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2005 Yang–Baxter deformations of quandles and racks
Michael Eisermann
Algebr. Geom. Topol. 5(2): 537-562 (2005). DOI: 10.2140/agt.2005.5.537


Given a rack Q and a ring A, one can construct a Yang–Baxter operator cQ:VVVV on the free A–module V=AQ by setting cQ(xy)=yxy for all x,yQ. In answer to a question initiated by D N Yetter and P J Freyd, this article classifies formal deformations of cQ in the space of Yang–Baxter operators. For the trivial rack, where xy=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 cQ. In many cases this allows us to conclude that cQ 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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Michael Eisermann. "Yang–Baxter deformations of quandles and racks." Algebr. Geom. Topol. 5 (2) 537 - 562, 2005.


Received: 16 September 2004; Revised: 18 May 2005; Accepted: 3 June 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1160.17304
MathSciNet: MR2153117
Digital Object Identifier: 10.2140/agt.2005.5.537

Primary: 17B37
Secondary: 18D10, 20F36, 20G42, 57M25

Rights: Copyright © 2005 Mathematical Sciences Publishers


Vol.5 • No. 2 • 2005
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