Categorial connection between SU(2) and SU(3)

Question

I recall reading a proposition that there is a connection in terms and language of category theory between $SU(2)$ and $SU(3)$ and no other two ranks including each of the two I mentioned have a categorial connection of this kind, if at all.

My aim, to be clear, is to approach $SU(3)$ using $SU(2)$ which is easy to study and comprehend, and I cannot say the same about the 3rd rank group.

Any lead, answer, or link would be appreciated.

Answer 1

Important connections between $SU(2)$ and $SU(3)$ would undoubtedly be mentioned in the wiki on the topic. While their individual importance in physics is outlined there, I do not see any strong connection between the two, categorical or not.

There is, towards the bottom, a relationship $SU(n)\supset SU(p)\times SU(n-p)\times U(1)$ where $1<p<n$, which would be in some sense unique for $n=3$ given those constraints. But if we admit this relationship as relevant then the idea that and no other two ranks including each of the two I mentioned have a categorial connection of this kind goes out the window.

There is, of course, a famous connection between $SU(2)$ and $SO(3)$ but that is not what you've asked.

To make any more progress, we're probably going to need more hints than a half-remembered statement about the topic.

Answer 2

The generators of the Quaternions $i,j,k$ can correspond to Pauli matrices multiplied to close as a Lie Algebra for $SU(2)$, as anti-Hermitian.

$U(3)$ has $9$ generators and tensor product $SU(2)\times SU(2) \equiv U(3)$

Every finite representation can be multiplied by a phase to become an element of $SU(3)$. So

$$SU(3)\subset \frac{SU(2)\times SU(2)}{U(1)}$$

Perhpas this is an equality.

Couplets of Pauli matrices are always traceless, but need to multiply by an imaginary unit, and that representation has that

$$su(3) \subset su(2)\times isu(2) $$

This is consistent with the above, and this is a connection between the algebras, and hence between the groups.