Abstract
We prove an estimate on denominators of rational Drinfeld associators. To obtain this result, we prove the corresponding estimate for the p-adic associators stable under the action of suitable elements of \({{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})}\) . As an application, we settle in the positive Duflo’s question on the Kashiwara–Vergne factorizations of the Jacobson element J p (x, y) = (x + y)p − x p − y p in the free Lie algebra over a field of characteristic p. Another application is a new estimate on denominators of the Kontsevich knot invariant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseev, A., Enriquez, B., Torossian, C.: Drinfeld associators, braid groups and explicit solutions of the Kashiwara–Vergne equations, preprint arXiv:0903.4067
Alekseev, A., Torossian, C.: The Kashiwara–Vergne conjecture and Drinfeld’s associators, preprint arXiv:0802.4300
Alekseev, A., Torossian, C.: Kontsevich deformation quantization and flat connections, preprint arXiv:0906.0187
Bar-Natan, D.: Non-associative tangles. Geometric topology (Athens, GA, 1993), 139–183, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, (1997)
Bar-Natan D.: On associators and the Grothendieck–Teichmuller group. I. Selecta Math. (N.S.) 4(2), 183–212 (1998)
Drinfeld V.G.: On quasi-triangular quasi-Hopf algebras and a group closely connected with \({ {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}\) . Leningrad Math. J. 2(4), 829–860 (1991)
Duflo, M.: private communications
Etingof P., Kazhdan D.: Quantization of Lie bialgebras. I. Selecta Math. (N.S.) 2(1), 1–41 (1996)
Furusho, H.: Pentagon and hexagon equations, preprint arXiv:math/0702128
Furusho H.: p-adic multiple zeta values. I. p-adic multiple polylogarithms and the p-adic KZ equation. Invent. Math. 155(2), 253–286 (2004)
Furusho, H.: Double shuffle relation for associators, preprint arXiv:0808.0319
Kashiwara M., Vergne M.: The Campbell-Hausdorff formula and invariant hyperfunctions. Invent. Math. 47(3), 249–272 (1978)
Kontsevich M.: Vassiliev’s knot invariants. Adv. in Sov. Math. 16(2), 137–150 (1993)
Lazard M.: Groupes analytiques p-adiques. Publ. Math. IHES 26, 5–219 (1965)
Le T.Q.T.: On denominators of the Kontsevich integral and the universal perturbative invariant of 3-manifolds. Invent. Math. 135(3), 689–722 (1999)
Le T.Q.T., Murakami J.: The universal Vassiliev-Kontsevich invariant for framed oriented links. Compositio Math. 102, 41–64 (1996)
Podkopaeva, M.: On the Jacobson element and generators of the Lie algebra \({\mathfrak{grt}}\) in nonzero characteristic, preprint arXiv:0812.0772
Tamarkin D., Tsygan B.: Noncommutative differential calculus, homotopy BV algebras and formality conjectures. Methods Funct. Anal. Topol. 6(2), 85–100 (2000)
Torossian, C.: La conjecture de Kashiwara-Vergne (d’après Alekseev et Meinrenken). Séminaire Bourbaki. Vol. 2006/2007. Astérisque No. 317, Exp. No. 980, 441–465 (2008)
Ševera, P., Willwacher, T.: Equivalence of formalities of the little discs operad, preprint arXiv:0905.1789
Author information
Authors and Affiliations
Corresponding author
Additional information
Pavol Ševera: on leave from FMFI UK Bratislava, Slovakia.
Rights and permissions
About this article
Cite this article
Alekseev, A., Podkopaeva, M. & Ševera, P. On rational Drinfeld associators. Sel. Math. New Ser. 17, 47–65 (2011). https://doi.org/10.1007/s00029-010-0035-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-010-0035-x