# M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra

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## Abstract

We show that the zeroth cohomology of M. Kontsevich’s graph complex is isomorphic to the Grothendieck–Teichmüller Lie algebra \(\mathfrak {{grt}}_1\). The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by \(\mathfrak {{grt}}_1\), up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) \(E_n\) operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for \(\mathfrak {{grt}}_1\)-elements implies the hexagon equation.

## Notes

### Acknowledgments

I am very grateful to Anton Alekseev, Vasily Dolgushev, Pavel Etingof, Giovanni Felder, Benoit Fresse, David Kazhdan, Anton Khoroshkin, Sergei Merkulov, Pavol Ševera and Victor Turchin for many helpful discussions and their encouragement. In particular Anton Alekseev and Pavol Ševera helped me a lot in shaping my understanding of \(\mathfrak {{grt}}\) and the graph complex. I am highly indebted to Vasily Dolgushev who carefully read the manuscript and pointed out several mistakes in the original version, one of which was relatively severe and made some changes to this manuscript necessary. For the present revised version I added some more details, streamlined the presentation, fixed the mistakes and changed the notation a bit to adhere more to standard conventions. I apologize for potential referential inconsistency with the older version. I am very grateful for support by the Harvard Society of Fellows during most of the writing of this work. As final note, let me also point out the more detailed account [15] on some of the results of this paper given by V. Dolgushev and C. Rogers.

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