# $SU(N)$ subalgebras and Dynkin diagrams?

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 $$SU(5)\rightarrow SU(3)\times SU(2) \times U(1)$$.
• 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. Jan 11, 2021 at 18:42
• Ok, thanks for the clarification. Do you think the paper i linked talks about extended Dynkin diagrams? Jan 11, 2021 at 18:52
• 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. Jan 11, 2021 at 18:54

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.