Is SU(3) always contains SU(2) groups?

In summary, the conversation discusses the relationship between different Lie groups, specifically SU(3), SU(2), and U(1). It is possible to find SU(2) as a subgroup of SU(3), and there are many subgroups of SU(3) such as SU(2) and U(1). The concept of compatibility between these groups is also mentioned, with the understanding that the tensor product of two independent groups contains one group as a subgroup.
  • #1
Ans
22
2
Hi,

I trying to understand. If there is non-trivial SU(3) group, is it always possible to find SU(2) as part of SU(3)?
And same question about SU(2) and U(1).
 
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  • #2
Sure, there are many ##SU(2)## subgroups of ##SU(3)## just like there are many ##O(2)## subgroups of ##O(3)##.
 
  • #3
Thanks.

And another question about same.
SU(3) seems as have less number of parameters than SU(3)xSU(2)xU(1).
If there is SU(3) group, is is possible to say it is compatible with SU(3)xSU(2)xU(1) because SU(3) always contains SU(2) subgroups, and SU(2) always contains U(1)?
 
  • #4
The group, ##G_1\times G_2## is the tensor product of two independent groups. ##G_1## is contained in ##G_1\times G_2## by the projection ##\pi : (g1,g2)\rightarrow g1##. The subgroup inclusion discussed in my previous reply is a subgroup of a different sort. (Mathematics is not my strongest subject) It would help to know what you mean by "compatible?"
 
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1. What is the relationship between SU(3) and SU(2) groups?

The SU(3) group is a higher-dimensional generalization of the SU(2) group. This means that the SU(2) group is a subgroup of SU(3) group, meaning that all elements of the SU(2) group are also elements of the SU(3) group.

2. Is it always true that SU(3) contains SU(2) groups?

No, this is not always true. While SU(2) is always a subgroup of SU(3), there are other subgroups of SU(3) that do not contain SU(2) groups. It ultimately depends on the specific structure and elements of the SU(3) group.

3. Can SU(3) and SU(2) groups coexist in the same system?

Yes, SU(3) and SU(2) groups can coexist in the same system. In fact, many physical systems involve both SU(3) and SU(2) groups, such as in quantum mechanics and particle physics.

4. How are SU(3) and SU(2) groups used in science?

SU(3) and SU(2) groups are used in various fields of science, particularly in theoretical physics and mathematics. These groups have applications in understanding the fundamental forces and particles in the universe, as well as in developing mathematical models and theories.

5. Are there any other related groups to SU(3) and SU(2)?

Yes, there are other related groups to SU(3) and SU(2), such as the SU(N) group which is a generalization of both SU(3) and SU(2) groups. There are also other groups that are closely related, such as the SO(N) and Sp(N) groups, which have important applications in physics and mathematics as well.

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