# Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately.

A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra has three commuting generators. But I also read elsewhere that $SO(N)$ has rank $N(N-1)/2$, ie, equal to its number of generators, here 15. Similarly I see it stated that $SU(4)$ is rank 3 (as it better be, under some understanding, since $SO(6) \sim SU(4)$) but again I am more familiar with its rank being defined as the number of its generators, ie, $4^2-1=15$ here, $N^2-1$ for general $SU(N)$.

I suppose these are just two different uses of the word rank? Is one usage sloppy? (edit: It sounds like one usage is not just sloppy, but actually wrong)

Perhaps the number of generators is more properly called the group's dimension. But then there is also the dimension of a representation, which is not in general the same as the number of generators for the group. Correct? Like $SU(2)$ in its fundamental representation acts on a vector space of dimension 2, but it has 3 generators.

I know in some places the rank is specifically defined as the number of generators in the Cartan subalgebra. If that is the case, how else could one determine the rank? I wonder because the author states the rank of these groups is 3 and then deduces from that the number of Cartan generators.

Sorry for any sloppy terminology or general ignorance, mathematicians!

If there's some place where all this is gathered together and made clear, I'll happily take any references you might give.

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Rank is not the same as dimension. Rank is the dimension of a Cartan. Also, general warning about talking about Lie theory with mathematicians: don't mix up Lie groups and Lie algebras! We don't like it when physicists do that. – Qiaochu Yuan Apr 12 '13 at 0:12
Hi, Yuan! So in this reference docs.google.com/file/d/… when they say that the rank of SO(N) is N(N-1)/2 on p. 15, they are just wrong? That's certainly believable to me, I just wanted to be sure. – gn0m0n Apr 12 '13 at 3:53
Given that, how can one deduce the rank of a Lie group? I know the common formulas for what I guess is properly called the dimension of SU(N) and SO(N) but am not as familiar with finding ranks. Is it correct that SU(N) has rank N-1, while SO(2r) and SO(2r+1) have rank r? This is what I've been able to piece together. – gn0m0n Apr 12 '13 at 4:05
As for mixing up algebras and groups- I know, I know, it's a huge faux pas :) I actually do try and pay attention to the correct usage but I'm on unfamiliar ground here. I think the paper I was reading is not careful because he talks about the group $SO(6)$ and then its Cartan subalgebra rather than its subgroup or $so(6)$ and its subalgebra. And I realized I was not 100% sure whether technically the term rank applied only to groups or algebras or both. Apologies! I will learn. – gn0m0n Apr 12 '13 at 4:14
And is it correct that "dimension" applies both to a group (as the number of its generators) and to a representation of it (as the dimension of the vector space)? – gn0m0n Apr 12 '13 at 4:19