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What do 1, 2, 3 represent in $\operatorname{U}(1)+\operatorname{SU}(2)+\operatorname{SU}(3)$?
If they are dimensions, how they can be added? or plus has another meaning?

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    I don't quite understand what you're asking.2011-02-11

1 Answers 1

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They indicate the dimension of the complex vector space on which the group acts. The unitary group $\text{U}(n)$ and the special unitary group $\text{SU}(n)$ both act on $\mathbb{C}^n$. The "plus" actually indicates the direct product $G \times H$ of groups $G, H$, which is the set of pairs $(g, h), g \in G, h \in H$ under pointwise multiplication. This is sometimes conflated with the direct sum, but in my opinion it is better to avoid the sum notation for non-abelian groups.

Edit: Ah, you might be talking about the Lie algebras $\mathfrak{u}(n), \mathfrak{su}(n)$, in which case the answer is still that $n$ is the dimension of the complex vector space on which the Lie algebra acts, except that the notion of an action is different.

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    Generally, lower case letters indicate the Lie algebras. Of course, this doesn't really effect the content of what you wrote, but I would add that + or $\oplus$ is often used as the direct sum operation for lie algebras (though I agree with avoiding the sum notation for non-abelian groups!). *edit* - I just noticed this is tagged Lie-groups and not Lie-algebras. Now I'm just really confused ;-)2011-02-10