# Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence

Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a Gerstenhaber algebra.

If two $A_\infty$ algebras are Morita equivalent, are their Hochschild cochain complexes isomorphic in (i) the category of $k$-dg-algebras and (ii) the category of $k$-dg-Lie algebras, both up to quasi-isomorphism? Are they isomorphic in some category that feels both structures together?

Now suppose that $\mathcal{C}$ is a dg-category over a field $k$. We say that the $k$-dg-algebra $CH^*(\mathcal{C}) = End(id_\mathcal{C})$ is the Hochschild cochain complex. Does $CH^*(\mathcal{C})$ have a bracket that generalizes the known one in the case that $\mathcal{C}$ is a (derived) category of modules? If two dg-categories are quasi-equivalent are their Hochschild cochain complexes quasi-isomorphic?

Is there a point of view that clarifies these issues?

• I'm not entirely sure I understand the question, but perhaps Keller's paper people.math.jussieu.fr/~keller/publ/dih.pdf is what you are looking for. – Aaron Bergman Feb 12 '10 at 3:51
• Keller's result tells us that if $A$ and $B$ are ($A_\infty$-)quasi-isomorphic then $CH(A)$ and $CH(B)$ are quasi-isomorphic (prehaps even in the $L_\infty$ sens, not only as complexes). But what do you mean by two $A_\infty$-algebras to be Morita equivalent. For a genuine algebra $A$, the Hochschild complex is (I think) the deformation DGLA for the representation category $A-mod$. So it seems that Morita equivalent algebras will have quasi-isomorphic Hochschild DGLAs. – DamienC Jul 26 '10 at 14:59
• The paper in A. Bergman's comment contains the answer to my question. Indeed, the Hochschild cochain complex of a small dg category has a B-infinity structure, which is a structure that lies between the structure of a Gerstenhaber algebra and a strong homotopy Gerstenhaber algebra. According to Keller's paper, if two small dg-categories are Morita equivalent, then their Hochschild cochain complexes are isomorphic in the homotopy category of B-infinity algebras. This implies that their Hochschild cohomologies are isomorphic as Gerstenhaber algebras. – Ian Shipman Aug 31 '10 at 15:25
• Actually, Ian's comment and the various solutions to Deligne's conjecture in this context show that the Hochschild complexes are isomorphic as $E_2$-algebras. The operad $E_2$ is a cofibrant replacement of the operad of chains on the little squares operad. – Fernando Muro Oct 9 '13 at 9:23
• @Jay what you want, among many other things, is proven in this very futuristic paper arxiv.org/pdf/hep-th/9403055.pdf by Getzler and Jones – user36931 Dec 8 '13 at 11:35