×

M. Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra. (English) Zbl 1394.17044

Summary: 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.

MSC:

17B55 Homological methods in Lie (super)algebras
18D50 Operads (MSC2010)
53D55 Deformation quantization, star products
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alekseev, A., Enriquez, B., Torossian, C.: Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations. Publ. Math. Inst. Hautes Études Sci. 112, 143-189 (2010) · Zbl 1238.17008 · doi:10.1007/s10240-010-0029-4
[2] Alekseev, A., Torossian, C.: Kontsevich deformation quantization and flat connections. Comm. Math. Phys. 300(1), 47-64 (2010) · Zbl 1204.17015 · doi:10.1007/s00220-010-1106-8
[3] Alekseev, A., Torossian, C.: The Kashiwara-Vergne conjecture and Drinfelds associators. Ann. Math. 175(2), 415-463 (2012) · Zbl 1243.22009 · doi:10.4007/annals.2012.175.2.1
[4] Arone, G., Tourtchine, V.: Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots (2011). arXiv:1108.1001. · Zbl 1321.11088
[5] Arone, G., Tourtchine, V.: On the rational homology of high dimensional analogues of spaces of long knots (2011). arXiv:1105.1576 · Zbl 1257.17019
[6] Bar-Natan, D.: On Associators and the Grothendieck-Teichmüller Group I. Selecta Math. (N.S.) 4(2), 183-212 (1998) · Zbl 0974.16028 · doi:10.1007/s000290050029
[7] Bar-Natan, D., McKay, B.: Graph Cohomology—An Overview and Some Computations. unpublished preprint. http://www.math.toronto.edu/drorbn/Misc/index.php · Zbl 1058.53065
[8] Brown, F.: Mixed Tate motives over \[\mathbb{Z}\] Z. Ann. Math. 175(2), 949-976 (2012) · Zbl 1278.19008 · doi:10.4007/annals.2012.175.2.10
[9] Calaque, D., Rossi, C.A.: Lectures on Duflo isomorphisms in Lie algebra and complex geometry. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2011) · Zbl 1220.53006 · doi:10.4171/096
[10] Conant, J., Gerlits, F., Vogtmann, K.: Cut vertices in commutative graphs. Q. J. Math. 56(3), 321-336 (Sept. 2005) · Zbl 1187.05029
[11] Dolgushev, V.: A proof of Tsygan’s formality conjecture for an arbitrary smooth manifold (2005). arXiv:math/0504420. · Zbl 0866.57008
[12] Dolgushev, V.: A formality theorem for Hochschild chains. Adv. Math. 200(1), 51-101 (2006) · Zbl 1106.53054 · doi:10.1016/j.aim.2004.10.017
[13] Dolgushev, V., Rogers, C.L., Willwacher, T.: Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields (2012). arxiv:1211.4230 · Zbl 1329.14093
[14] Dolgushev, V., Willwacher, T.: Operadic twisting—with an application to Deligne’s conjecture (2012). arXiv:1207.2180. · Zbl 1305.18032
[15] Dolgushev, V.A., Rogers, C.L.: Notes on algebraic operads, graph complexes, and Willwacher’s construction. In Mathematical aspects of quantization, volume 583 of Contemp. Math., pages 25-145. Amer. Math. Soc., Providence, RI (2012) · Zbl 1327.18017
[16] Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \[{Gal}(\overline{\mathbb{Q}}/{\mathbb{ Q}})\] Gal(Q¯/Q). Algebr. i Anal. 2(4), 149-181 (1990) · Zbl 0718.16034
[17] Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras. I. Sel. Math. 2(1), 1-41 (1996) · Zbl 0863.17008 · doi:10.1007/BF01587938
[18] Fresse, B.: Rational homotopy automorphisms of E2-operads and the Grothendieck-Teichmüller group. work in preparation, http://math.univ-lille1.fr/fresse/E2RationalAutomorphisms.pdf · Zbl 1375.55007
[19] Furusho, H.: Pentagon and hexagon equations. Ann. Math. 171(1), 545-556 (2010) · Zbl 1257.17019 · doi:10.4007/annals.2010.171.545
[20] Furusho, H.: Double shuffle relation for associators. Ann. Math. 174(1), 341-360 (2011) · Zbl 1321.11088 · doi:10.4007/annals.2011.174.1.9
[21] Furusho, H.: Four groups related to associators. Technical Report. Report on a talk at the Mathematische Arbeitstagung in Bonn, June 2011 (2011). arXiv:1108.3389 · Zbl 1321.11088
[22] Hinich, V.: Tamarkin’s proof of Kontsevich formality theorem. Forum Math. 15, 591-614 (2003) · Zbl 1081.16014 · doi:10.1515/form.2003.032
