$SU(N)$ subalgebras and Dynkin diagrams?

Question

Reading this paper (Handout VIII for the course  SYMMETRIES IN PHYSICS, Michael Flohr, Subgroups and Unified Theories), I had my ideas confused: it is said that we can take any $SU(N)$ with its related Dynkin diagram, separate the last dot from the whole diagram and obtain two diagrams made of $N-2$ dots and $1$ dot respectively. So my questions. Am I right thinking that:

I can associate an $SU(2)$ algebra to that single dot.
As a consequence I can write that $su(3) supset su(2)oplus su(2)$. Note that $su(3)$ Dynkin diagram has two dots, then two $su(2)$ algebras.
Why instead that single dot is always associated with an $U(1)$ algebra? For example $SU(5)rightarrow SU(3)times SU(2) times U(1)$.

Matteo Brini · Answer

Studying more in detail the subject I figured out the problem. Considering $SU(3)$, it is true that we can find two subgroups (both of them are $SU(2)$) that are contained in it. This of course doesn't mean that $SU(3)=SU(2)otimes SU(2)$, the two subgroups may overlap. $U(1)$ came out from that dot that we left out as it only needs a diagonal generator and no additionals creation-annihilation operators that may be already included in some other subgroup.

Answered by Matteo Brini on May 7, 2021

$SU(N)$ subalgebras and Dynkin diagrams?

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