Fake Proof for Dimension of $SO(n)$ (rotations)?

Because there are $\binom{n}{2}$ distinct planes spanned by the elements of any given orthonormal basis for $\mathbb{R}^n$, it seems to follow readily (because apparently planar rotations generate all rotations, cf. [1][2][3]) that $SO(n)$ has $\binom{n}{2}$ generators (as a group).

For example, given the plane spanned by $e_i$ and $e_j$, if $i < j$, then let the corresponding planar rotation be the one taking $e_i$ to $e_j$ (the other rotation taking $e_j$ to $e_i$ is just the group inverse).

This seems like a very simple, straightforward, intuitive, combinatorial, etc. argument for why the dimension of $SO(n)$ "should" be $\binom{n}{2}$. However all proofs of the fact I've seen are much more involved, which leads me to suspect a major flaw in the reasoning.

In particular, the argument seems to be implicitly making the following

Claim: The following numbers are the same:

• The number of generators (as a group) of a Lie group
• The dimension (as a manifold) of the Lie group
• The dimension (as a vector space) of the associated Lie algebra

The equality of the last two is largely tautological, to the extent that the definition of the dimension of a manifold is the vector space dimension of its tangent spaces, and the Lie algebra is the tangent space at the identity.

So obviously it is the purported equality of the first with the last two that is questionable. Any counterexamples, or if it's actually true then references to proofs, would be appreciated.

Related question: cf. Dimension of $SO_n(\mathbb{R})$

Solution 1:

The second and third numbers are the same as you point out, essentially due to the standard isomorphism $\mathfrak{g}\cong T_eG$ and the basic fact about manifolds $\dim(T_pM)=\dim(M)$.

The first, however, is never the same in dimension $>0$. in fact, all Lie groups of poisitive dimension are infinitely generated, if we define "generator" in the sense used for abstract groups. The confusion seems to arise from the term "generator", which is often used loosely in the context of Lie groups, to the extent that I would not recommend using the term at all without first providing a definition.

Often, (in physics especially,) "a set of (infinitesimal) generators of a Lie group $G$" just means "a basis of the group's Lie algebra $\mathfrak{g}$". Used in this sense, the number of generators is equal to the dimension, but these "generators" do not correspond to any kind of generating set in the group theoretic sense (since, again, positive-dimensional Lie groups are not finitely generated). Alternately "an (infinitesimal) generator of a Lie group $G$" sometimes simply means "an element of the Lie algebra $\mathfrak{g}$", in which case the "number of generators" isn't a particularly meaningful piece of information. I suspect the term "generator" is used because a basis of $\mathfrak{g}$, together with the structure constants associated to it, give a complete description of the local structure of the group $G$. This correspondence, however, is more complicated that the group-theoretic notion of a generating set.