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Yang–Baxter deformations of quandles and racks

Michael Eisermann

Algebraic & Geometric Topology 5 (2005) 537–562

arXiv: math.QA/0409202


Given a rack Q and a ring A, one can construct a Yang–Baxter operator cQ: V V V V on the free A–module V = AQ by setting cQ(x y) = y xy for all x,y Q. 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.

Yang–Baxter operator, $r$–matrix, braid group representation, deformation theory, infinitesimal deformation, Yang–Baxter cohomology
Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 18D10, 20F36, 20G42, 57M25
Forward citations
Received: 16 September 2004
Revised: 18 May 2005
Accepted: 3 June 2005
Published: 19 June 2005
Michael Eisermann
Institut Fourier
Université Grenoble I
38402 St Martin d’Hères