# What do 1, 2, 3 represent in $\operatorname{U}(1)+\operatorname{SU}(2)+\operatorname{SU}(3)$?

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.
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