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


SIGMA 2 (2006), 080, 3 pages      math.QA/0404262      http://dx.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.

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References

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  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.

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