dg-lie structure on $HH^*$ and Koszul duality This is shamelessly close to my other question: A Question on Koszul duality and $B(\infty)$ structures on $HH^*$. Maybe this one will get a better response. Rather than rewrite that one, I am going to ask about a specific aspect of it in more detail. As in that question, it is known that for a space simply connected space M:
$HH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q})) \cong HH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))$
as Gerstenhaber algebras.
In particular, this implies that $HH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q}))[1] \cong HH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))[1]$ as Lie algebras.
Question:When are the dg-Lie algebra structures on Hochschild cochains: 
$HCH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q}))[1] \cong HCH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))[1]$ quasi-isomorphic. 
As I mentioned in that question: this follows from more general results of Keller in the case M is formal and coformal (i.e. the d.g. algebra $C^*(M)$ is equivalent to a graded Koszul algebra). 
Now suppose that g is a graded finite dimensional Lie algebra and work over $\mathbb{C}$(or $\mathbb{R}$), which corresponds to M be a $\mathbb{C}$ coformal space, with finite dimensional $\mathbb{C}$ homotopy groups. Let $C^*(g)$ be the Chevalley complex which is a model for $C^*(M)$. Here is an approach for proving the result: 
Step 1. We know from this MO question Extension of the formality theorem? that $HCH^*(C^*(g), C^*(g)) \cong (T_{poly},[v,])$ as $L(\infty)$ algebras. Here v is a vector field which corresponds to the d on $C^*(g)$(see that question for a detailed explanation of notation). These notes http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf by Damien Calaque are also extremely useful.  
Step 2. now $(T_{poly},[v,])$ is canonically isomorphic as a complex $C^*(g,Sym(g))$, that is the Chevalley complex in the Lie algebra module $Sym(g)$.
Step 3. By PBW $Sym(g) \cong Ug$ as g modules. 
Step 4. Just as in Step 1, we have an isomorphism between $C^*(g,Ug) \cong HH^*(U(g),U(g))$. To obtain this we think of Ug as the deformation quantization of $Sym(g)$ given by the Kirillov Poisson structure on $g^*$. Ordinarily, this exists as a formal deformation but just like in Step 1, there is no problem setting the formal parameter t=1.
Just as in that question, there is an induced $L(\infty)$ map on tangent cohomology groups that is an iso.
Question: Can these steps be generalized to dg-Lie algebras with finite dimensional homology? Note it follows from the cited question that step one generalizes. A generalization of Step 3 is given here in this paper of Baranovsky 
http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1396v1.pdf, but it seems tricky to make this work out with the other steps above.
 A: Hi,
Your Question:When are the dg-Lie algebra structures on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1] quasi-isomorphic ?
this is always true.
Step 1:
From my paper with Felix and Thomas, looking at the proof, you can see that
dg-Lie algebra structures on Hochschild cochains: $HCH∗(\Omega C_*(M),\Omega C_*(M))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1]$ are quasi-isomorphic
Here $\Omega C_*(M)$ is the Adams Cobar construction on the coalgebra C_*(M).
Step 2:
There is an quasi-isomorphism of chains algebras called Adams cobar equivalence
$\Theta:\Omega C_*(M)\rightarrow C∗(Ω(M)$.
In our paper, we prove (very short proof)
that this quasi-isomorphism $\Theta$ induces an isomorphism of
Gerstenhaber algebras between $HH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))$ and
$HH∗(\Omega C_*(M),\Omega C_*(M))$.
In particular, we have an isomorphism of graded Lie algebras.
You want a dg-Lie algebra isomorphism on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1]
and HCH∗(\Omega C_(M),\Omega C_(M)).
This is true. One of my coauthor had a proof.
But it is not in our paper, since I thought it was not interesting and too complicated. 
But if I remember well, Hamilton and Lazarev proved it in a paper following our paper.
I think that Keller proved also in the paper you quote "Derived Invariance of Higher Structures of the Hochschild complex".
ps: There is two versions of my paper with Felix and Thomas, the published squezeed version
valid only over a field, and the arxiv longer version with more details. 
