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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Yang-Baxter deformations and rack cohomology


Author: Michael Eisermann
Journal: Trans. Amer. Math. Soc. 366 (2014), 5113-5138
MSC (2010): Primary 16T25, 20F36, 18D10, 17B37, 57M27
Published electronically: May 20, 2014
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Abstract: In his study of quantum groups, Drinfeld suggested considering set-theoretic solutions of the Yang-Baxter equation as a discrete analogon. As a typical example, every conjugacy class in a group or, more generally, every rack $ Q$ provides such a Yang-Baxter operator $ c_Q \colon x \otimes y \mapsto y \otimes x^y$. In this article we study deformations of $ c_Q$ within the space of Yang-Baxter operators over some complete ring. Infinitesimally these deformations are classified by Yang-Baxter cohomology. We show that the Yang-Baxter cochain complex of $ c_Q$ homotopy-retracts to a much smaller subcomplex, called quasi-diagonal. This greatly simplifies the deformation theory of $ c_Q$, including the modular case which had previously been left in suspense, by establishing that every deformation of $ c_Q$ is gauge equivalent to a quasi-diagonal one. In a quasi-diagonal deformation only behaviourally equivalent elements of $ Q$ interact; if all elements of $ Q$ are behaviourally distinct, then the Yang-Baxter cohomology of $ c_Q$ collapses to its diagonal part, which we identify with rack cohomology. This establishes a strong relationship between the classical deformation theory following Gerstenhaber and the more recent cohomology theory of racks, both of which have numerous applications in knot theory.


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Additional Information

Michael Eisermann
Affiliation: Institut für Geometrie und Topologie, Universität Stuttgart, Germany
Email: Michael.Eisermann@mathematik.uni-stuttgart.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05785-1
PII: S 0002-9947(2014)05785-1
Keywords: Yang--Baxter equations, Yang--Baxter operator, tensor representation of braid groups, deformation theory, infinitesimal deformation, Yang--Baxter cohomology
Received by editor(s): December 18, 2008
Received by editor(s) in revised form: December 31, 2011
Published electronically: May 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.