[23] Kontsevich, M.: Formal (non)commutative symplectic geometry. In Proceedings of the I. M. Gelfand seminar 1990-1992, pp. 173-188. Birkhauser (1993). · Zbl 0821.58018
[24] Kontsevich, M.: Feynman diagrams and low-dimensional topology. Progr. Math., 120:97-121. First European Congress of Mathematics, Vol. II, (Paris, 1992) (1994) · Zbl 0872.57001
[25] Kontsevich, M.: Formality conjecture. In Sternheimer, D. et al. (eds.). Deformation Theory and Symplectic Geometry, pp. 139-156 (1997). · Zbl 1149.53325
[26] Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35-72 (1999) · Zbl 0945.18008 · doi:10.1023/A:1007555725247
[27] Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157-216 (2003) · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf
[28] Kontsevich, M., Soibelman, Y.: Deformations of algebras over operads and the Deligne conjecture. In Conférence Moshé Flato 1999, Vol. I (Dijon), volume 21 of Math. Phys. Stud., pp. 255-307. Kluwer Acad. Publ., Dordrecht (2000) · Zbl 0972.18005
[29] Lambrechts, P., Turchin, V.: Homotopy graph-complex for configuration and knot spaces. Trans. Am. Math. Soc. 361(1), 207-222 (2009) · Zbl 1158.57030 · doi:10.1090/S0002-9947-08-04650-3
[30] Lambrechts, P., Volic, I.: Formality of the little N-disks operad (2008). arXiv:0808.0457. · Zbl 1308.55006
[31] Le, T.T.Q., Murakami, J.: Kontsevich’s integral for the Kauffman polynomial. Nagoya Math. J. 142, 39-65 (1996) · Zbl 0866.57008
[32] Loday, J.-L., Vallette, B.: Algebraic operads. Number 346 in Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2012).
[33] Merkulov, S., Vallette, B.: Deformation theory of representations of prop(erad)s. II. J. Reine Angew. Math. 636, 123-174 (2009) · Zbl 1191.18003
[34] Paljug, B.: Action of derived automorphisms on infinity-morphisms (2013). arXiv:1305.4699. · Zbl 1377.55008
[35] Schneps, L.: The Grothendieck-Teichmüller group gt: a survey. In Geometric Galois actions, 1, volume 242 of London Math. Soc. Lecture Note Ser., pp. 183-203. Cambridge Univ. Press, Cambridge (1997) · Zbl 0910.20019
[36] Schneps, L.: Double shuffle and Kashiwara-Vergne Lie algebras. J. Algeb. 367, 54-74 (2012) · Zbl 1302.17043 · doi:10.1016/j.jalgebra.2012.04.034
[37] Shoikhet, B.: Vanishing of the Kontsevich integrals of the wheels. Lett. Math. Phys. 56(2), 141-149 (2001). arXiv:math/0007080 · Zbl 1018.53043 · doi:10.1023/A:1010842705836
[38] Tamarkin, D.: Another proof of M. Kontsevich formality theorem (1998). arXiv:math/9803025. · Zbl 1009.17012
[39] Tamarkin, D.: Action of the Grothendieck-Teichmueller group on the operad of Gerstenhaber algebras (2002). arXiv:math/0202039. · Zbl 1243.22009
[40] Tamarkin, D.: Quantization of Lie bilagebras via the formality of the operad of little disks. IRMA Lect. Math. Theor. Phys. 1, 203-236 (2002) · Zbl 1009.17012
[41] Tamarkin, D.E.: Formality of chain operad of little discs. Lett. Math. Phys. 66, 65-72 (2003). arXiv:math/9809164 · Zbl 1048.18007 · doi:10.1023/B:MATH.0000017651.12703.a1
[42] Turchin, V.: Hodge-type decomposition in the homology of long knots. J. Topol. 3(3), 487-534 (2010) · Zbl 1205.57023 · doi:10.1112/jtopol/jtq015
[43] Ševera, P., Willwacher, T.: The cubical complex of a permutation group representation - or however you want to call it (2011). arXiv:1103.3283. · Zbl 1048.18007
[44] Ševera, P., Willwacher, T.: Equivalence of formalities of the little discs operad. Duke Math. J. 160(1), 175-206 (2011) · Zbl 1241.18008 · doi:10.1215/00127094-1443502
[45] van der Laan, P.P.I.:. Operads up to homotopy and deformations of operad maps (2002). math/0208041. · Zbl 1191.18003
[46] Willwacher, T.: A Note on Br-infinity and KS-infinity formality (2011). arXiv:1109.3520. · Zbl 1191.18003
[47] Willwacher, T.: Stable cohomology of polyvector fields (2011). arxiv:1110.3762. · Zbl 1318.53104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.