# “Exactness” of operadic cohomology

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.)

## 1 Answer

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.