Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 080, 3 pages      math.QA/0404262      https://doi.org/10.3842/SIGMA.2006.080
Contribution to the Vadim Kuznetsov Memorial Issue

A Formula for the Logarithm of the KZ Associator

Benjamin Enriquez a and Fabio Gavarini b
a) IRMA (CNRS), rue René Descartes, F-67084 Strasbourg, France
b) Universitá degli Studi di Roma ''Tor Vergata'', Dipartimento di Matematica, Via della Ricerca Scientifica 1, I-00133 Rome, Italy

Received October 03, 2006, in final form November 10, 2006; Published online November 13, 2006

Abstract
We prove that the logarithm of a group-like element in a free algebra coincides with its image by a certain linear map. We use this result and the formula of Le and Murakami for the Knizhnik-Zamolodchikov (KZ) associator Φ to derive a formula for log(Φ) in terms of MZV's (multiple zeta values).

Key words: free Lie algebras; Campbell-Baker-Hausdorff series; Knizhnik-Zamolodchikov associator.

pdf (173 kb)   ps (143 kb)   tex (7 kb)

References

  1. Drinfeld V., On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(`Q/ Q), Leningrad Math. J., 1991, V.2, 829-860.
  2. Enriquez B., Quasi-reflection algebras, multiple polylogarithms at roots of 1, and analogues of the group GT, math.QA/0408035.
  3. Le T.T.Q., Murakami J., Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J., 1996, V.142, 39-65.
  4. Reutenauer C., Free Lie algebras, London Mathematical Society Monographs. New Series, Vol. 7, Oxford Science Publications, New York, The Clarendon Press, Oxford University Press, 1993.

Previous article   Next article   Contents of Volume 2 (2006)