dg-lie structure on $HH^*$ and Koszul duality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:27:37Z http://mathoverflow.net/feeds/question/70493 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70493/dg-lie-structure-on-hh-and-koszul-duality dg-lie structure on $HH^*$ and Koszul duality Daniel Pomerleano 2011-07-16T11:25:56Z 2011-07-18T07:21:28Z <p>This is shamelessly close to my other question: <a href="http://mathoverflow.net/questions/70151/a-question-on-koszul-duality-and-b-infty-structures-on-hh" rel="nofollow">http://mathoverflow.net/questions/70151/a-question-on-koszul-duality-and-b-infty-structures-on-hh</a>. 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:</p> <p><code>$HH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q})) \cong HH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))$</code></p> <p>as Gerstenhaber algebras.</p> <p>In particular, this implies that <code>$HH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q}))[1] \cong HH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))[1]$</code> as Lie algebras.</p> <p>Question:When are the dg-Lie algebra structures on Hochschild cochains: <code>$HCH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q}))[1] \cong HCH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))[1]$</code> quasi-isomorphic. </p> <p>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 <code>$C^*(M)$</code> is equivalent to a graded Koszul algebra). </p> <p>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 <code>$C^*(g)$</code> be the Chevalley complex which is a model for <code>$C^*(M)$</code>. Here is an approach for proving the result: </p> <p>Step 1. We know from this MO question <a href="http://mathoverflow.net/questions/56145/extension-of-the-formality-theorem" rel="nofollow">http://mathoverflow.net/questions/56145/extension-of-the-formality-theorem</a> that <code>$HCH^*(C^*(g), C^*(g)) \cong (T_{poly},[v,])$</code> as $L(\infty)$ algebras. Here v is a vector field which corresponds to the d on <code>$C^*(g)$</code>(see that question for a detailed explanation of notation). These notes <a href="http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf" rel="nofollow">http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf</a> by Damien Calaque are also extremely useful. </p> <p>Step 2. now <code>$(T_{poly},[v,])$</code> is canonically isomorphic as a complex <code>$C^*(g,Sym(g))$</code>, that is the Chevalley complex in the Lie algebra module <code>$Sym(g)$</code>.</p> <p>Step 3. By PBW $Sym(g) \cong Ug$ as g modules. </p> <p>Step 4. Just as in Step 1, we have an isomorphism between <code>$C^*(g,Ug) \cong HH^*(U(g),U(g))$</code>. 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.</p> <p>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 <a href="http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1396v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1396v1.pdf</a>, but it seems tricky to make this work out with the other steps above.</p> http://mathoverflow.net/questions/70493/dg-lie-structure-on-hh-and-koszul-duality/70594#70594 Answer by menichi for dg-lie structure on $HH^*$ and Koszul duality menichi 2011-07-18T05:59:30Z 2011-07-18T07:21:28Z <p>Hi,</p> <p>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 ?</p> <p>this is always true.</p> <p>Step 1: From my paper with Felix and Thomas, looking at the proof, you can see that dg-Lie algebra structures on Hochschild cochains: <code>$HCH∗(\Omega C_*(M),\Omega C_*(M))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1]$</code> are quasi-isomorphic Here <code>$\Omega C_*(M)$</code> is the Adams Cobar construction on the coalgebra C_*(M).</p> <p>Step 2: There is an quasi-isomorphism of chains algebras called Adams cobar equivalence <code>$\Theta:\Omega C_*(M)\rightarrow C∗(Ω(M)$</code>. In our paper, we prove (very short proof) that this quasi-isomorphism $\Theta$ induces an isomorphism of Gerstenhaber algebras between <code>$HH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))$</code> and <code>$HH∗(\Omega C_*(M),\Omega C_*(M))$</code>. 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_<em>(M),\Omega C_</em>(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".</p> <p>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. </p>