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This question already has an answer here:

According to https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory:

Yang–Mills theory is a gauge theory based on [...] any compact, semi-simple Lie group.

But the symmetry group of Standard Model, i.e. ${SU(3)}\times{SU(2)}\times{U(1)}$, is not semisimple, due to the $U(1)$ factor, right? What am I missing here?

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marked as duplicate by Qmechanic Mar 13 '17 at 6:09

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    $\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/52452/2451 $\endgroup$ – Qmechanic Mar 12 '17 at 18:40
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    $\begingroup$ You're missing nothing, it's not semisimple, having an Abelian factor is a special case that's not included in the Wikipedia definition of a Yang-Mills theory. What is the physics question here? $\endgroup$ – ACuriousMind Mar 12 '17 at 18:40
  • $\begingroup$ "On what kind groups of Lie groups are Yang-Mills theories based?" $\endgroup$ – user3257624 Mar 12 '17 at 18:44