There are two somewhat widely known theorems which say

  • if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_{Harr}(A, A) \to H^*_{Hoch}(A, A)$ is injective

  • if $L$ is a Lie algebra in char $0$, then $H^*(L, L) \to H^*(L, UL)$ is injective

Are they related at all? Is it a part of more general phenomenon occuring for exact sequences of (probably Koszul) operads?

(Also, it's obvious that latter theorem does not care about grading at all, just because $L \to UL$ is split injection of modules. Is it true for former one? I gess not.)


The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as an $L$-module, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a sufficiently abstract version of the PBW theorem. This is much less well known and I haven't seen it written down anywhere.

(In particular the phenomenon is not something general for any sequence of Koszul operads - you really need a PBW theorem in the background.)

Here is how it goes. Via the map $\mathsf{Lie} \to \mathsf{Ass}$ we may consider $\mathsf{Ass}$ as a bimodule over the Lie operad. Recall that if $P$ is an operad, then a $P$-bimodule is the same thing as a $P$-algebra in the category of right $P$-modules, so we may think of $\mathsf{Lie} \to \mathsf{Ass}$ as a morphism of Lie algebras in a certain tensor category. In fact we may identify $\mathsf{Ass}$ with the universal enveloping algebra of $\mathsf{Lie}$ in this category, considered as a Lie algebra. By a general form of the PBW theorem (for Lie algebras in abstract symmetric tensor categories over a field of characteristic zero) it then follows that there exists a splitting $\phi \colon \mathsf{Ass} \to \mathsf{Lie}$ in the category of modules over the Lie operad, considered as an algebra over itself. Unraveling this, this means that $\phi$ is a map of infinitesimal bimodules.

Explicitly, the fact that $\phi$ is a morphism of infinitesimal bimodules means that we have two commutative pentagons. The first says that the composition $$ \mathsf{Lie} \circ_{(1)} \mathsf{Ass} \to \mathsf{Ass} \circ_{(1)} \mathsf{Ass} \to \mathsf{Ass} \stackrel\phi\to \mathsf{Lie}$$ coincides with $$ \mathsf{Lie} \circ_{(1)} \mathsf{Ass} \stackrel{\mathrm{id}\circ_{(1)} \phi}\to \mathsf{Lie} \circ_{(1)} \mathsf{Lie} \to \mathsf{Lie}.$$ The other says the same thing except with $\mathsf{Lie}$ acting on $\mathsf{Ass}$ on the right.

Now let $A$ be a $C_\infty$-algebra. The $C_\infty$-structure is given by a Maurer--Cartan element $\mu$ in the pre-Lie algebra $\mathfrak g := \mathrm{Hom}_{\mathbb S}(\mathsf{coLie},\mathsf{End}_A)$, which is essentially the Harrison chain complex. The pre-Lie product $f \star g$ of two Harrison chains is given by the composition $$ \mathsf{coLie} \to \mathsf{coLie} \circ_{(1)} \mathsf{coLie} \stackrel{f \circ g}\to \mathsf{End}_A \circ_{(1)} \mathsf{End}_A \to \mathsf{End}_A.$$ We also have the associative version $\mathfrak h := \mathrm{Hom}_{\mathbb S}(\mathsf{coAss},\mathsf{End}_A)$ with analogously defined pre-Lie product. Via the map $\mathsf{coAss} \to \mathsf{coLie}$ we can think of $\mathfrak g$ as a pre-Lie subalgebra of $\mathfrak h$. Now the dual of $\phi$ induces a map $\phi^\ast \colon \mathfrak h \to \mathfrak g$, which is not in general a morphism of pre-Lie algebras. However, the fact that $\phi$ is a map of infinitesimal bimodules is exactly the condition needed for $\phi^\ast$ to satisfy a "projection formula": for $f \in \mathfrak g \subset \mathfrak h$ and $g \in \mathfrak h$ we have $\phi^\ast(f \star g) = f \star \phi^\ast(g)$ and $\phi^\ast(g \star f) = \phi^\ast(g) \star f$. Indeed if we write out the definitions of the pre-Lie product then the conditions we need for the projection formula to hold become exactly that we have a map of infinitesimal bi-comodules $\mathsf{coLie} \to \mathsf{coAss}$.

In particular we have the Harrison differential on $\mathfrak g$ given by $df = f \star \mu - (-1)^{\vert f\vert}\mu \star f$ and the analogous Hochschild differential on $\mathfrak h$. The previous paragraph says in particular that the splitting $\mathfrak h \to \mathfrak g$ is compatible with these differentials, so that Harrison chains are a direct summand of Hochschild chains.


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