the words ``center of group''.
It is defined to be the subgroup of some parent group and consists of elements which
commutes with all elements from the parent group. But what is the center of SU(3)
group? I need concrete answer as follows instead formal definition,

For SU(2) group, fundamental representation, the center consists the following
two matrices
c1=diag{1,1}, c2=diag{-1,-1}

What is the case for SU(3) group, fundamental representation?

Thank you very much!
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  • #2
The center of [tex]SU(n)[/tex] is [tex]\mathbb{Z}_n[/tex]. It is generated by

[tex]C = \alpha I_n,[/tex]

where [tex]\alpha = \exp(2\pi i/n)[/tex] is an [tex]n^\text{th}[/tex] root of unity. Note that for [tex]n=2[/tex], [tex]\alpha= -1[/tex], so we have elements [tex]I, -I[/tex].
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1. What does the term "center" mean in the context of SU(3) group?

The center of a group is the set of elements that commute with all other elements in the group. In other words, if an element from the center is multiplied by any other element in the group, the result will always be the same regardless of the order of multiplication.

2. How is the center of SU(3) group defined?

In SU(3) group, the center is defined as the set of elements that are scalar multiples of the identity matrix. This means that any element in the center of SU(3) can be written as a scalar multiple of the identity matrix, where the scalar is a complex number with unit magnitude.

3. Is the center of SU(3) group a subgroup?

Yes, the center of SU(3) group is a subgroup. It is a normal subgroup, which means that it is closed under conjugation by any element in the group and is also an abelian subgroup.

4. How does the center of SU(3) group affect the group's structure?

The center of SU(3) group plays a crucial role in determining the group's structure. It is a normal subgroup, which means that it can be used to define quotient groups. The center also affects the group's irreducible representations and its character table.

5. What is the significance of the center in SU(3) group theory?

The center of SU(3) group is significant in many areas of group theory. It is essential in understanding the group's structure, its subgroups, and its representations. The center is also used in the study of quantum chromodynamics, a fundamental theory in particle physics that uses SU(3) group to describe the strong interaction between quarks.

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