Cohomology of coproduct of dgLie algebras

Question

What results are known about the cohomology of the coproduct of two differential graded Lie algebras (with differential of degree $1$, of course)?

Namely, let $\mathfrak{g},\mathfrak{h}$ be dgLie algebras. The coproduct $\mathfrak{g}\sqcup\mathfrak{h}$ is a quotient of the free dgLie algebra over the direct sum of the underlying vector spaces (the identifications are generated by the relations $[x,y]_{\mathfrak{g}\sqcup\mathfrak{h}} = [x,y]_\mathfrak{g}$ whenever $x,y\in\mathfrak{g}$, and similarly for $\mathfrak{h}$). I would like to have a description of $H^\bullet(\mathfrak{g}\sqcup\mathfrak{h})$ in terms of $H^\bullet(\mathfrak{g})$, $H^\bullet(\mathfrak{h})$, and the (homology of) the inclusion maps.

Intuitively, I could think of something like $$H^\bullet(\mathfrak{g}\sqcup\mathfrak{h})\cong H^\bullet(\mathfrak{g})\sqcup H^\bullet(\mathfrak{h}),$$ where the second coproduct is taken in the category of $L_\infty$-algebras (which is probably the most natural place for cohomology of dgLie algebras). However, I wouldn't know how to try to prove it.

Any comment, idea or reference is welcome.

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Remark: A special case is known: Let $M=k\cdot x\oplus k\cdot y$ be the cochain complex (over the field $k$ of characteristic $0$) with $|x|+1 = |y|$ and $dx = y$. Denote by $Lie(M)$ the free Lie algebra over $M$. Then the inclusion map (of dgLie algebras) $$\mathfrak{g}\longrightarrow \mathfrak{g}\sqcup Lie(M)$$ is a quasi-isomorphism at the level of chain complexes. Would something like that be true when, say, $\mathfrak{h}$ above has zero cohomology?

Answer 1

My guess was wrong, and also the case where $\mathfrak{h}$ has trivial cohomology doesn't give what I thought:

Let $$\mathfrak{h} = k\cdot x\oplus k\cdot [x,x]$$ with $|x|=1$ and $dx = [x,x]$. Take any element $y\in\mathfrak{g}$ of odd degree with $dy = 0$. Then you can check that $$d[x,y] = 0$$ in the coproduct (hint: use the Jacobi identity). Of course, this cannot be exact (there's no way to make the $x$ appear out of thin air), so we have a new cohomology class in the coproduct that doesn't come from the cohomology of $\mathfrak{g}$.

Of course, the question

What is the cohomology of a coproduct of dgLie algebras?

remains wide open (at least for me), but now I don't think it has an easy answer anymore.