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I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some bosonic grading compatible with the Lie bracket; in fact, the image of my functor lands in Lie algebras generated by their degree-$1$ part.) The functor first takes any (unital, associative) monoid $G$ to is "monoid ring" $kG$, which is the vector space with basis $G$, with multiplication induced by the multiplication in $G$, and with comultiplication $g\mapsto g\otimes g$ for $g\in G$; it is a cocommutative (unital, associative, counital, coassiciative) bialgebra.

Second, let $A$ be any (unital, associative, counital, coassiciative) bialgebra. Then it has a distinguished augmentation ideal, and hence a distinguished filtration by powers of the augmentation ideal. The associated graded bialgebra is generated (as an algebra) by its degree-$1$ part, any everything in the degree-$1$ part is primitive; hence $\operatorname{gr} A$ is a universal enveloping algebra for a graded Lie algebra which is generated by its degree-$1$ part (at least in characteristic $0$).

So my functor takes a monoid $G$ to the graded Lie algebra $\mathfrak g$ such that $\operatorname{gr}(kG) = \mathcal U\mathfrak g$.

I learned in this question that, at least when $G$ is a group, this operation has a simple algebraic description, due to Quillen (1968). Namely, any group $G$ is filtered by its lower central series $G_1 = G \supseteq G_2 = [G,G_1] \supseteq G_3 = [G,G_2] \supseteq \dots$, and the associated graded group $\operatorname{gr} G$ is a graded abelian group and in fact a graded Lie ring (Lie algebra over $\mathbb Z$). The algebra I care about is $\mathfrak g = k \otimes_{\mathbb Z}\operatorname{gr}G$.

My question is for a geometric description of this functor. I have not succeeded in computing small nontrivial examples. A few are: if $G$ is abelian, then so is $\mathfrak g = k\otimes_{\mathbb Z}G$. If $G$ is a free group, I believe that $\mathfrak g$ is a free Lie algebra on the same generators. If $G = \langle x,y | x^2 = y^2 = 1\rangle$, then $\mathfrak g = 0$. If $G$ is the pure braid group, then $\mathfrak g$ is Bar Natan's Lie algebra of "infinitesimal braids". My first thought was that $\mathfrak g$ was formed from $G$ by "zooming out", maybe in a Gromov-Hausdroff sense, but the case of $\langle x,y | x^2 = y^2 = 1\rangle$ shows that that's not quite right (unless I made an error somewhere).

I would also be interested in a geometric description of the intermediate step that completes the monoid algbera $kG$ at its augmentation ideal (finding a Hopf algebra in some pro category), or even the functor $G \mapsto (kG) / (\bigcap I^n)$.

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arxiv.org/abs/0903.2307 may be of interest to you, in particular section 3. –  Mark Grant Mar 8 '11 at 10:19

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