Yang-Mills algebra and lower central series of surface groups Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "Yang-Mills Algebra", arXiv:0206205 by Connes and Dubois-Violette, they construct the Yang-Mills algebra $\mathcal{A}$ and study some of its properties. On p. 7, they mention that $\mathcal{A}$ is the universal enveloping algebra of a certain graded Lie algebra $\bigoplus_{j \geq 1} \mathfrak{g}_j$, and give an explicit formula for the dimensions, which begins:
$$
4, \mathbf{6}, 16, 45, 144, 440, \dots
$$
This is sequence A072279 (In fact, I think the bold 6 in their paper and the OEIS entry is a typo, as their own formula yields 5). EDIT: No, as @DamienC points out, it seems as if the 6 is correct.
Second, I was recently looking at the ranks of the lower central factors of the surface group $S_2 = \langle a, b, c, d \mid [a,b][c,d]=1 \rangle$, i.e. the factor groups in the lower central series of $S_2$. Curiously, I noticed that this is given by the same formula as in the case above (though with a 5 instead of a 6). There are known ways to get there in high-powered ways, as e.g. in this answer using Koszul duality (which I do not understand). Furthermore, this question and the answer to it are of course relevant, especially given the paper linked in there by Papadima and Yuzvinsky. So I suppose everything on this side seems fairly well understood.
My question is then:
Q. Is there a deeper connection between surface groups and the (Lie algebra associated to the) Yang-Mills algebra $\mathcal{A}$?
Of course, the answer might be "no, the dimensions just satisfy the same relation", or "yes, by general facts about Koszul algebras", either of which I suppose I would be satisfied with, too. I've added the reference-request tag, just in case.
 A: This is a long comment, rather than an answer.
As far as I understand, on the one hand we have the Lie algebra appearing in the paper of Connes and Dubois-Violette, that is the graded Lie algebra $\mathfrak{g}$ generated by $x_0,x_1,x_2,x_3$ (in degree $1$) with cubic relations:
$$
[x_i,[x_i,x_l]]+[x_j,[x_j,x_l]]+[x_k,[x_k,x_l]],\quad\mathrm{with}~\{i,j,k,l\}=\{0,1,2,3\}.
$$
Dimensions of the graded pieces of $\mathfrak{g}$ are given by the sequence A072279 in the OEIS:
$$
4,6,16,45,144,440,\dots
$$
$$
~
$$
On the other hand, we have the Lie algebra $\mathfrak{L}$ generated by $a,b,c,d$ (in degree $1$) with quadratic relation:
$$
[a,b]+[c,d]=0. 
$$
You seem to claim that dimensions of the graded pieces of $\mathfrak{L}$ are also given by the sequence A072279, but with a 5 instead of a 6:
$$
4,5,16,45,144,440,\dots
$$
I couldn't find a reference for this [actually, the sequence with a 5 instead of a 6 doesn't appear in the OEIS].
$$
~
$$
A naive way to understand such a coincidence of dimensions would be to find a surjective graded lie algebra morphism $\mathfrak{g}\to\mathfrak{L}$ that has one-dimensional kernel that sits in degree $2$.
I don't know if this is a reasonnable thing to ask or not.
