Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) such that category of conilpotent reps of UC(T) is equivalent to the category of conilpotent reps of T. It can be constructed explicitly as $$\operatorname{colim} \,(U(T^\*)/I^k)^\*$$


Here $U(T^\*)$ is a usual enveloping algebra of a dual complete pro-finite dimensional pronilpotent Lie algebra $T^\*$. $I$ is its augmentation ideal.




---


There's a canonical filtration on any coaugmented conilpotent coalgebra C, with terms being kernels of maps $\bar{\Delta}^k : C \to (C/1)^{\otimes k}$. Let's call that filtration $J\_k UC$




---


There's an increasing filtration $V'\_iT$ by Lie subcoalgebras on every conilpotent Lie coalgebra, corresponding to a lower central filtration by ideals on the dual (complete, ..., etc) Lie algebra. Inclusions of filtration terms give us a **possibly** different filtration $V\_i UC(T)$; its terms are images of homomorphisms $UC(V'\_i T) \to UC(T)$, obtained by functoriality of $UC$. (Sidenote: those homomorphisms are usually injective, at least when T is flat over base ring they are.)


If I'm not mistaken, it's always true that $J\_i \subseteq V\_i$.


*Q: Is it true that $J\_i = V\_i$?*



Two (or less) filtrations on coenveloping coalgebra
Yes, both of those filtrations should be equal, at least when the ground field has characteristic $0$. I don't know if there is a good standard reference in the literature, but I did have to grapple with the dual version of this problem in a paper I wrote a couple of years back (see the appendix of arXiv:1909.05734, especially sections A.3 and A.4.1).


The result we ended up proving there concerned pro-nilpotent Lie algebras, i.e. Lie algebra objects in $\mathrm{pro}{-}\mathbf{Vec}$ (the pro-category of finite-dimensional vector spaces) whose descending central series is separated. Since $\mathrm{pro}{-}\mathbf{Vec}$ is dual to the category of all vector spaces, we know that the category of pro-nilpotent Lie algebras is dual to the category of conilpotent Lie coalgebras. Every pro-nilpotent Lie algebra $L$ has an associated completed universal enveloping algebra $U(L)$, which is a cocommutative Hopf algebra object in $\mathrm{pro}{-}\mathbf{Vec}$ which is pro-nilpotent (meaning that the filtration of $U(L)$ by powers of its augmentation ideal is separated).


From this dual perspective, the main result we want is then the following.


**Theorem:** For any pro-nilpotent Lie algebra $L$ (in characteristic $0$), there is a bijection between increasing filtrations on $L$ (by subobjects in $\mathrm{pro}{-}\mathbf{Vec}$, compatible with the Lie bracket, and indexed by non-positive integers) and increasing filtrations on $U(L)$ (by subobjects in $\mathrm{pro}{-}\mathbf{Vec}$, compatible with the Hopf algebra operations, and indexed by non-positive integers). This bijection sends a filtration on $L$ to the induced filtration on $U(L)$, and sends a filtration on $U(L)$ to the restriction of this filtration along the inclusion $L\hookrightarrow U(L)$. In particular, the descending central series on $L$ induces the filtration on $U(L)$ by powers of its augmentation ideal (since these are the smallest possible filtrations in either case).


This result is a consequence of a suitably completed and enriched version of the Milnor--Moore Theorem, which we proved as Theorem A.3.6. This says that if $\mathcal C$ is a $\mathbb Q$-linear complete tensor category, then there is an equivalence of categories between the category of pro-nilpotent Lie algebra objects in $\mathcal C$ and the category of pro-nilpotent cocommutative Hopf algebra objects in $\mathcal C$. This equivalence is given in one direction by $L\mapsto U(L)$, and in the other by $H\mapsto H^{\mathrm{prim}}$, where $H^{\mathrm{prim}}$ denotes the Lie algebra of primitive elements.


This can be applied to the category $\mathcal C$ of increasing non-positively filtered objects in $\mathrm{pro}{-}\mathbf{Vec}$, which tells us that there is an equivalence of categories between non-positively filtered pro-nilpotent Lie algebras and non-positively filtered pro-nilpotent cocommutative Hopf algebras, given by $L\mapsto U(L)$ with its induced filtration, and $H\mapsto H^{\mathrm{prim}}$ with its induced filtration. This in particular gives the claimed bijection between filtrations on $L$ and on $U(L)$.




---


**Some remarks**


* This feels like overkill to prove this particular result, and perhaps there is an easier way. But I don't think I found one when I thought about this.
* This idea of proving theorems for Lie algebra objects enriched in certain categories comes from Fresse's *Homotopy Theory of Operads* book. Though the results there don't seem to be directly applicable, since Fresse discusses these enriched results in categories with well-behaved colimits, which $\mathrm{pro}{-}\mathbf{Vec}$ does not have.


