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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:

  1. I can associate an $SU(2)$ algebra to that single dot.
  2. 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.
  3. Why instead that single dot is always associated with an $U(1)$ algebra? For example $SU(5)\rightarrow SU(3)\times SU(2) \times U(1)$.
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    $\begingroup$ I think you may be conflicting extended and unextended Dykin diagrams here. A single dot is the unextended diagram of SU(2), but the extended diagram has two dots (and four edges, if memory serves me right). The extended diagram of SU(N) has N dots, forming a circle. Dynkin's method of maximal subalgebras involves both extended and unextended diagrams. You may want to read about this method in a more specific source, a book on simple Lie algebras. $\endgroup$
    – AccidentalFourierTransform
    Jan 11, 2021 at 18:42
  • $\begingroup$ Ok, thanks for the clarification. Do you think the paper i linked talks about extended Dynkin diagrams? $\endgroup$
    – Matteo Brini
    Jan 11, 2021 at 18:52
  • $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic ♦
    Jan 11, 2021 at 18:54

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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.

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