Two (or less) filtrations on coenveloping coalgebra

Question

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.

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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 $J_k UC$

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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 $J_i = V_i$?

Answer 1

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 $U(L)$.

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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.