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 $SU(5)\rightarrow SU(3)\times SU(2) \times U(1)$.