dg-lie structure on $HH^*$ and Koszul duality

Question

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.

Answer 1

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.