Pentagon and hexagon equations following Furusho

Authors:
Dror Bar-Natan and Zsuzsanna Dancso

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1243-1250

MSC (2010):
Primary 17B37

Published electronically:
August 5, 2011

MathSciNet review:
2869109

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Abstract | References | Similar Articles | Additional Information

Abstract: H. Furusho proved the beautiful result that of the three defining equations for associators, the pentagon implies the two hexagons. In this paper we present a simpler proof for this theorem (although our paper is less dense and hence only slightly shorter). In particular, we package the use of algebraic geometry and Groethendieck-Teichmüller groups into a useful and previously known principle, and, less significantly, we eliminate the use of spherical braids.

**[BN]**Dror Bar-Natan,*On associators and the Grothendieck-Teichmuller group. I*, Selecta Math. (N.S.)**4**(1998), no. 2, 183–212. MR**1669949**, 10.1007/s000290050029**[Dr1]**V. G. Drinfel′d,*Quasi-Hopf algebras*, Algebra i Analiz**1**(1989), no. 6, 114–148 (Russian); English transl., Leningrad Math. J.**1**(1990), no. 6, 1419–1457. MR**1047964****[Dr2]**V. G. Drinfel′d,*On quasitriangular quasi-Hopf algebras and on a group that is closely connected with 𝐺𝑎𝑙(\overline{𝑄}/𝑄)*, Algebra i Analiz**2**(1990), no. 4, 149–181 (Russian); English transl., Leningrad Math. J.**2**(1991), no. 4, 829–860. MR**1080203****[F]**Hidekazu Furusho,*Pentagon and hexagon equations*, Ann. of Math. (2)**171**(2010), no. 1, 545–556. MR**2630048**, 10.4007/annals.2010.171.545**[LM]**Tu Quoc Thang Le and Jun Murakami,*Representation of the category of tangles by Kontsevich’s iterated integral*, Comm. Math. Phys.**168**(1995), no. 3, 535–562. MR**1328252****[W]**T. Willwacher:*M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra,*arXiv:1009.1654.

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Additional Information

**Dror Bar-Natan**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Email:
drorbn@math.toronto.edu

**Zsuzsanna Dancso**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Email:
zsuzsi@math.toronto.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-10996-1

Keywords:
Pentagon equation,
hexagon equations,
associators,
Furusho’s theorem

Received by editor(s):
October 4, 2010

Received by editor(s) in revised form:
December 11, 2010, and January 5, 2011

Published electronically:
August 5, 2011

Communicated by:
Gail R. Letzter

Article copyright:
© Copyright 2011
By the authors under
Creative Commons Attribution 3.0 License (CC B4 3.0)