Yang–Baxter deformations and rack cohomology
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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.References
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Additional Information
- Michael Eisermann
- Affiliation: Institut für Geometrie und Topologie, Universität Stuttgart, Germany
- Email: Michael.Eisermann@mathematik.uni-stuttgart.de
- Received by editor(s): December 18, 2008
- Received by editor(s) in revised form: December 31, 2011
- Published electronically: May 20, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 5113-5138
- MSC (2010): Primary 16T25, 20F36, 18D10, 17B37, 57M27
- DOI: https://doi.org/10.1090/S0002-9947-2014-05785-1
- MathSciNet review: 3240919