Yang-Baxter deformations of quandles and racks

2004/09/30 by Michael Eisermann · 1 citation
Mathematics · #math.QA #math.GT #msc:17B37 #msc:18D10 #msc:20F36 #msc:20G42 #msc:57M25

paper · pdf · doi:10.2140/agt.2005.5.537

published as Algebr. Geom. Topol. 5 (2005) 537-562 · Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-23.abs.html

arxiv created 2005/06/30 · arxiv updated 2014/09/30

Abstract

Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of Yang-Baxter operators. For the trivial rack, where x^y = x for all x,y, one has, of course, the classical setting of r-matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of c_Q. In many cases this allows us to conclude that c_Q is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.

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