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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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$U(n)$ is the unitary group of $n\times n$ matrices. And $SU(n)$ is the special unitary group of $n\times n$ matrices. Should the + be a $\times$? – Joe Johnson 126 Feb 10 '11 at 14:23
Are you talking about Lie groups or Lie algebras? – t.b. Feb 10 '11 at 14:52
i guess Lie groups as i am not sure whether it is in gauge theory, how dimension 1x1 times 2x2 times 3x3? or U is matrix 1x2 times SU 2x3 times SU 3x3? – Agul Feb 11 '11 at 1:11
I don't quite understand what you're asking. – Qiaochu Yuan Feb 11 '11 at 2:39

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 ;-) – Jason DeVito Feb 10 '11 at 18:13

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