ID  49095 
FullText URL  
Author 
Soudères, Ismaël

Abstract  Associators were introduced by Drinfel’d in [Dri91] as a
monodromy representation of a KnizhnikZamolodchikov equation. Associators
can be briefly described as formal series in two noncommutative
variables satisfying three equations. These three equations yield a
large number of algebraic relations between the coefficients of the series,
a situation which is particularly interesting in the case of the original
Drinfel’d associator, whose coefficients are multiple zetas values. In
the first part of this paper, we work out these algebraic relations among
multiple zeta values by direct use of the defining relations of associators.
While wellknown for the first two relations, the algebraic relations we
obtain for the third (pentagonal) relation, which are algorithmically explicit
although we do not have a closed formula, do not seem to have
been previously written down. The second part of the paper shows
that if one has an explicit basis for the barconstruction of the moduli
space M0,5 of genus zero Riemann surfaces with 5 marked points
at one’s disposal, then the task of writing down the algebraic relations
corresponding to the pentagon relation becomes significantly easier and
more economical compared to the direct calculation above. We discuss
the explicit basis described by Brown and Gangl, which is dual to the
basis of the enveloping algebra of the braids Lie algebra UB5.
In order to write down the relation between multiple zeta values, we
then remark that it is enough to write down the relations associated
to elements that generate the bar construction as an algebra. This
corresponds to looking at the bar construction modulo shuffle, which
is dual to the Lie algebra of 5strand braids. We write down, in the
appendix, the associated algebraic relations between multiple zeta values
in weights 2 and 3.

Published Date  201301

Publication Title 
Mathematical Journal of Okayama University

Volume  volume55

Issue  issue1

Publisher  Department of Mathematics, Faculty of Science, Okayama University

Start Page  1

End Page  52

ISSN  00301566

NCID  AA00723502

Content Type 
Journal Article

language 
英語

Copyright Holders  Copyright©2013 by the Editorial Board of Mathematical Journal of Okayama University

File Version  publisher

Refereed 
True

Submission Path  mjou/vol55/iss1/1

JaLCDOI 