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 he 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.
