Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:59:58Z http://mathoverflow.net/feeds/question/14860 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14860/regarding-the-gerstenhaber-bracket-on-hochschild-cohomology-and-morita-equivalenc Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence Ian Shipman 2010-02-10T05:36:24Z 2010-02-10T05:48:20Z <p>Associated to any <code>$A_\infty$</code> $k$-algebra $A$ the Hochschild cochain complex <code>$CH^*(A)$</code> has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a Gerstenhaber algebra. </p> <p>If two <code>$A_\infty$</code> 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?</p> <p>Now suppose that $\mathcal{C}$ is a dg-category over a field $k$. We say that the $k$-dg-algebra <code>$CH^*(\mathcal{C}) = End(id_\mathcal{C})$</code> is the Hochschild cochain complex. Does <code>$CH^*(\mathcal{C})$</code> 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?</p> <p>Is there a point of view that clarifies these issues?</p>