# Products of Lie-Groups versus Lie-Group Extensions in Physics

The Standard Model of elementary particle physics is a gauge theory based on the Lie group $$U(1) \times SU(2) \times SU(3)$$.

From the mathematical perspective I read that:

1. Simple Lie groups have been classified.

2. The problem of classifying group extensions is hard (even for finite groups). For finite groups one can classify all possible (nonisomorphic) groups of small order by working very hard.

My physical question is now: In some sense the product $$U(1) \times SU(2) \times SU(3)$$ is a minimal extension of the factors $$U(1)$$, $$SU(2)$$, $$SU(3)$$, but I could presumably also take for the gauge group $$G$$ some semi-direct products or even more general extensions.

1. What would be the physics of such gauge groups $$G$$, if they are not direct products?

2. Is the classification of the possible extensions containing $$U(1)$$, $$SU(2)$$, $$SU(3)$$ tractable? (I searched the literature on this somewhat, but I found results only for Lie Algebras only up to dimension 4 or 5)

Remark: I do not think this is necessarily a GUT question, because I do not want to obtain $$U(1) \times SU(2) \times SU(3)$$ by breaking down some larger group. I would also like to know if there are extensions $$G$$ of these components which are in some sense close to the direct product.

• Slightly related: physics.stackexchange.com/q/301468 – alphanum Feb 1 '19 at 14:38
• It is in a sense dual to the search of a GUT: in a GUT one searches for a group of which the standard model group is a subgroup, whereas you are looking for a group the standard model group is a quotient of – doetoe Feb 1 '19 at 19:17
• @doetoe: This is not what I want. Think of the difference between the Lorentz group ${\mathscr L} \ltimes {\mathbb R^4}$ and the direct product ${\mathscr L} \times {\mathbb R^4}$. I do not think one of those is a quotient of the other. – alphanum Feb 1 '19 at 19:23
• Now thinking of it, I believe it could perhaps mean physically something like introducing an additional electric charge for gluons with an independent coupling. – alphanum Feb 1 '19 at 19:26
• I see, you are not looking for extensions of the standard model group, but you want to study general extensions of some of its factors by the others – doetoe Feb 1 '19 at 19:36